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ABSTRACT ALGEBRAIC GEOMETRY

I. V. Dolgachev UDC 513.6

The present article consists mainly of papers reviewed in Referativnyi Zhurual ."qVlatematika ~ (Soviet Mathematical Abstracts Journal) during 1960-1971 and is concentrated around questions connected with abstract algebraic geometry. By the latter we mean first of all the foundations as well as those sections which arose in the mid-Fifties under the influence of Serre and Grothendieck. The ideas and methods of the latter turned out to have an immense influence on almost all sections of algebraic geometry (and on many other branches of mathematics) and gave a start to "Modern Algebraic Geometry. ~ A fundamental role in the latter is played by precisely those sections to which the present art icle is devoted. The year 1958, when Grothendieck announced at the Edinburgh congress his program of investigations in algebraic geometry, serves as the starting date for this art icle. In it we have not included those papers on commutative algebra and analytical geometry which were written under the direct influence of Grothendieck's paper. We have also left aside such important sections of algebraic geometry as K-theory, the theory of algebraic cycles, the resolution of singuLarities, the theory of modules, and theory of intersections; the theory of group schemata* and of formal groups has been touched on only incidentally, and they deserve a separate survey.

w F o u n d a t i o n s of A l g e b r a i c Geometry

The establishment of the fundaments of classical algebraic geometry began comparatively long ago. Beginning with Hilbert and his successors (Noether, Krull, van der Waerden) algebraic geometry was based on the theory of polynomial ideals. The results in the papers from this school were summarized in Brog- ~er's book [216]. After the appearance in 1946 of Well's book [507] valuation theory and field theory (the language of Weil's "generic points ~) became the commonly-accepted fundaments of algebraic geometry. Weft also introduced new objects of study in algebraic geometry, namely, abstract algebraic varieties. The powerful methods of commutative and, in particular, local algebra were introduced into abstract algebraic geometry (signifying at that time the study, of abstract algebraic varieties over an arbitrary field of constants) by Zariski and his school (Samuel, Cohen, etc.). An account of these methods can be found in Sam- uel [459].

Serre ' s paper [468] on coherent algebraic sheaves served as the source of a subsequent process of reorganization of the fundaments of algebraic geometry ("Serre 's language"). In it for the f i rs t time there was introduced into algebraic geometry the ideas and methods of homological algebra and also was extended the notion of algebraic variety (Serre 's ~algebraic space~). Another point of view on algebraic varieties ("Chevalley's schemata ~) was developed by Chevalley [154] and Nagam [392, 297].

Finally, in 1958 Grothendieck, by developing and generalizing Ser re ' s ideas, introduced into algebraic geometry the language of functors and of the theory of categories and also essentially extended the notion of an algebraic variety by laying the beginnings of the theory of schemata. Starting with the publication of the first chapters of Grothendieck and Dieudonne's treatise [240-248], the language of the theory of schemata solidly became the custom of algebraic geometers and is now most widespread and commonly accepted. The orderl iness, completeness, and geometricity of this theory permitted algebraic geometers not to dwell any longer on the foundations and to re turn to solving the concrete problems of algebraic geometry, be- queathed by previous generations, as well as to develop the connection of this science with other areas of mathematics.

*Translator;s note: I have prefer red to use "schema ~ (plural ' schemata ~) even though there is a tendency nowadays to use "scheme ~ (plural "schemes ' ) .

TransLated from Itogi Nauki i Tekhniki, Seriy~ Matamatika (Algebra, Topologiya, Geometriya), Vol. !0, pp. 47-112, 1972.

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I. Development of the Notion of Algebra ic Var ie t 7. As we see f rom the preceding, the development of the foundations of a lgebra ic g e o m e t r y pa ra l l e l ed the development of the notion of an a lgebra ic variety.. Beginning with the notion of Wel l ' s a b s t r a c t va r i e t y and ending with the notion of Ar t i n ' s and Moishezon ' s notion of an a lgebra ic space, this development was s t imula ted chiefly by var ious const ruct ions in a lgebra ic g e o m e t r y (the Jacobi variety., the Po inca re schema , the space of modules , min imal modules , etc.: see w The c l a s s i ca l definition of an a lgebra ic va r i e ty was r e s t r i c t e d to affine or projec t ive k - s e t s over an a l - gebra ica l ly c losed field k, i .e. , c losed in the Za r i sk i topolog'y by subse ts of an affine or projec t ive space over k. The idea of c a r r y i n g ove r the cons t ruc t ion of different iable va r i e t i e s (with the aid of pasting together) onto a lgebra ic va r i e t i e s was due to Well. In [507] he defined a b s t r a c t a lgebra ic va r i e t i e s as a s y s - tem of affine a lgebra ic va r i e t i e s (Vot) in each of which there a r e picked out open subse t s ~X/~V~ cons i s - tently i somorphic with the open subse t s W~cV~. Well succeeded in extending onto these va r i e t i e s a l l the fundamental concepts of c l a s s i ca l a lgebra ic geomet ry . In 1950 L e r a y [336] introduced the notion of a sheaf on a topological space. Carman's 1950/51 s e m i n a r in P a r i s was devoted to the development of the theory of sheaves . This notion pe rmi t t ed the definition of different iable and analyt ic va r i e t i e s f rom one point of view, including them within the gene ra l notion of a r inged topological space . In 1955 Se r re [468] d i scovered that a s i m i l a r definition was appl icable a lso in a lgebra ic geomet ry . A r inged space local ly i somorphic to an affine va r i e ty with a sheaf of g e r m s of r e gu l a r functions on it came to be ca l led an a lgebra ic va r i e ty (an a lgeb ra i c space in S e r r e ' s terminology) . The addit ional s t ruc tu re of a r inged space on an a lgebra ic v a r i e - ty p e r m i t s not only the s impl i f ica t ion of var ious cons t ruc t ions with a b s t r a c t va r i e t i e s but a lso introducing in the i r study the powerful methods of homologica l a lgeb ra , connected with the theory of sheaves . At the Edinburgh cong r e s s in 1988 Grothendieck sketched the pe r spec t ive for the fur ther genera l iza t ion of the notion of a lgebra ic va r i e ty connected with the theory of s chemata [14]. The f i r s t definitions of s chemata were p re sen ted in his r e p o r t a t the Bourbaki s e m i n a r in 1959 [219]. The idea of affine s chema ta was s ta ted independently a lso by C a r t i e r (unpublished) and by IC/hler [299].

Le t X be an affine va r i e ty over a field k with a coordinate r ing k [X]. Its points {in the c l a s s i ca l sense) a r e found in one- to -one co r r e spondence with the h o m o m o r p h i s m s f : k [X] -" [~, where [c is the a lgebra ic c losure of field k. The ke rne l of such a h o m o m o r p h i s m is a m a x i m a l ideal j / of r ing k [3[]. The c o r r e - spondence f - - j f defines a bi ject ion of the se t of points X (~':) of va r i e ty X w~th coordina tes in k (with iden- t if ication of points conjugate over k) and with the se t Spec m (k iX]) of max ima l ideals of r ing k [X]. F u r t h e r - m o r e , the Za r i sk i topology on X (k) co r r e sponds to the spec t r a l topology on Spec m (k [X]) in which c losed se ts a r e se t s of m a x i m a l ideals containing a fixed ideal I c ~ [X]. The lat t ice of a r inged space on X ([~) c o r - r e sponds to an analogous la t t ice on Spec m (k iX]} in which the f iber at the point m~S0ec rn (~ [X]) of the la t - t ice sheaf is a local izat ion of r ing k [X] re la t ive to a mul t ip l ica t ive c losed se t S----~[X]\m. Converse ly , each k - a l g e b r a of finite type without nilpotent e l emen t s is i somorph ic to the coordinate r ing of some affine va r i e ty in the sense of Ser re , and the co r re spondence A - - Spectra (A) is a b i iect ive co r respondence between a k - a l g e b r a of the type being cons ide red and affine a lgebra ic va r i e t i e s (to within i somorphism}, Grothen- dieck genera l i zed this co r r e spondence in two e s sen t i a l r e s p e c t s . F i r s t of al l he noted that it should define a functor with values in the ca t ego ry o f r inged spaces . Fo r e v e r y homomorph i sm o f k - a l g e b r a s ~: A --- B, the only r easonab le method of defining the m o r p h i s m $pec m (B) - - Spec m (A) is that to a max ima l ideal mcB there m u s t c o r r e s p o n d its p r e image =-L (m) in A. However , this ideal no longer has to be ma x ima l , although it a lways r e m a i n s p r ime . Grothendieck sugges ted that space Spec m ~A) be rep laced by the space Spec (A) of al l p r ime ideals with analogous spec t r a l topology and with the latt ice of a r inged space . This genera l iza t ion yields the functor ia l i ty of the co r re spondence A - - Spec (A) and is analogous to Wel l ' s idea of cons ider ing the points of va r i e ty X with coord ina tes in an a r b i t r a r y extension K/k of the ~round field k. The nex-t r e m a r k of Grothendieck was that we need to d i sca rd e v e r y condition on k-a lgebra A and to cons ider that A is an a r b i t r a r y commuta t ive r ing with unit ( some t imes Noetherian}. Here field k is r ep l aced by an a r b i t r a r y subr ing B cA. This genera l iza t ion is of a meaningful nature since it al lows us to explain ce r t a in c l a s s i ca l phenomena in I ta l ian a lgebra ic g e o m e t r y "by the p re sence of nilpotent e l ements in r ing A." A r inged space Spec (A) is cal led the affine schema cor respond ing to r ing A and is a natural genera l iza t ion of an affine a lgebra ic var ie ty . The s tandard cons t ruc t ion of past ing together r inged spaces now p e r m i t s us to ~ v e the definition of a schema as a r inged space local ly i somorph ic to an affirm schema. F r o m a new point of view a lgebra ic va r i e t i e s a r e reduced schemata of finite-type over a field.

The c h a r a c t e r i s t i c f ea tu res of this new theory in a lgebra ic geome t ry a re the following:

I. A commuta t ive a lgebra becomes a pa r t of a lgebra ic geometTy. Namely, this is a theory, of local obiects of a lgebra ic geomet ry , i .e. , of affine schemata . The advantage of ~ such a viewpoint is mvo-fold. F i r s t ly , it p e r m i t s us to c a r r y over all the concepts of commuta t ive a lgebra into ~eomet r ic language and by

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the same token puts a powerful tool in the hands of the a lgebrais t , namely, geomet r ic intuition. Secondly, by examining affine schemata within the f ramework of the genera l theory of schemata, it a t t rac t s to thair study the powerful methods of a lgebraic geometry .

2. The ~ o u n d field k is rep laced by an a r b i t r a r y ~oTound schema S, i .e. , to examine schemata X for which there is given a rnorphism f : X - - S (lattice morphism of an S-schema X). A r b i t r a r y S -morph i sms , namely, the mappings ~: X 1 - - X 2 for whichf2o~ = f~, where f i : Xi -" S (i = 1, 2) are latt ice morphisms , become morph i sms of S-schemata . Each schema X is a Z-schema , i .e. , a schema over the affine schema S = Spec (Z). The c lass ica l notion of the extension of the field of constants, due to Zar iski and Weil, is rep laced by notion of a change of base. For any S-schema S' and S-schema X we can consider the "change of base ," namely, the S ' - schema X(s.)=XXS'~$', where XXS" denotes a d i rec t product in the ca tegory

S S

of S-schemata . Special cases of this operat ion a re such concepts as the reduct ion of a va r ie ty by a pr ime module, the f iber of a morphism, etc. Class ica l vers ions of the definitions of these notions were only s l igh t - ly geometr ic and r a the r awkward.

3. The introduction of nilpotent e lements . The p resence of nilpoteat e lements in coordinate r ings of a r b i t r a r y schemata proved to be natural enough and of f requent occu r r ence in a lgebra ic geometry . For example, the f ibers of a morphism of the usual nonsingular a lgebra ic var ie t i es a re schemata with nilpotent e lements (for example, Kodai ra ' s "multiple f ibe r s" in the theory of a lgebra ic sur faces) . The p resence of nilpotent e lements in a "schema of modules" or in " P o i n c a r e ' s schema" allowed us to explain the previous- ly not well unders tood phenomena of c lass ica l Italian a lgebra ic geomet ry as well as ce r t a in pathologies of a lgebraic var ie t ies over a field of posit ive cha rac t e r i s t i c (see Mumford [376, 378, 382]). The theory of schemata with nilpotent e lements plays an impor tant ro le in the study of the "~nfinitesimal" p roper t i e s of a lgebraic var ie t i es and se rved as the foundation of Grothendieck 's fo rmal geomet ry (see w

We r e m a r k that o ther a t tempts , which did not become prevalent , were made to genera l ize the notion of an a lgebra ic var ie ty , in which a cent ra l ro le was played by the concept of a local r ing of the field of algebraic functions (Chevalley 's schema [154, 392, 397, 266, 159, 509]). The idea of cons ider ing a lgebra ic var ie t ies over a r b i t r a r y Dedekind r ings is due also to Nab'am [392].

A number of general iza t ions of the concept of a schema a rose in ce r t a in concre te problems of a l - gebraic geometry . A natural genera l iza t ion of schemata consis ts in the immers ion of the ca tegory (Sch/S) of S-schemata into some l a rge r category. F o r example , the examinat ion of the ca tegory of pro jec t ive ob- jec ts of the ca tegory (Sch/S) leads to the notion of a formal schema (Grothendieck [219, 240]) and to S e r r e ' s p ro -a lgebra ic groups. The dual concept of an inductive sys tem turned out to be useful in the a lgebra ic theory of uniformizat ion of Pyate tsk i i -Shapi ro and Shafarevich [45]. An analysis of the ca tegory (Sch/S) of cont ravar ian t functors on (Sch/S) (or i ts subcategory consist ing of sheaves on a ce r t a in Grothendieck topology on S; see w further) permi t ted us to identify S--schemata with r ep resen tab le functors . This point of view proved to be par t icu la r ly useful for the theory of group S-schemata [169-171]. Grothendieck [236] also suggested that a r b i t r a r y r inged spaces (or even the Grothendieck topology) be cons idered as ground schemata. The theory of such schemata ("re la t ive schemata ' ) was developed in Hakim's d i sser ta t ion [255]. This notion proved useful in analyt ical geometry .

An essent ia l genera l iza t ion of the concept of a schema a re a lgebra ic spaces introduced independently by Art in [84] and by Moishezon (he cal led them "min ischemata" [38-401). In Well 's language an analog of such a c o n c e ~ was introduced in 1965 by Matsusaka (Q-var ie t ies ; see [360]) in connection with the theory of modules of polar ized a lgebra ic var ie t i es . The major i ty of definitions and resu l t s of the theory of schemata was c a r r i e d over to a lgebra ic spaces by Knutson [314]. See w for details on this theory.

2. Theory of Schemata ~ F rom !960 on, Grothendieck (in col laborat ion with Dieudonne) s t a r t ed to pablish the monumental t r ea t i se on a lgebra ic geomet ry in which it was proposed to es tabl ish the foundations of a lgebraic geomet ry within the f ramework of schemata theory. Although no fur ther chapters of this t rea t i se have been published, the major i ty of the c lass ica l r e su l t s on the foundations have found in it their nanlral genera l iza t ion and c lar i f ica t ion in the theory of schemata . We r e m a r k that the degree of genera l i ty in this theory has proved to be so grea t that the publications of Grothendieck and Dieudonne r ema in a lmost unique in schemata theory proper . P a p e r s E98-!00, 107, 143, 146, 180, 205, 251-254, 274, 275, 320, 328, 329, 333, 335, 386, 428, -~84, 509, 5101 were devoted to ce r ta in unessent ia l general iza t ions of Grothendieck 's resu l t s . A detailed analysis of Grothendieck 's resu l t s in the theory of schemata would take up too much space. We r e s t r i c t ourse lves only to a su rvey of the individual highlights of this theory.

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Chapter I of the treatise [240, 248] was devoted to the language of schemata; Chapters II and IV [241, 244-247] were devoted to the study and definition of various classes of morphisms of schemata. The cohomology theory of coherent sheaves on schemata (see w was developed in Chapter HI [242, 243]. The ter - minology introduced by Grothendieck in relation with the propert ies of morphisms of schemata includes more than a hundred terms. All properties of Grothendieck morphisms separam into local and global ones, and in their own turn, those are subdivided into three categories: finiteness conditions (morphism of local finite type, locally finitely presented morphism of finite type, finite morphism, quasifinite morphism, etc.), topological conditions (Noetherianness, quasicompactness, open and universally submersive morphisms, etc.), and properties which can be called good (smoothness, flatness, formal nonramifiability, etc.). Among the most importan~ results in the theory of morphisms of schemata are theorems of the following types: cr i ter ia of fulfillment of a given property, construction conditions (the set of points at which a given property is fulfilled, constructively}, preservat ion properties af ter a change of base, descent of a given property, etc. As a rule the proofs of these theorems reduce to the affine case, where they represent sometimes profound results from commutative algebra. Here the technique of passing to the projective limit, developed by Grothendieck, permits us to pass from arb i t ra ry rings to Noether rings and even to algebras of finite type over Z. To the global propert ies of morphisms re fe r the propert ies of projectivity, of the properness of a morphism, investigated in detail in Chapter II.

Grothendieck's results gave a powerful impetus to the development of commutative algebra, by introducing new methods, ideas, and problems. We list some of them:

1. The concept of flatness of a module (introduced by Serre in 1955 [478]) received further development, was given a geometric interpretation, and its role in algebraic geometry and in commutative algebra was stressed.

2. The creation of the technique of passing to the projective limit, mentioned above.

3. The connection of the notion of the depth (or homologic dimension} of a module, introduced and developed by Serre [49], with cohomology theory and, in particular, with local cohomology theory [227,234].

4. The creation of the theory of excellent rings, generalizing and systematizing the resul ts of Zariski and Nagata on local Noetherian rings [401].

5. The theory of Hensetian rings, f i rs t established by Nagata, was essentially developed and was made the foundation of the theory of gtale cohomologies, of smooth morphisms, etc. [247, 453, 467].

6. The theory of descent (see w further on) provided an influx of new ideas and problems into commutative algebra (for example, see [455]).

7. The application of global methods of algebraic geometry and of cohomology theory permitted the solving of certain problems of factorial rings [15-17, 227, 344].

Concrete problems of algebraic geometry made it necessary to study schemata of a more special type. For example, the theory of singularities of algebraic varieties was connected with the study of local schemata, i.e., of open sets of schemata of the form Spec (A), where A is a local ring. A natural generalization of the concept of an algebraic group into the language of schemata (group schema, see [169-171]) proved to be suitable for the study of the reduction of an Abetian variety. ~n this language Raymaud [444] gave a very simple and natural definition of the Neron model of an Abelian variety. This concept, introduced in Well's language by Neron [414], plays an important role in the arithmetic of Abelian varieties. Lichten- baum [340] and Shafarevich [463] developed the theory of two-dimensional regular schemata over a Dede- kind ring and, in particular, the theory of "arithmetic surfaces �9 (also see [112, 128]). Other aspects of the application of schemata theory in the arithmetic of algebraic varieties are the theory of finite ~oToup schemata [421, 424, 425, 497], the theory of p-divisible ~oups [51], the theory of Abelian schemata (Mum- ford [380, 385]), Greenberg schemata [209, 210].

The technique of blowing down sheaves of ideals to schemata, due to Gro~hendieck [241] (also see [321]), proved to be a powerful ~echnical tool in the resolution of the singularities of algebraic varieties over a field of zero character is t ic (Hironaka [269]). The concept of an ample sheaf, introduced by Grothen- dieck in [241], played an important role in the theory of projective immersions of abstract algebraic varieties (see w See [20, 32, 173, 174, 194, 348, 476] for surveys and textbooks on schemata theory.

3. General Facts on Algebraic Varieties. Paral lel with the development of schemata theory investigations were continued on the general properties of algebraic varieties in the spirit of classical algebraic

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geomet ry . Kuan 's r e s u l t s [317-319] re la te to t h e o r e m s of Ber t in i type: if a va r i e t y p o s s e s s e s a ce r t a in p roper ty , then its common hyper f la t sect ion a lso p o s s e s s e s this p roper ty . An analog of B e r t i n i ' s theorem for local domains was obtained by Chow [157]. C a r t i e r [148] in t roduced the notion of a local pr inc ipa l d iv isor ( C a r t i e r ' s divisor) and ~tve a c r i t e r i on for the ra t iona l i ty of a d iv isor . Cheva l l ey ' s 1958/59 s e m i - car [466] was devoted to the theory of divisors. The last section of Chapter 4 of Grothendieck's "Elements" [447] was devoted to the general properties of Cartier's divisors and their connection with V~Teil's divisors. Papers [134, 178, 267, 305, 408, 420, 461, 462, 466] also are devoted to the theory of divisors. Interesting results have been obtained on endomorphisms of algebraic varieties. Ax [96, 97] proved that any injective endomorphism is surjective. Another proof of this fact was given by Borel [133]. The lattice of an algebraic group was introduced in Ramanugam's paper [437] on the automorphism group of a complete algebraic variety. Later Matsumura and Oort [358] reproved this result in a stronger form (see w Many papers have been devoted to the differential properties of algebraic varieties. Each variety X over a field

O" ~ " ~ k defines a sheaf ~X/k of ~,erms of one-dlmenslonal regular differential forms on X. The fiber of this sheaf at a point x 6 X is the 8x = -module O~x j~ of the IC~hler differentials of the ring 8x..~ over field k.

The general properties of the sheaves n /k <and, more generally, even of sheaves n /S for an arbitr=y S-schema X) were studied by Grothendieck in [247] (also see [467]). Many properties of algebraic varieties

I were expressed in terms of the properties of sheaf f~X/k. Thus, for example, the smoothness of X over a perfect field k is equivalent to the local freedom of she'af ~X/k (Grothendieck [247, 467], Nakai [405], Kunz [322]). The case of an imperfect field k was investigated by Asaeda [90]. Grothendieck [247] obtained analogous criteria for the smoothness of a morphism within the framework of schemata theory. Other properties of varieties (for example, normality), expressed in terms of the properties of ~X/k, were investigated in [185, 304, 325, 342, 498, 503]. The book [111] is devoted to the analogs of these results in analytical geometry. In [342] Lipman conjectured that if char (k) = 0, then variety X is nonsingular if and only if the dual sheaf e~----Horn~x (o~i~, ~x) of tangent vectors is locally free. Papers [316, 342, 343] are devoted to this conjecture. Arima [75, 76], Kodama [315], and Kunz [323] addressed themselves to differential forms of the second kind on an algebraic variety over a field of positive characteristic. The general properties of finite coverings of algebraic varieties are taken up by Abhyankar [69] and Popp [435], One of the central results of this theory is the Zariski-Nagata purity theorem for the ramification set. This theorem states that the set of ramification points of a finite covering/: V -- W is a divisor if V is normal and W is nonsingular. In the case when the ground field k has a zero characteristic, this result was proved by Zariski in 1958 [514]. Nagata [396] proved it simultaneously (for arbitrary k). Later on, this theorem (in a different version) was proved over again by various authors [18, 93, 139, 195, 227, 325, 326]. Various analogs and generalizations of the Zariski-Nagata purity theorem were obtained by Grothendieck (for the Brauer group [231] and for a biratlonal morphism [247]), Dolgachev [18, 179] (for the property of smoothness of a family of curves), and Artin (SGA4). For textbooks on algebraic geometry (without schemata theory) see [55, 101, 154, 195, 297, 331, 459, 460].

w Cohomologies of Algebraic Varieties and of Schemata

As is well known, the foundation for the topological and transcendental methods of studying complex algebraic varieties is the presence, on the one hand, of the theory of cohomologies with complex coefficients and, on the other hand, of the theory of cohomologies with coefficients in a coherent analytic sheaf. The profoundness of the results, generalizing the classical results of Picard, Poincare, Lefschetz, obtained by thesemethods at the beginning of the Fifties (Hodge, Hirzebruch, Kodaira, Chern), raised the natural question of the presence of analogs of cohomology theories for abstract algebraic varieties. The necessity for such a theory was stressed, on the other hand, by Well in connection with his significant conjectures in diophantine geometry. In 1955 Serre constructed a theory of coherent algebraic sheaves. In 1962 there appeared the theory of ~tale (or covering) cohomologies of Grothendieck - the first of the theories of the Well cohomologies destined to replace cohomologies with complex coefficients. Later on, other theories of Well cohomologies were proposed by a number of authors (Lubkin, Monsky, Washnitzer, Grothendieck, Verthelot).

1. Cohomologies of Coherent Algebraic Sheaves. In the fundamental paper [46, 468] Serre included algebra{c va r ie t i e s in the genera l ca tegory of local ly r inged spaces and for the la t te r defined the concept of a coherent sheaf of modules over a lattice sheaf of r ings . He a lso developed the cohomology theory of such sheaves. In the case when X [s an algebraic variety over the complex number field C, any coherent algebraic sheaf F on X induces on the corresponding analytic space X an a coherent analytic sheaf F an. Serre's classical results [478] (well known under the designation "GAGA") show that for projective varieties

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there holds the canonic isomorphism of cohomology spaces H i (X, F) - H i (X an, Fan). Moreover, Serre has proved that for any coherent algebraic sheaf 5- on X an there holds the isomorphism F an ~-- ~ where F is some coherent algebraic sheaf on X. Later on, these results were generalized by Grothendieck [217, 442] to the case of complete varieties (not necessarily projective) and by Hartshorne [261, 264] to the case of quasiprojective varieties. Grothendieck transferred Serre's theorems to a formal geometry (see w An essential development of Serre's work were Grothendieck's results on the cohomologies of coherent sheaves on arbitrary schemata [242, 243]. The following finiteness theorem is a central result. For any proper morphism /: X -- S and coherent sheaf F on X, the sheaves iRq):, (F) associated with the presheaf U -- H q (F i I~U), F//-ICU)) are coherent.

In the case when S is the spectrum of field k and X is a projective varie~ over k, this result (estab- lishing in this case the finite dimensionality of the k-spaces H q (X, F))was proved by Serre. Serre's af- fineness criterion [469], contained in the characterization of affine varieties by the property of H i (X, D = 0 for any sheaf of ideals I, was carried over to the case of affine schemata by Grothendieck [242] and by Nastold [412] (also see [194]). The techniques of Grothendieck's formal geometry allowed us to give a cohomological generalization of the connection theorem and of Zariski's theory of holomorphic functions (see ~6). The general theory of duality for coherent algebraic sheaves is another direction in the generalization of Serre's results. Grothendieck =~ave an outline of this theory at the Edinburgh congress [14] and in a report at the Bourbaki seminar [218]. Serre's theorem [513] states that for any nonsingular projective variety over a field k of dimension n and for a locally free sheaf L on X the spaces H I (X, L) and H n-~ (X, L@~x) are dual to each other. Here o~ X is the sheaf of germs of regular n-forms on X, and L is a sheaf dual to L.

In report [218] Grothendieck =~ave certain generalizations of this theorem and then at the Edinburgh congress [14] formulated and discussed the general duality theorem for arbitrary complete varieties. Hartshorne's seminar [259] was devoted to this theorem (and its generalizatior/for an arbitrary proper S- schema). The general formalism of duality, established in the language of arbitrary Weil categories, developed in [259, 501], was then used in other duality theories [118, 417, 499]. Another approach to the proof of the general duality theorem is due to Deligne (see [259], Appendix). The connection between these approaches was shown by Verdier [502]. The Altman-Kleiman seminar [71] was devoted to the specialization of Grothendieck's results for the case of projective varieties. A very beautiful approach to the theory of duality on a curve was indicated by Tare [52]. Also see Nastold [411, 413]. The language of sheaf theory proved to be very convenient for the formulations and proofs of many results of classical algebraic geometry, namely, the questions connected, first of all, with the Riemann-Roch theory and the theory of linear systems (see [31, 513]).

Sampson and Washnitzer [457, 458] obtained important results on the behavior ofH i CK, F) relative to monoidal transforms and the BYdnneth formula. The la~er was significantly generalized by Grothendieck [242]. The behavior of cohomologies relative to proper mappings and algebraic correspondences was studied by Snapper [492, 495] and Matsumura [357]. The first of them proved a very important theorem stating that for any coherent sheaf F on a nonsingular variety X and Euler divisors D I ..... D k the characteristic 7. (X, F ~ ~ (~D~ +... + =~D~)) is a polynomial in n I ..... n k [493]. This result was used by him to define the intersection number for divisors [494] and was then generalized to quasiprojective varieties by Cartier [149]. Borelli [134] introduced the concept of a divisoral variety a special case of which is a nonsingular or quadiprojective variety and extended Snapper's theorem to this variety. A generalization of Snapper's theorem is due also to ~oishezon [37].

While the cohomologies of projective and affine varieties have been studied in sufficient detail, there have remained many questions on the cohomologies of arbitrary quasiprojective varieties. The first result in this direction was Grothendieck's theorem [234] stating that l-/n(X, F) = 0 for any coherent sheaf F on an n-dimensional variety X all of whose irreducible components are improper. An elementary proof of this theorem (not using local cohomology theory) was given by Kleiman [310]. Fundamental results on the cohomologieal dimension of quasiprojective varieties were obtained by F/artshorne [260, 261]. In particular, he has proved that cd (P~ \ ,C)= l if C is a curve, and this put an end to the a t t empts to give a cohomological p roof to K n e s e r ' s old prob lem [318]. Together with Goodman he g'ave a cha rac t e r i za t i on of va r i e t i e s the cohomolo~y g'roupe on which a r e f in i te -d imensional .

The main technical tool in ~he preced ing theory is +.he theory of local cohomologies of coherent sheaves , due to Grothendieck [227, 234]. By genera l iz ing the c l a s s i ca l definition of cohomologies with compact sup- p o r t s , h e defined groups r respec t ive ty , s h e a v e s ) o f local cohomologies H~ (X, F) ( respec t ive ly , J{w (F)) of sheaf : re la t ive to a c losed subschema Zc_X. Here a r e the mos t impor tant r e su l t s of this theory:

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a) A. cohomological cha rac te r i za t ion is given of the impor tant concept of the depth, depth Z (F), of a sheaf F re la t ive to subschema Z. This concept is the global var iant of the notion of the depth (or homological dimension) of an A-module M, due to Se r r e [49]. The re holds the theorem depthz(F) ~ k ~,~r i < k .

b) The exact sequence

. . . . n z ' ( X , F ) - , . . . .

allows the connection of groups Hi(X, F) and H i ( X \ Z , F ) .

on local duality is proved permit t ing the computation of the group Extk (M, A) for an C) A theorem i A-module M in terms of the local cohomologies ,h~ (A4)----H{m ~ (Spec(A), ~W) (under certain restrictions on

ring A).

d) The theorems on the finiteness of sheaves ' ~z iF), due to Grothendieck [227] and then generalized by Hartshorne [262] and Peskins [430].

We mention the most important applications of this theory. The concept depth Z (F) is closely connected with the question of the coherent extension of a sheaf FIX- Z. In particular, results in item a) permit us to prove again a part of the assertions of M. Baldassarri and his students [101'-106, 351-355] on the characterization of torsion-free sheaves as well as one of Hartshorne's results [256]. The results in item d) were applied to the theorems on the finiteness of cohomologies for quasiprojective varieties [261, 428]. Sheaves of local cohomologies play an important role in duality theory [259]. With their aid we can construct the canonic injective resolvent of a sheaf (the Cousin complex), which plays one of the central roles in this theory. Sharp [485] has studied this complex in the case of affine schemata. The theorems of finiteness of local cohomologies were applied for the generalization of the comparison and existence theorems (see w to the case of not necessarily proper morphisms [227]. In [262] Hartshorne proved the affine duality theorem stated as a conjecture by Grothendieck in [227].

Important applications of local cohomology theory relative to "Lefschetz-type" theorems on the comparison of the Picard group (or the fundamental group) of schema X and of its subschema Y of codimension I were obtained in [227]. This comparison is carried out in three stages. At first Pic C~) and Pic (U) are compared, where U is an open set containing Y while X is the formal completion of X along Y (see w Next, Pic (U) and Pic (X) are compared, and finally Pic (X) and Pic (Y). Local cohomology theory has a bearing only on the first two stages, the last stage being studied by the methods of formal geometry. The investigation of the second stage is connected with the condition of factoriality of the rings of points x 6 X/U. Here Grothendieck obtained important results, one of which is the affirmative answer to the following conjecture of Samuel: a local Noetherian ring A, being a complete intersection and factorial in codimension m 3, is a factorial ring (see Giraud's report [199] for a survey of these results). In the investigation of the analogous stage for a fundamental group Grothendieck generalized the Zariski-Nagata purity theorem (see w I) to the case of local Noetherian rings of dimension z 3, being complete intersections.

Grothendieck obtained both local as well as global versions of Lefschetz's theorems on hyperflat sections. A special case of these results is, for example, the following theorem. For any complete intersection X in P~ of dimension >- 3 (respectively, ~- 2), Pic (X) ~ Z and is induced by a hyperflat section (re- s cUvel �9 The r pe " Y, I (X) = 0). se esults generalize the classical facts true for the case when X is nonsingular and complex (cf. [41, 197, 198]). A generalization of Grothendieck's theorem was given by Rayuaud [438]; also see Hartshorne [261].

2. Well Cohomolo~es of Schemata ~ In 1949 Weil pointed out the need for some theory of the cohomologies of abstract algebraic varieties, analogous to the theory of complex cohomologies, which would possess all the formal properties (Poincare duality, IGinneth formula) for the derivation of an analog of Lefschetz's formula on fixed points. More precisely, such a theory should be described by the following assignments and axioms:

There exists a contravariant functor X -- H* (X) from the camgory of smooth projective k-varieties into the category of finite-dimensional anticommutative graded algebras over a field K of zero character- isric. Here the following properties should be fulfilled:

A. Poincare dually. Namely: a) groups H i (X). = 0 for [ > 2n (n = dimX), b) the or~ntation isomorphism H m ~) -~ K exists, c) the canonical coupling H ~ iX) x H 2n-i (X) --- H m ~X) ~ K is nondegenerate.

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B. K'dnneth formula . Let Pl: Xi x X 2 - - Xi (i = 1, 2) be projec t ions . The canonic mapping a $ b ~ p~ [a).p~.~b) defines the h o m o m o r p h i s m H* (Xt)~H '~ (X~.) --~ H ~ (X, X X~).

C. Gysin hom om orph i s m . There ex i s t s the h o m o m o r p h i s m of ~oToups 7X: C p (X) - - H 2p (X), where cP (X) is the ~oT0up of a lgebra ic cyc les of codimension p of va r i e t y X. Such a h o m o m o r p h i s m should be functor ia l with r e s p e c t to X, mul t ip l ica t ive (Txxr (Z X ~ ' ) = T x (Z)| (V/)), and centTal (if P is a point, then C* (P) ~- Z - - H* (P) - K is a canonic imbedding).

The p re sence of such a theory p e r m i t s us in a fo rma l manne r to der ive L e f s c h e t z ' s formula for the fLxed points of the co r re spondence and, in pa r t i cu la r , if k is the a lgebra ic c losure of a finite field, to ob- rain proofs of the f i r s t two conjec tures of Well on the i - func t ion ( ra t ional i ty and functional equation) (see [311]). tn case k = C the funcmr X - - H* (X an, C) yie lds the theory of Well cohomologies .

S e r r e ' s a t t empt s to cons t ruc t such a theory, s t a r t ing f rom the theory of coheren t sheaves , was not crowned with success . To be p r ec i s e , in [471] Serve p roposed to examine the cohomology H~(X, V/(#x)) with coeff ic ients in the sheaf of r ings of Witt vec to r s ove r local r ings X. Although these cohomologies p roved useful for many quest ions connected with al~oebraic va r i e t i e s over a field of posit ive c h a r a c t e r i s t i c s (for example , see [381]), they turned out to be unfit for Wei l ' s theory. Even for Abetian va r i e t i e s the cohomologies N ~ (X, ~ (#x)), cons ide red as modules over the Witt r ing W (k), a re not finitely genera ted (Serve [479]). We have not succeeded even in obtaining the "true ~ Betti numbers of va r i e ty X with the aid of co- he ren t sheaves [286, 471].

A comple te ly di f ferent approach to the definition of Weil c o h o m o l o ~ e s was f i r s t sugges ted by Gro- thendieck [141]. The f i r s t publicat ion on this theory. (Grothendieck 's cover ing or ~tale cohomologies) appea red only in 1962 in the fo rm of r e p o r t s a t A r t i n ' s s e m i n a r at H a r v a r d [78]. Surveys [1, 30, 79, 83, 200] a r e devoted to sketching the mi le s tones in this theory. The Ar t in -Gro thendieck s e m i n a r in 1963/64 (SGA 4) as well as V e r d i e r ' s pape r [499] ~vas devoted to the proofs of the fundamental t heo rems on ~tale cohomologies (with the aid of these t h e o r e m s it was es tab l i shed that a var ian t of the theory of ~tale cohomologies , i .e, , of l -ad ic cohomologies , is the theory of Weil cohomologies) . These r e su l t s as well as the appl icat ions to $- func t ions on a lgeb ra i c va r i e t i e s were the subject of Grothendieck ' s s e m i n a r of 1965 (SGA 5). Tate [50], Kleiman [311], Grothendieck [228], and Deligne [164] (cf. the su rvey [44]) have d i scussed the appl icat ions of this theory to the a r i thmet ic quest ions of a lgebra ic va r i e t i e s . Other methods of de te rmin ing Well cohomologies were sugges ted by Lubkin in 1967 [345] and by Grothendieck [237] (see l a te r on about them).

3. Grothendieck Cohomologtes of Schemata . The cons t ruc t ion of these cohomologies is based on the genera l notion of Grothendieck ' s topology. The l a t t e r s ignif ies the ~ v i n g of a ce r t a in ca tegory T with f iber bundles and of a col lect ion Coy (T) of fami l ies of m o r p h i s m s {U~U}0.Er ' ,6r cal led the cover ings of object U. Here ce r t a in natura l ax ioms should be fulfilled, turning into the usual ax ioms for a topological space if as T we take the ca t egory of open se ts of some topological space (morph i sms turn into embeddings) . The genera l f o r m a l i s m of Gro thendieck ' s topology and of its var ian t , leading to the notion of a topotogized ca t egory (site), v,'as developed in (SGA Q (also see Giraud [200] and [468, 469]). At the p resen t t ime this theory plays an independent ro le in homological a lgebra , ca tegory theory, and l o ~ c . I f X is a schema , i fT is

some subca tegory of the ca t egory of X - s c h e m a t a (Sch/X), and if the fami l ies {U~ ~U}~ E~ of X - m o r p h i s m s

into T such that U=[J/,(U~) are cover ings , then we obtain the notion of Grothendieck 's topology of s chema

X, a s soc i a t ed with the ca tegory T. F o r example , if as T we take the ca tegory of ~tale m o r p h i s m s , we then obtain the ~tale topology for s chema X,which we denote Xet. The idea for such a mpotogy was suggested by Grothendieck in Serve [470]. If T is the ca t egory of s t r i c t ly flat quas icompact m o r p h i s m s , then we obtain the fpqc- topo logy of schema X, denoted Xfpqc. Other Grothendieck topologies of schemata - the quas i - finite Xqf , the s t r i c t ly flat f initely p re sen ted X /pp f , e tc . - a r e obtained analogously. In pa r t i cu la r , if T is the ca t egory of Z a r i s k i l o p e n subse ts of X, then we obtain the usual Za r i sk i topology XZa r of schema X, La te r on, Grothendieck genera l i zed the notion of a sheaf with values in a ca tegory C to a topological space by defining a sheaf on the Grothendteck topology T as a ce r t a in con t rava r i an t functor F: T - - C sa t is fying a ce r t a in axiom (the sheaf axiom). For any Abelian sheaf (C is the ca tegory of Abellan ~oToups) the coho- rnotogy ~voups H i (T, F) a re defined by the usual methods. (in the non-Abelian case, cohomology theory was developed by Giraud [202, 204]). In pa r t i cu la r , for schema X there have been defined var ious Grothendieck cohomo[ogies: the ~tale cohomolo~o~/H i fXet , F), the fpqc -cohomology H [ (Xf~qc, F), etc. All of the usual f o r m a l i s m of sheaf theory takes place within ~ e f r a m e w o r k of the theory of sl ieaves on Grothendieck

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topologies [78, 169] (SGA 4). When X = Spec (k), where k is a field (i.e., X is a "point"), the ~v ing of a sheaf F on Xet is equivalent to the ~v ing of the GaI (k/k)-module F(,%)= IimF(K), where K/k a re all possible

z extensions of the Galois field k. The groups of cohomologies H i(Xet, F) a re p rec i se ly the Galois cohomologies H i (Gal (k/k), F ([Q). F rom this point of view the theory of Galois cohomologies, set forth, for example, in S e r r e ' s book [48], can be cons idered as a point cohomolo~o'y theory. Commutat ive group X-schemata [421] furnish examples of Abelian sheaves on Xet and X / p q c . For example, the mult ipl icat ive group schema G m defines the sheaf O~,x (U-~Q~,.x(U)=.F ~U, 9~)); the constant group A defines the constant sheaf A X CU-" A~0(U)). A genera l iza t ion of "I-Iilbert's theorem 90," proved by Grothendieck, s ta tes that H' (Xet, Gm~x) = Pic (X). The group H 2 (Xet, GIn,X) turned out to be c lose ly re la ted to the B r a u e r group Br (X) of schemaX. The la t te r concept, general iz ing the c lass ica l concept of the B r a u e r group of a field, was introduced by Auslander and Goldman [95] for the case of commutat ive r ings and was extended to the case of a r b i t r a r y schemata by Grothendieck. The foundations of this theory and cer ta in impor tant computat ions a re contained in three fundamental papers [299-301] of Grothendieck. Other computations of the Brau e r group a r e to be found in [23, 366, 464]. The f i r s t computat ions.of ~tale cohomologies for a lgebra ic curves and sur faces were made by Artin. In this case the groups H I (Xet ,(Z/n)x) ((n, char (kn)) = I) a re comple te ly analogous to the c lass ica l groups H i (X an, Z/n) if X is a complex curve or sur face . La te r on Art in proved a compar i son

i theorem H (Xet, (Z/u)X)=~ I-I i CX an, Z/n) for a lgebra ic va r ie t i e s over field C (SGA 4) (also see [200]). The computation in [443] of cohomologtes Hi(X,t, ~), where ~ is the sheaf assoc ia ted with the Neron model of an Abelian va r ie ty A over the field of ra t ional functions of curve X, allowed us to in te rpre t and genera l ize the resu l t s of Ogg [416] and of Shafarevich [53] on pr incipal homogeneous spaces over A. When X is a two- dimensional r egu la r schema over a homogeneous r egu la r schema B, group Br CO proves to be c lose ly r e - lated with the Shafarevich group H'(B,,, ~9, where ~ is the Neron model of the Jacobi va r ie ty of the common f iber X. In the geomet r ic case when X is an ell iptic surface , this connection was d i scpvered by Shafa- revich [54]. tn the case of a lgebraic sur faces X the Betel numbers , calculated as dimFz(H 1, }let, (z/~)~) (5 is a p r ime not equal to char k), coincide with those de te rmined previous ly by Igusa [287]. Fur thermo' re , in this case the group Br (X) (its divisible part) can be in te rpre ted as an analog of the group of t ranscenden- tal cycles on X. We also note the decisive ro te of the B r a u e r group in the Art in and Mumford resolu t ion of L u r o t ~ ' s conjecture [165]. The fundamental p roper t i e s of 4tale cohomologies were proved by Art in and Grothendieck (SGA 4). Art in proved a theorem on homological dimension (cdX -~ dim X for a lgebra ic va r ie - ties) and a f ini teness theorem: the group H i (X, F) is finite for any per iodic sheaf F e i the r on a p rope r schema X or on an exce l len t schema X of cha rac t e r i s t i c zero . To Grothendieck a re due a " theorem on change of base ," a Klinneth formula, and a theory of cohomologies with compact supports . Verd ie r [499, 500] announced general duality theorems and Lefschetz formulas analogous to the Atiyah-Singer-Bot~ formulas for el l iptic complexes . Lefschetz formulas for ~tale cohomologies were cons idered by Raynaud [439, 441]. For any algebraic var ie ty X over k and for a pr ime Z ,' char k the 6taie cohomologies define t - ad i c cohomologies of X. By definition, H'(X, Z,) = H_m H'(X,t,(Z/I~)). The Q ! - spaces kl' (X, Q,) = H ~ (X. Zz)~Q z play the ro le of

Weft cokomologies for X. The in teres t ing quest ion of the independence of the Betti numbers b i (X; Z) = dimQ~ H 1 (X, QZ) f rom Z has been answered only in special cases (for example, d imX <- 2). See [50, 228]

as well as the survey [44] for the applicat ion of Z-adic cohomologies to a r i thmet ic .

Using the concept of the Grothendieck topology, we can give a cons t ruc t ion of the c lass i fying space for an algebraic or a d i sc re te group G [236]. A detai led exposi t ion of the cor responding theory is contained in Giraud's book [204]. Grothendieck also gave an a lgebra ic definition of Chern c lasses of the r e p r e s e u t a - tion of a d i sc re te group G over an a r b i t r a r y field k [236]. The ~ n e r a l i z a t i o n of these ideas to a r b i t r a r y Lie groups and thei r r epresen ta t ions was ~ v e u by Geronimus [9-11].

Shatz [486-490] and Mazur [363, 364] devote themselves to the study of the general p rope r t i e s of o ther Grothendieck cohomologies as well as to ce r ta in computations. Mumford [379] applied the Grothendieck topology to module problems. Mazur ' s papers (cf. [27]) contain a computation of flat cohomologies (/pqc or /pp/~ for number schemata as well as applications to the a r i thmet ic of a lgebraic var ie t ies . The topologies X/pqc and X f p p / h a v e proved to be par t icu la r ly useful in the theory of group schemata [168-171]. F o r ex-

ample, we mention the in terpre ta t ion of =~'oups of principal homogeneous spaces of a group S-schema G as the group H i ( S ~ p / , G) [34, 369, 490]. We r e m a r k that by vir tue of Grothendieck 's theorem [231], if G is a

"~ i S i S G % smooth schema, then H ( f p p f , G) = H ( e t ' ) . . par t of paral le l resu l t s on Grothendieck cohomologies in another language was obtairied in [151-153, 177].

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4 : Other Cohomology Tll.eories of Schemata. a) De Rham cohomologies of a lgebra ic var ie t ies . The beginnings of this theory were set forth in Groth.endieck's fundamental paper [232]. For any S-schema X Grothendieck defines the de Rham cohomology H~R (X/S) as the hypercohomology H n (X, 0~"/S) of a complex of sheaves of re la t ive different ia ls of schema X. When S = Spec (C) and X is a smooth a lgebraic var ie ty over S, Grothendieck, developing the ideas of Atiyah and Hodge [91], proved the compar i son theorem H~R (X/C) = H n (X an, C). This theorem pe rmi t s us to compute the cohomologies of a complement to a hy- pe r su r face in a pro jec t ive space with the aid of the cohomology c lasses induced by ra t ional different ial fo rms with poles on this hypersur face . This r e su l t was used by Atiyah, Bott, and G;Irding in [6].

More general ly , Grothendieck defined the sheaves of re la t ive de Rham cohomologies as ~{~R ( X / S ) = Rl / , (~2}/s), where on the r ight stand the values of the i- th hyperder iva t ive functor of the d i rec t image functor of sheaf/~, on the de Rham complex i]~L/S. I f / : X - - S is a p rope r smooth morphism of complex algebraic var ie t ies , then from Grothendieck 's compar i son theorem follows the exis tence of a canonic i somor - phism of analytic coheren t sheaves ~ R (X/S) ~' ~- iPl/a, n (C)~c~san" Katz andOda [25] gave ana lgebra ic definition

of a canonic integrable connection (the C-auss-Manin connection) on the sheaves 3g~ (X/S). In the preceding situation this definition reduces to the connection on a vec tor bundle definable by a local ly f ree sheaf 3g~R (X/S) an whose sheaf of hor izontal sect ions coincides with /~/~n (C). When X/S is an algebraic curve over a functional field, the Katz-Oda const ruct ion reduces to the one proposed ea r l i e r by Manin in [28]. In a slightly less genera l case an a lgebra ic definition of the Gauss-Manin connection was ~ v e n by Grothendieck [237], Katz gave an a lgebra ic proof of the r egu la r i ty theorem for the Gauss-Manin connection [302]. Ana- Iytical proofs were given by Griffith [215] and Deligne [163]. The concept of a regu la r connection genera l izes the c lass ica l notion of a different ial equation with a r egu la r s ingular point and is the subject of Deligne's book [163] (also see [302, 350]). A genera l iza t ion of Grothendieck 's compar i son theorem also is given in it.

Katz [300, 301] ~ v e an in terpre ta t ion of Dwork~s resu l t s [182] with the aid of de Rham cohomologies and of the Gauss-Manin connection on them. Oda [415] studied the connections between H~) R (X/k), where X is a complete var ie ty over a per fec t field k of cha rac te r i s t i c p > 0, and the Dieudonne modules c o r r e - sponding to the P i c a r d schema P tCx/k .

Since the spaces H~) R (X/k) hard been defined, genera l ly speaking, over a field of zero cha rac te r i s t i c , the de Rham cohomologies a re not the Well cohomology theory. Never the less , Har tshorne [261] has proved Po inca re duality for H~R (X/k), while in the case of a zero cha rac te r i s t i c he has given an a lgebraic definition of the Gysin homomorphism and of the tCdnneth formula. H e r r e r a and Liebermann [265] proved that for a smooth subschema Y c X of an a lgebra ic var ie ty over field C there holds the i somorphism /-f* (Y, C) "~I!m_H~u(Y~)/C ), where y(k) is the k- th inf ini tesimal neighborhood of subschema Y. This r e su l t was ob-

tained independently a lso by Deligne. The la t ter has also investigated in detail the connection between de Rham cohomologies and Hodge cohomologies for va r ie t i e s over a field of zero cha rac te r i s t i c [162].

b) Crys ta l cohomologies. The de Rham cohomologies found their natural general izat ion in the f r amework of c rys t a l cohomologies. The basic construct ions of this theory and also the program for fuatre invest igat ions in this a rea formed the subject of Grothendieck 's 1966 lec tu res [237]. The fundamental r e - sults of this theory, establ ishing that c rys t a l cohomologies a re Well cohomologies, were announced la te r on by Berthelot [!13-121]~

For any S-schema X Grothendieck proposed to examine topologized category (X/S)cry s whose objects are all nilpotent S-imbeddings UC-.T, where U is a Zariski-open subset of X, while the ideal of U in T is provided with a lattice of separated degrees. Morphisms of the pair (U, T) are defined in the natural way, and the Grothendieck topology on (X/S)CrFs is induced by the Zariski topology on X. The sheaf on (X/S)CrFs is given by the system of sheaves F(U ' T) on T. The lattice sheaf ~x/s is defined by the system F(u,r ) =~r.

If X is smooth over S, then each coherent sheaf ~ on schema X, provided with an integrable connection, defines a sheaf of modules on (X/S)CrFs. Berthelot [115] proved the isomorphism H* ((X/S)crF s, ~'Crys)~

H* (Xz~, J r | where on the r ight stand the hypercohomologies of the complex ~r@o~/s of coherent sheaves

on X, defined by the connection on ~'. When the S-schema is of zero cha rac te r i s t i c , this r esu l t was proved by Grothendieck [237]. in par t icu lar , if J r = ~ x , this r e su l t shows that ?/~R (X/S)~/-I* ((X/S)crys,~vx/~) and, in

par t icular , the de Rham cohomotogies can be computed without differential forms (!). In o rde r to obtain

the V~eil cohomoiogies from c rys t a l cohomologies, for any schema X 0 over a per fec t field k we need to set

273

H~rys(Xo/k) = li, m H* ((Xo/S=)crys,#x~,s~), where S n = Spec (w/pn+iw) is the spect rum of the quotient r ing of

the Witt r ing W (k). If X 0 is proper , smooth on k, and is l ifted up to the v e ry same schema X over S = Spec (W (k)), then there holds the i somorphism HCry s (X0,/k) HDR (X/S). This r e su l t shows the invariance of the de Rham cohomologies I-I~R (X/S) re la t ive to the different tiftings of schema X 0 up to schema X. This r e su l t was obtained by other methods by Lubkin [345] and Monsky-Washni tzer (cf. [237]). The cohomology groups H~vis (Xo/lr) ~ K (K is the field of par t icu la r r ings W (k)) sa t i s fy the conditions of Well cohomologies W/'t/t)

[121]. In [116] Ber the lo t cons t ruc ted a theory of local c rys t a l cohomolo~es and in [122, 123], together with !l lusie, defined the Chern c lasses of a sheaf with value in c rys t a l cohomologies. A compar i son theorem of c rys t a l cohomotogies with c lass ica l ones over field C is due to Deligne (unpublished). Grothendieck 's work on represen ta t ions of profinite groups [239] has applicat ion also to c rys t a l cohomologies. The application of c rys t a l cohomologies to Dieudonne modules and to p-divis ible groups is contained in Grothendieck 's r epo r t at the Nice congress and also in Mess ing ' s d i sser ta t ion [365].

We r e m a r k that a theory of c rys t a l cohomologies is a p-adic theory of cohomologies in the sense of Grothendieck [237]. That is, in con t ras t to the theory of Z-adic cohomologies, it gives information on the p - to r s ion in the cohomologies (p = char k). Other Grothendieck cohomologies (the fpqc- or the f p p f - c o h o - mologies) ~ v e good p-adic cohomologies only in smal l dimensions (since by virtue of Ar t in ' s theorem, ~Ii(Xfpp/, Z /p k) = 0, i > d imX + 1).

c) The p-adic cohomologies of Monsky-Washni tzer and of Lubkin. Other possible approaches to the theory of p-adic cohomotogies were suggested by Monsk'y-Washnitzer [372-375] and by Lubkin [346, 347]. In a b r i e f note [374] in 1964 the f i r s t authors proposed to define cohomologies in the following way. F i r s t of all they defined a ce r ta in c lass of smooth a lgebras over a Witt r ing W(k) (the Monsky-Washni tzer ~w.c.f .g.~-algebras) and proved that for any smooth k -a lgebra A 0 there exis ts a functorial lifting up to an

, algebra A with A | ~-----A o. The ass ignment A 0 - - HDR (A/W(k)) defines a cohomological functor f rom the IV(~)

ca tegory of smooth k-a lgebras into the ca tegory of W (k)-modules. By localizing this funetor re la t ive to the Zar iski topology, Monsk T and Washni tzer obtain the cohomology sheaves ~ z (X) for any smooth schema X ove r a perfec t field k. Global sect ions of these sheaves furnish ~good ~ invariants in smal l dimensions. For example, for a curve X of genus g, Ho(X, g ~ v (X)) is a f ree W(k)-module of rank 2g. Working only with k -a lgebras of finite type, Monsky succeeded in using his own theory to prove a f ixed-point formula (using complete ly continuous opera tors ) and the ra t ional i ty of the ~-function of smooth schema [372, 373]. The question of the gtobalization of the MonskT-Washnitzer idea for obtaining a theory of Weil cohomologies remains open until now (see [237] for a discussion of this).

In [346] Lubkin, on the basis of the Monsk-'y-Washnitzer ideas, gave a definition of p-adic Welt cohomologies for the subcategory of var ie t ies liftable to cha rac t e r i s t i c zero . We r e m a r k that by vir tue of S e r r e ' s example [475], such var ie t ies do not exhaust the c lass of all smooth project ive var ie t ies . Also, as Grothendieck 's c rys t a l cohomologies, these cohomologies can be computed on the basic only of the usual Zar iski topology of schema X. If X is a lifting of a nousingular project ive var ie ty X over a field k up to a flat p roper ~ - s c h e m a , where the res idue field of r ing # is k, while the f rac t ion field K has zero c h a r a c t e r - istic, then Lubkin assumed H* (X)==H'oa(X~K/A') and proved that this is not dependent on the lifting of X to

X. In this paper the author developed a genera l technique of hypercohomologies I-I t (X, U, F* ) of finite complexes of sheaves on a topological space X with r e s p e c t to a module of an open set U.

d) Combinatorial Lubkin cohomologies. In [345] Lubkin, using an Stale topology of a schema X, as - sociated with each such schema a ce r t a in pros impl ic ia l complex S(X)=limS(X, U) and, a f te r this, defined

5" H*(X, M)=lirnH* (S (X, U), :14) for any Abelian group M. The author ' s main theorem a s s e r ~ that these -y

cohomologies coincide with the Grothendieck ~tale cohomologies t-I* (Xet, (M)x). The author announced theorems analogous to the fundamental Ar t in-Grothendieck theorems and, f rom these theorems , der ived in a s tandard manner the f i r s t two Well conjec tures as well as proved that the c lasses of a lgebra ic cycles a re finitely genera ted with r e spec t to numer ica l equivalence. The la t te r fact follows formal ly f rom any theory of Well cohomotogies [311].

e) Grothendieck 's theory of motifs . The presence of so many different cohomology theor ies of schemata posed the quest ion of the crea t ion of some ~universal cohomology theory. ~

274

The idea of such a theory was suggested by Grothendieck in 1967 (unpublished). The first publication on this theory, in which the main constructions of Grothendieck were set forth, belongs to Manin [33]. Later on, this theory as well as Manin's results,which we take up below, was the subject of Demazure's report [IS7]. Grothendieck's construction is extremely sharp and, roughly speaking, consists of the following. The category V/k of algebraic varieties over a field k is immersed at first into an additive category CV/k (by an introduction of supplementary morphisms, i.e., of all possible correspondences relative to some theory of intersection C) and next into a pseudo-Abelian category CV/k (by an association formally of the kernels and images of all possible projectors into CV/k). Every object (motif) of category CV/k is re.m- resented as a direct sum of "n-dimensional pieces of varieties," i.e., of objects of the form (X, p), where X is an n-dimensional variety and p is a projector from C n (X x X). The motif h (X) = (X, id) is called the

2a motif of variety X, and its expansion into the direct sum h(X) = ~-0 h~(X) yields the definition of the "groups

of motif cohomologies h i (X)" of variety X. Any functor of cohomologies X --- H* (X) passes through functor h. Grothendieck called H* (X) a realization of motif h (XS. Unfortunately, the proof that the functor X -- h (~ itself is a theory of Well cohomologies is based on the as yet unproved assumptions on algebraic cycles (the standard conjectures; see [238, 311]).

Nevertheless, Manin succeeded in showing in [33] that, without using the hypothetical part of theory of motifs, we can progress rather far in their computations and obtain important applications. This paper clarifies the behavior of motifs under monoidal transformations and finite coverings and also defines the "motif intermediate Jacobian" of a three-dimensional unirational variety. Using the functoriality of the latter, Manin proved the Well conjectures for such varieties. The articles [56, 57] of Shermenev serve as con- tinuations of Manin's work, in which Shermenev computes in explicit form the motif of a cubic hypersurface, of an Abelian variety, and of a Weft hypersurface (the latter has not been published).

In conclusion we remark that a cohomology theory of schemata, established chiefly in Grothendieck's fundamental papers, has turned into a fascinating area of algebraic geometry with a lot of unsolved problems and with r i ch applications.

w Fundamental Group and Homotopic I n v a r i a n t s

o f S c h e m a t a

As the foundation of the a lgebra ic definit ion of the fundamental group of a s chema we take the c l a s - s ica l fact that this group should c l a s s i fy nonrarnif ied cover ings . By the l a t t e r in s chema ta theory we mean an a r b i t r a r y finite ~tale m o r p h i s m . When X is a no rma l i r reduc ib le a lgebra ic va r ie ty , the group ~r 1 (X) was defined by Abhyankar [63] as a Galois group (a profini te group) of a max ima l non.ramified extension of the ra t iona l function field k (X) of va r i e t y X. By vir tue of R i e m a n n ' s c l a s s i ca l ex is tence theo rem, when the ground field k is the complex number field, ~'i (X) is a profinite completion of the usual fundamental group (for example, see [442]).

In 1960 Grothendieck [467] gave a definition of ?r i (X) for any schema X, which reduces to the above definition if X is normal and irreducible. For this he defined the concept of a Galois covering of schema X, proved the prorepresentability of the functor X' -- Horn X (~, X') from the category of such coverings into the category of sets (~ Ks a geometric point of schema X), and then set ~:, (X, i) = lira Aut (P~/, where (P~)~61 is the

"r

prorepresentative object of the functor being considered. For a connected schema X the group ~I (X, ~) does not depend (to within isomorphism) on ~. Later Grothendieck [170] modified the definition of a fundamental group (groupe fondamental ~lar~) so as to take into cohsideration.the infinite coverings of schema X. When schema X is one-branched (for example, is normal), these groups coincide. In the general case the extended group '~! CK) is not profinite (for example, for a rational curve X with a regular double point, "~i CK) = Z). Murre's lectures [391] were devoted :o Grothendieck's definition.

Another definition of the fundamental group (coinciding with Grothendieck's extended fundamental group) was ~ven by Lubkin [345].

The series of papers [63-68] by Abhyankar was devoted to the first algebraic computations of the fundamental group. In the main they were devoted to the computation of the group ~, (V\R7), where i~TcV is a divisor on a nonsingular algebraic variety V of dimension -~ 2. In case char k > 0 there has actually been computed the subgroup ~ (V\ V/)c~., (V\LV) classifying the coverings which are prolonged upto tamely r ami f i ed cover ings of V. Abhyandar ' s r e su l t s genera l i ze the c l a s s i ca l computat ions of the Za r i sk i gToup

275

~.1 (P~\C), where C is a curve . S e r v e ' s a r t i c l e [473] and P o p p ' s book [435] su rvey Abhyauka r ' s work. Ed- rounds [182] also concerns h i m s e l f with this s ame group of quest ions . In his fundamental pape r [219] Grothendieck computed the group =~ ( X \ (P~ . . . . . P,}) for a nonsingular point curve X. This r e s u l t p layed a cen t ra l ro le in the pape r s of Shafarevich [53] and Ogg [416]. Gro thendieck ' s computat ions were based on the lifting of the curve to c h a r a c t e r i s t i c zero and on the appl ica t ion of the topological r e s u l t s in [440]. An analogous computat ion for a comple te curve X (the f i r s t of which computa t ion was proved without the use of the methods of fo rma l geometry) was c a r r i e d out a l so by Popp [431, 432]. The a s sumpt ion of t ame r a m i f i - cat ion was r e m o v e d by Fulton [191]. A pure ly a lgebra ic computat ion of =~ ( X \ { P 1 . . . . . P=}) is as ye t not known, tn [435] Popp showed that every th ing r educes to the three-poin t p rob lem, i .e. , to the case when n= 3 (also c o m p a r e with [176]).

The method for comput ing the fundamental group with the aid of l ift ing to c h a r a c t e r i s t i c ze ro is based on the following theo rem of Grothendieck. If f : X - - Spec (A) is a p r o p e r m o r p h i s m with a c losed connected f iber X0, where A is a full Noether ian local r ing , then vl CA') - v i (X0). This s ame theo rem, with A r ep l aced by an arbitraz-y I-Ienseltan Noether ian local r ing, was p roved by Art in [85]. F r o m it follows the t heo rem on the change of base in Gro thendieck ' s ~tale cohomologies . Ar t in a lso c o m p a r e d the groups r: t CC) and ~l (U), where U is an open se t in Spec (A) and U is its p r e image in Spec (A) [81, 85].

See w on the Lefschetz theorem for the fundamental group. A var ian t of the Lefsohetz t heo rem for the case of a quas ipro jec t ive v a r i e t y V was proved by Popp [434]. In the c l a s s i c a l case , when V is an open set in a p ro jec t ive space, this r e s u l t was proved (with gaps) by Z a r i s k i in 1937. De Bru in ' s pape r [144] a lso was devoted to Z a r i s k i ' s theorem. In [433] (also see [435]) Popp studied the behav ior of the fundamental group of the complemen t of a curve on a smooth su r face as the cu rve v a r i e s within a family .

Quest ions connected with the local fundamental ~roup, i .e. , with the ca lcula t ions of ~t (U), where U is an open se t of the spec t rum of a local Hense l ian r ing, turned out to be v e r y in te res t ing . An impetus t o th e se quest ions was provided by Mumford [26], who computed the fundamental group of the boundary 8V of some "c-ne ighborhood" of a no rma l s ingular point P on a complex a lgebra ic su r face . In pa r t i cu la r , Mumford proved a c r i t e r i o n for s impl ic i ty (conjectured by Abhyankar) : point P is nonsingular if and only if ~r 1 (SV) = 0. An a lgebra ic analog of se t 8V (as was noted apparen t ly for the f i r s t t ime by Grothendieck; cf. [227], exp.XID is the schema X ' = S p e c A \ { m } ~ where A is the Hense t iza t ion (or completion) of the local r ing of a point P of an a r b i t r a r y normal a lgeb ra i c su r face and m is a m a x i m a l ideal of r ing A. Using A r t i n ' s theo- r e m [82] on the lifting of a two-d imens iona l s ingular i ty to c h a r a c t e r i s t i c ze ro and his r e s u l t s f rom [81], we can apply Mumford ' s topological computat ions to compute the group $1(X')(P ), namely , the fac tor group of $1 (X') by the normal d iv i sor genera ted by a Sylow p - subgroup (p = cha rk ) . Grothendieck and Mur re [249] p roved the topological f in i t e -generab i l i ty of the group ~t (X')(P). We note that Mumford ' s c r i t e r i o n of s i m - pl ici ty does not c a r r y ove r d i rec t ly to the case of a posi t ive c h a r a c t e r i s t i c p. Indeed, by v i r tue of Naguta ' s example [398] (also see [82]) there ex i s t an i r r e g u l a r r ing R and a r ad ica l m o r p h i s m f : Spec (R') - - Spec (R), where t%' is a r egu l a r r ing. Hence it follows that =~ (Spec (_R)\ {m})---=l {Spec(R')k (m})-~ 0 (the spaces Spec (R) and Spec (R') a r e homeomorphic ) . It is as ye t not known whether the condition that X ' be homeomorphic to the point s pec t rum of a r egu l a r r ing is a n e c e s s a r y one for the vanishing of ~r 1 (X').

We note S e r r e ' s r e s u l t [472] on the s impte--connectedness of a unira t ional va r i e ty and Gro thendieck ' s t heorem on the b i ra t ional invar iance of ":l (X) [467]. Using the theory of descent , Grothendieck es tab l i shed many impor tan t p rope r t i e s of vl~ in pa r t i cu la r , he cons t ruc ted the s t a r t of the exac t sequence of homotopy groups (the f i r s t six t e rms) for a f lat p rope r m o r p h i s m .

The computat ion of the fundamental gToup of an Abel tan v a r i e t y (actually, contained in Lung and Serve [332]) can be found also in [467]. Se r re [474] computed 7r I for an a r b i t r a r y a lgebra ic group.

We r e m a r k that the Galois theory of r ings (for example , see [151]) in commuta t ive a lgebra was developed in pa ra l l e l with the "geomet r i c " study of the fundamental group of schemata . Many r e s u l t s of this theory a re ea s i ly in t e rp re t ed and reduced to the co r respond ing r e s u l t s on the a lgebra ic fundamental group or on the gtale cohomologies of affine schemata . Takeuchi devoted his pape r [496] to the connect ion of these theor ies .

The definition of higher homotopic invar ian ts of schemata was p roposed independently by Lubkin [345] and by Ar t i n -Mazur [83, 88]. A detai led account of the l a t t e r theory is contained in [89]. Both these theo- r i e s a re v e r y close to each other and a r e based on the Verd i e r -Lubk in cons t ruc t ion of a functor f rom the ca tegory of local ly Noether ian schemata into the ca t egory of p ro -ob j ec t s of the homotopy ca t egory of s i m - pliclal se t s . Taking the composi t ion of this functor with the functor of "geometTiC rea l i za t ion , " Ar~in and

276

Mazur obtained a certain pro-CW-complex Xet canonically comparable to schema X and called the homotopy type of schema X. The Verdier-Cartier construction of hypercoverings (SGA4, exp.V, Appendix) permits us to generalize the preceding construction and to determine the homotopy type of an arbitrary Grothen- dieck topology. The generalization of homotopy theory to pro-objects allows us to determine the homotopy groups ~i(Xet). For example, ~r i (Xet) is an "extended" Grothendieck fundamental g-roup. Artin and Mazur introduced important notions of a profiuite completion K of a cell complex K and, by generalizing Riemann's existence theorem, proved that C/Xet = Xcl for a normal schema X over a field. Here Xcl is the usual CW- complex defined by the complex space X an. From this theorem it follows, in particular, that ~ (X,t)

/\ /\ ~-- -~(X~z), and for s imply connected X, ~.~ (X:t) "" ~(Xct), i > 1, where a profini te comple t ion of the group occu r s eve rywhe re on the r ight . Among the other r e s u l t s ~n [89] we note, for example , the following fact. If .::, z : ~ ~, C a re dis t inct i m m e r s i o n s of an a lgebra ica l ly c losed k into a field C, then Xcl = X~cl holds for the co r respond ing complex va r i e t i e s X t, X 2 obtained f rom the k - v a r i e t y X by a change of base . By vir tue of the wel l -known example of Se r r e [477], he re it is imposs ib le , in genera l , to r e m o v e the A.

w A m p l e S h e a v e s . P r o j e c t i v e I m b e d d i n g s

o f A b s t r a c t V a r i e t i e s

E v e r since Well int roduced the notion of an a b s t r a c t a lgebra ic var ie ty , the natura l quest ion was posed of the i r p ro jec t ive immers tb i l i t y . As Well h imse l f proved, the Jacoblan va r i e t y of a curve (its cons t ruc t ion was the or ig ina l purpose for the introduction of an a b s t r a c t var ie ty) is a lways pro jec t ive . L a t e r on,Chow proved that any homogeneous space is p ro jec t ive [156]. The f i r s t example of an incomplete a b s t r a c t v a r i e - ty not i m m e r s i b l e into a p ro jec t ive space was p resen ted by Nagata in 1956 [393], Next he cons t ruc ted an example of a comple te nonproject ive a lgebra ic su r face (with s ingular points) [394]. Final ly, in 1958 he cons t ruc ted and example of a nonsingular nonquasiproject ive su r face (which, m o r e o v e r , is even rat ional) [395]. La t e r on, examples of nonproject ive va r i e t i e s were cons t ruc ted by Hironaka [268].

In 1961 Moishezon [35] announced a c r i t e r i on of p ro jec t iv i ty of nonsingular a b s t r a c t va r i e t i e s . This c r i t e r i o n was next genera l ized to s ingular va r i e t i e s [36], and a detai led account of p rev ious ly obtained r e - sul ts appea red in 1964 [37].

As is well known, each b i ra t iona l mapping ~: X --- p n is ~ v e n by some l inear sys t em, i .e . , by an inver t ib le sheaf ~ on X and by a se t of i ts sec t ions s 1 . . . . . s k. The sheaf ~ is sa id to be v e r y ample if the comple te l inear s y s t em l-~l defines the c losed imbedding ? : X c . . P " ( n = d i m ~H0(X, ~ ) - - 1). The sheaf is said to be ample if some t en s o r power ~| of it is ve ry ample . The p ro jec t ive c r i t e r i on is in fact an a m p l e n e s s c r i t e r i o n for the sheaf. The f i r s t such c r i t e r ion , coinciding with the co r respond ing Moishezon c r i t e r ion , was ~ v e n for nonsingular pro jec t ive su r f aces by Nakai in 1960 [404]. In 1963 he succeeded in extending his own r e s u l t to a r b i t r a r y p ro jec t ive s chema ta [406]. At the p r e sen t t ime this c r i t e r i o n (genera l - ized to comple te s chem a t a by Kleiman [307]) is cal led the Nakai -Moishezon c r i t e r i o n and is s ta ted as follows. A sheaf ~ o n a comple te k - s c h e m a X of d imension n > 1 is ample if and only if (D, . . . . . D~.~, ~ ) > 0 . for any effect ive d iv i so r s D l . . . . . Dn_ t on X. He re the in te r sec t ion number is de te rmined with the aid of Snappe r ' s theorem (see w

A beautiful amp lenes s c r i t e r i o n is due to Seshardi (cf. [261]).

A n u m e r i c a l c r i t e r i on for a v e r y ample sheaf on a nonsingular su r face is due to Mumford [381]. Beautiful cha r ac t e r i z a t i ons of ve ry ample sheaves in the language of the g e o m e t r y of cones of amp le sheaves a re contained in K l e i m a n ' s p a p e r [309] and in M u m f o r d ' s book [381].

The following Kle iman ' s theorem (expressed as a consequence of ChevaUey"s conjecture) s e r v e s as an example of a "nonnumer ica l" p ro jec t iv i ty c r i t e r i on [309]. A nonsingular va r i e t y X is p ro jec t ive if and only if each finite se t of c losed points of X is contained in an open affine set . An example of a "nonnumer i ca l a m p l e n e s s c r i t e r i on" is G r a u e r t ' s c r i t e r i o n [341].

t f an effect ive d iv i sor D on schema X is ample (i.e., the sheaf ~ = ~ x (D) is ample) , then the comple - ment X \ D is affine. Gizatull in [12] and Goodman [206] independently proved that the conver se a lso is t rue in the case of su r faces all of whose s ingular points lie on D. Hence we obtain a new proof (also see [309]) of the old Za r i sk i theorem stat ing that an a lgebra ic sur face X is pro jec t ive if all its s ingular points lie in an affine piece (Zar i sk i had fur ther a s s u m e d the normal i ty of X). By vir tue of Ar t i n ' s r e su l t K771 each normal sur face over a finite field is p ro jec t ive . The Gizatul l in-Goodman theorem does not c a r r y over d i rec t ly to the case of va r i e t i e s of dimension g r e a t e r than two (for example , the Za r i sk i variet3r; of. [206]). A genera l iza t ion of this ",heorem is due to Goodman and Har t sho rne [206, 261].

277

Har t sho rne [258] defined an ample vec to r bundle (or local ly f ree sheaf) . One of the equivalent defini- t ions of an ample sheaf ~ cons i s t s of the r e q u i r e m e n t that the tautological inver t ible sheaf ~p(]) on P = P x ( ~ ) be ample . Var ious c h a r a c t e r i z a t i o n s and p r o p e r t i e s of an ample sheaf a r e proved in [258]. Fo r example , if c h a r k --- 0, then the t ensor power of an ample sheaf is ample . Bar ton [109] succeeded in proving this r e - sult in the case of a posi t ive c h a r a c t e r i s t i c .

tn [261] H a r t s h o r n e d i s cus sed var ious poss ib le genera l i za t ions of the notion of an ample d iv isor on a subvar ie ty W'c.X of l a r g e r codimension. These definit ions a r e c lose ly r e l a t ed to the quest ion of the cohomologica l d imension of the c o m p l e m e n t X \ W " and a l so to the a m p l e n e s s of the no rma l bundle to W. Gr i f - fiths [215] sugges ted that analogous and o ther poss ib le p r o p e r t i e s of a m p l e n e s s of a subvar i e ty be taken as a definition in the analyt ic case .

Block and Gie seke r [130] proved H a r t s h o r n e ' s con jec tu res in the z e r o - c h a r a c t e r i s t i c case : if a sheaf is ample , then its Chern c l a s s e s c~ (~) a r e numer i ca l l y posi t ive. The la t t e r s ignif ies that (ct (W).I~ > 0 for

any effect ive cycle Y of c o d l m e n s i o n n - i . Kleiman [312] (and Barenbaum [7], independently) has p roved this r e s u l t e a r l i e r for the case of su r f aces ove r an a r b i t r a r y field.

Chow's r e s u l t s on the p ro jec t iv i ty of homogeneous spaces were genera l i zed to a cons iderab le degree by Raynaud to the case of a r b i t r a r y group schema ta [449]. We note, for example , that even an Abel ian S- s chema for an a r b i t r a r y s chem a S p roves to be not n e c e s s a r i l y p ro jec t ive (we need to r equ i r e the n o rma l i - ty of S).

By genera l iz ing his own re su l t of [399], Nagata p roved a theorem which is ve ry useful for appl ica t ions : each S - s c h e m a of finite type can be imbedded as an open subvar i e ty into a p rope r S - schema .

w C o n s t r u c t i o n T e c h n i q u e s in A l g e b r a i c G e o m e t r y

1. Genera l Resu l t s and Technica l Tools . a) Represen tab le functors . In a s e r i e s of r e p o r t s [220-226] at the Bourbaki s e m i n a r in 1959/62 Grothendieck subjected to detai led ana lys i s , s ignif icant ly c la r i f ied , and genera l i zed ce r t a in c l a s s i ca l cons t ruc t ions in a lgeb ra i c g e o m e t r y (the Po inca re var ie ty , the Chow var ie ty , etc.) . The fundamental concept in t roduced for this purpose by Grothendieck was the concept of a r e p r e s e n t - able functor . We cons ider a c e r t a i n ca t egory % and we let @ be the ca tegory of con t r ava r i an t functors on

with value in the ca t egory of se t s . Gro thend ieck ' s fundamental idea cons i s t s in that the functor h: ~- ,@, defined as h(X): Y-~Hom~(Y, X) = X ( l 0 for any X ~ , e f fec ts the imbedding of ca t ego ry ~ onto the comple te subca tegory @. Objects f rom ~ of the fo rm h (X) a r e cal led r e p r e s e n t a b l e functors . F o r example , in appl i - cation to s chema theory , when ~=(Sch/S) , this r e m a r k of Grothendieck p e r m i t s us to identify the S - s c h e m a X with the functor "of a point with value in an S - s c h e m a " h (X): Y - - Horn s (Y, X) = X (Y). In this context Grothendieck succeeded in showing that a l l the known cons t ruc t ions in a lgebra ic g e o m e t r y a r e examples of r e p r e s e n t a b l e con t r ava r i an t functors on the subca tegory (Sch/k) cons is t ing of k - v a r i e t i e s . Fo r example , the c l a s s i ca l Po inca re cons t ruc t ion cons is t s of a s soc ia t ing with each nonsingular va r i e ty X over an a l - geb ra i ca l ly c losed field k the fac tor group Pic ~ (X) of d iv i so r s a lgebra ica l ly equivalent to zero by the subgroup of pr incipal d iv isors . The p rob lem of the exis tence of the Po inca re va r i e ty is the following. Does there ex i s t for a given va r i e ty X a un ive r sa l va r i e t y ~(X) p a r a m e t r i z i n g the group P ic ~ (X)? The l a t t e r s ignif ies that there ex is t s a ~universal c l ass of d iv i sors R @ on X • such that for any D E Pic~ (X) we can find a unique point z~(X), for which D = np 1 (X • {z} .@). In the case of an a f f i rma t ive answer to this quest ion the group Pic ~ (X) is in an e s sen t i a l way provided with the s t ruc tu re of an a lgeb ra i c va r i e ty (the e l emen t s of Pic ~ (X) a r e found in bi ject ive co r respondence with the points of va r i e ty ~(X)). Grothendieck r e m a r k e d that under the exis tence condition the va r i e ty ~ (X) r e p r e s e n t s the following functor on the ca t s - go ry of k -var ie t i es : Y ~ Pic ~ (XX Y) (here the d iv isor @ c o r r e s p o n d s to the identity m o r p h i s m ~(X)-~(X)~.

The exis tence prob lem in a lgebra ic g e o m e t r y now becomes the problem of the r ep re sen t ab i i i t y of a con t rava r t an t functor on the ca tegory of schemata .

The Language of r e p r e s e n t a b l e functors turned out to be ve ry convenient for defining many c l a s s i ca l objects in a lgebra ic g e o m e t r y within the f r amework of s chemata theory (the G r a s s m a n n schema , vec tor bundles, the flag schema, etc. [248]).

The f i r s t profound theo rems on r ep resen tab i l i t y belong to Grothendieck. In 1964 Murre gave a c r i - ter ion for the r ep re sen tab i l i t y of an Abelian functor (i .e. , with value in Abel ian groups) and appl ied it to the cons t ruc t ion of the Po inca re schema [389]. The c r i t e r i on was genera l ized by Mat sumura and Oort [3B8] to a r b i t r a r y group functors (not n e c e s s a r i l y Abetian). M u r r e ' s r e p o r t [390] was devoted to one c r i t e r ion of Grothendieck. P a p e r [491] a lso r e l a t e s to genera l r ep re sen tab i l i t y c r i t e r i a .

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b) P r o r e p r e s e n t a b l e functors . An impor tan t n e c e s s a r y condition for the r ep re sen t ab i l i t y of a functor is i ts p ro rep re sen t ab i l i t y . The la t te r concept was introduced by Grothendieck in [221]. We consider the ca t egory of p ro - ~ - p r o - o b j e c t s o f a c a t e g o r y ~, i .e. , a r b i t r a r y p ro jec t ive s y s t e m s (X~)~EI of objects of ca te - gory ~ (the m o r p h i s m s of p ro jec t ive s y s t e m s a re defined in the natural way). Each func~or P :~-~(Ens) is continued by a natural method up to the functor /~: pro-~-~(Ens). To do this we need to se t /~[(X~)~EI)=IimP'(X~).

A functor F is said to be p r o r e p r e s e n t a b t e if F is r ep r e sen t ab l e , in other words , if there ex i s t s a p r o - o b j e c t (Pi)~s such that F (X) = tim Horn (X, P;) for any X6"~.

Grothendieck fo rmula ted impor tan t c r i t e r i a for the p r o r e p r e s e n t a b i l i t y of a functor F on ca tegory with f iber bundles and with a final object @. The following conditions a r e n e c e s s a r y conditions for p r o r e p - resentability: 1) F (~)) = {one point}, 2) (left-exactness) P(XX V)=F(PO X F(Y) for any objects X, Y, Z of

Z Z~F)

category ~. Grothendieck showed that in many important cases these conditions are also sufficient. Groth- endieck's program of research in this area was carried out by Levelt [339].

Particularly simple is the situation when category ~ is Noetherian or Artinian (i.e., the set of pro- objects of each object of category ~ possesses the ascending or descending chain condition) while functor F is Abelian. In this case Gabriel [193] proved that a contravariant functor on an Abelian Noetherian category defines a prorepresentable functor on the corresponding dual Artinian category. The application of this fact to the theory of local duality is contained in Grothendieck's seminar (227] (also compare [48, 234]).

The most important application of the theory of prorepresentable functors relates to the case when category ~ is the category C A of Artinian A-algebras, where A is a Noetherian local algebra over its own residue field k.

Schlessinger [SS] proved that the functor F: C A -- (Eus) is prorepresentable if and only if it is left- exact and dimkF (k is]) < ~, where k is] is the algebra of dual numbers. Besides this, he ==ave a method for verifying the left-exactness of functor F. In this case the prorepresenting object is identified with some complete A-algebra @; by the same token, F(A)=Hom~(A,d~) for any A E C A. If, furthermore, the functor F is formally smooth, i.e., the map F (A') -- F (A) is surjective for each surjection A'- A in CA, then algebra @ is isomorphic with A[[tl ..... tn]], n = dimkF (k [~]).

The application of the preceding situation to contravariant functors on the category of schemata is based on the following reasoning. Let G be a contravariant functor of the category of preschemata over Spec (A) and let e E G (Spec (k)). For any ring A E C A we denote by F (A)cO (Spec (A)) the set of ~ EG(Spec(A)) such that G(i)~ = e, where i is an imbedding of 5pec (k). Artin [86] calls functor G prorepresentable in e if the corresponding functor F is prorepresentable by some A-algebra W. If G is a representable functor and X is the schema representing it, then e defines a rational point x E X and functor F is prorepresentable by the ring ~x. =W.

Functor G is said to be effectively prorepresentable in e if there exists an element z E G (Spec (W)) which induces a compatible system of elements z~6g (Spec (W/m~)) with z 0 = e. Artin's theorem on the algebraization of formal moduli [86] asserts that in this case there exist a k-schema X of finite type, a closed point x EX, a k-isomorphism k(x) =~ k, and an element z E G(X) inducing e E G(Spec (k)) and such that the ring ~x.~.prorepresents the corresponding functor F (it is assumed, furthermore, that k-algebra A has a finite type). In many cases effective prorepresentability follows from prorepresentability and Grothendieck's existence theorem. {The application of this theory to formal moduli is given in the next section.)

Levelt~s papers [337, 338] are devoted to the connection between the prorepresentability criteria of Schlessinger and of Grothendieck and also to their generalization.

e) Grothendieck~s theory of descenz [201, 220, 467]. One of the first problems of the theory of descent was the following one, stated clearly by Well in [506]. We are given an algebraic variety V over a finite normal extension k' of field k. Can we ~lower n V to k (i.e., do there exist a k-variety W and a k'-isomorphism W| ~ = V)~ Weil gave an affirmative answer to this question for the case when V is quasiprojective

and satisfies natural necessary conditions for such a descent: there exist canonical k'-isomorphisms her: V- V ~ satisfying the ~pasting-together condition ~ h<r ~. = ~) -h~.. Here ~, r ~ Gal (k'/k), while V ~ is the image of V re la t ive to the act ion of o- on the coeff ic ients of the equations defining V. Ca r t i e r [148] examined ~ e analogous p rob lem for an inseparab le extension. Grothendieck noted that one and the same common- ca tegory s i tuat ion l ies at the base of the Weil and ~ e Ca r r i e r p rob lems .

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Let J~be a f iber category, ove r category. ~, i .e. , a ca te~ory K s is a s soc i a t ed with each object SEc~ and a functor ~*:~'s-->,.~'s, is a s soc ia t ed with each m o r p h i s m ~: S' - - S into c~. Fo r any such morph i sm and for an object Xff~" s. the "giving of a descent" onto X cons i s t s in the exis tence of an i s o m o r p h i s m u: p~ (X) "~

Wt P~ p: (X) (where S ' x S ' ~ S ' a r e m o r p h i s m s of a project ion) sa t i s fy ing the pas t ing- together condition p~2(u) �9

S Ps

P i J

p~ (u) = Pl3* (u) (where S" x s S ' x s S'p,~S' • S ' a re m o r p h i s m s of project ions) , The giving of a descen t is said to Pt*

be effect ive if there ex i s t s an object YEars such that ~* (Y) ~- X. Here the functor ~*:..~s~:~s, r e a l i z e s the equivalence of ca tegory S s and subca tegory ~'-s, cons i s t ing of objects with the effect ive ~ v i n g of a descent . The p rob lem of the theory of descen t cons i s t s in finding the conditions on m o r p h i s m f in o r d e r that a n y e o n - dition of descent on the object X62r s, be effect ive .

The appl icat ion of this c o m m o n - c a t e g o r y p rob lem to a lgebra ic g e o m e t r y r e f e r s to the ease when ce is the ca tegory of schemata . The m o r p h i s m f : S' - - S of s chemata is cal led a m o r p h i s m of effect ive descen t re la t ive to the f iber ca t egory J if any ~ v i n g of a descent on object XF.~ s, is ef fect ive . ]By genera l iz ing the r e su l t s of Weft and C a r t i e r , Grothendieck proved that an absolu te ly flat quas icompac t m o r p h i s m is a m o r - phism of effect ive descen t for the ca t ego r i e s of quas icoheren t sheafs , of affine or quasiaff ine S - schema ta , of fitale S - s c h e m a t a of finite type, and of some o the r s . The s ame is t rue for a finite su r j ec t ive m o r p h i s m and for the ca t egor i e s of unrami f i ed cover ings , of quas ip ro jec t ive s chema ta (if, m o r e o v e r , the m o r p h i s m is local ly free) , or of all s chema ta (if the m o r p h i s m is radica l ) [467].

Impor tan t c r i t e r i a for the descent of group schema ta were obtained by Rayuaud [449]. In the p r e sen ce of an effect ive descent we can cons ider the addit ional p rob lem: it is r equ i r ed to p r e s e r v e the p r o p e r t i e s of the lowered object (for example , the local f reedom of a lowered sheaf, the f l a m e s s of a lowered schema , etc.) . These p rob l em s were solved by Grothendieck in [467].

The ideas of the theory of descen t (pa r t i cu la r ly the l a s t problem) proved to have a g r e a t influence on commuta t ive a lgebra . Among the m a s s of paper s devoted to these p r o b l e m s we note [455]. Among the appl icat ions of the theory of descen t in a lgeb ra i c g e o m e t r y we note the following:

1) The r e p re s en t ab i l i t y of functors was proved f i r s t fo r the ca t egory (Sch/S), and next, by applying the theory of descent , the r ep r e s en t ab i l i t y of a functor on the ca t egory (Sch/S) was obtained. Here S t - - S is the m o r p h i s m of effect ive descent (see [223-225, 295]).

2) Cr i t e r i a for the ra t iona l i ty of d iv i sors (Ca, r t i e r [148], Oort [420]) and their genera l i za t ions , namely , ~-I i lber t ' s theorem 90" in ~tale cohomologies (SGA4).

3) P r o o f of the topological invar iance of the fitale topology (i.e., the ca t egor i e s Xet and (Xred)et a re equivalent).

4) Descent p r o p e r t i e s for var ious c l a s s e s of s chema m o r p h i s m [247, 467].

d) Equivalence re la t ions on schemata . The p rob lem of the theory of descent is a spec ia l ease of the genera l p rob lem of exis tence of a f ac to r of a schema by some equivalence re la t ion. The la t te r p rob lem is one of the m o s t impor tan t technical tools for proving the r ep r e sen t ab i l i t y of var ious functors .

In [222] Grothendieck defined an equivalence re la t ion on an object X of an a r b i t r a r y ca tegory * with f iber bundles as a ce r t a in subfunctor R c A ( X ) x h ( X ) such that R ( T ) c X ( T ) X X ( T ) is the graph of the equiv-

Pt alence re la t ion on the se t X (T). If R ~ X is the r e s t r i c t i o n of p ro jec t ion m o r p h i s m s to R, then the fac tor

X/R by the equivalence re la t ion R is cal led the eokernel (Pl, P2), i .e. , the object r e p r e s e n t i n g the covar ian t functor Z - - {~ 6 X(Z)I r = *P2}. The factor X/R = Y is said to be effect ive if the canonic m o n o m o r p h i s m /~ ~ X )< X is an i somorph i sm.

Y

A typical example of an equivalence re la t ion is the case when there ac ts on object X a group object G of category ~ (i.e., h (G) is a presheaf of groups on ~). By definition, the action of G on X is given by the morphism G x X -- X, and the image of the morphism G x X -" X x X ((g, x) -* (gx, x)) defines an equivalence relation on object X, the factor by which is denoted by X/G.

As is well known (for example, see [268]), the factor X/G does not always exist for the category, of schemata. Grothendieck proved the existence of factors X/R in ~e following cases: !) the projection

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pl: 1% - - X is a finite morph i sm, and each equivalence c l a s s re la t ive to R is contained in an open affine set; 2) X is quas ipro jec t ive over S, and Pl is a p rope r m o r p h i s m ; 3) X is quas ipro jec t ive over S, and R is c losed in XXX [169, 222].

S

Raynaud [451] proved the existence of an open dense set UcX for which U/R exists. Furthermore, if R is closed in XX2, then U contains points of codimension<-l. Some other criteria for the represent-

s ability of factor X/R by a flat equivalence relation (i.e., the projections Pl, P2: R --- X are flat) also were obtained by Raynaud [450, 452]. He has also obtained important applications of these results to the representability of the Neron-Severi schema, of the Poincare schema, and of the Neron model [451].

Corollaries of Grothendieck's results in the existence problem of the factor of a group schema by some subgroup are discussed in [168, 169]. See further on for applications to the Poincare schema. In [380] Mumford obtained profound theorems on the existence of the factor X/G of schema X by a reductive ~oup G.

A more general situation (equivalence prerelations) was studied with the aid of the formalism of groupoids in [168, 169, 450].

e) Artin-Moishezon algebraic spaces. The absence in the general case of a factor of a schema by an equivalence relation forces us to seek a natural extension of the category of S-schemata (Sch/S) in which this factor always exists. The obvious solution is to immerse (Sch/S) into the category of locally ringed spaces. However, this category is too broad to obtain any "reasonable" results in it. If S = Spec (C), then it is natural to seek the factor X/R in the category of complex spaces. Hironaka's example in [268] showed that such a factor can exist in this category but does not possess an algebraic structure (or even a Klihler- ian one). In 1965 Matsusaka developed the general theory of Q-varieties in Well's language [360]. In this category the factors of algebraic varieties by an algebraic equivalence exist ~cry definition." Another approach is due to Grothendieck and has been applied particularly successfully in the theory of group schemata. Here the category (Sch/S) is identified with the category of representable sheaves on the topology S f~l c. Fundamental to such an identification is the theory of descent for absolutely flat quasicompact morphisms.

The concept of an algebraic space, proposed by Artin in 1968 ("~tale schemata" or "varieties" in the original terminology) and, from other considerations, independently by Moishezon ("minischemata" in his terminology), connected up in a natural way the points of view of Matsusaka and Grothendieck. By definition, an algebraic space over a schema S is the name given to a sheaf on Set, being a factor of some S-schema U of locally finite type by an equivalence relation R whose projections R-~U are ~tale morphisms. We note that the factor is considered in the category of sheaves on Set.

When S = Spec (C), the algebraic space over S possesses in natural fashion the stracture of Moishe- zon's analytic space, and conversely, each such space is an algebraic space [40, 87].

The i m m e r s i o n of the ca t egory of S - s c h e m a t a into the ca t egory of a lgebra ic S - s p a c e s p e r m i t s us to examine the quest ion on the r ep r e s en t ab i l i t y of a functor in two s tages : f i r s t , to invest igate the question of i ts r ep r e sen t ab i l i t y by an a lgebra ic space, and next, to prove the r ep re sen t ab i l i t y of the l a t t e r by some schema . We r e m a r k that a s a t i s f a c t o r y answer to the f i r s t quest ion a l r eady yie lds a quite good solution to the p rob lem being cons idered , siz/ce the var ious a spec t s and the na tu ra lness of the concept of an a lgebra ic space, invest igated by Art in , Moishezon, Kautson [38-40, 86, 87, 314], allow us to cons ider the la t ter as a comple te ly reasonab le genera l i za t ion of an a lgebra ic var ie ty . The genera l p rope r t i e s and techniques of the theory of s chemata were t r a n s f e r r e d to a lgebra ic spaces by Knutson [314].

In [86] Ar t in proved a r ep r e s en t ab i l i t y c r i t e r i o n for a con t r ava r i an t functor by an a lgebra ic space. One of them is its effect ive p ro rep re sen tab i l i t y . The proof of the r ep re sen tab i l i t y c r i t e r i on is based on the following ubiquitous Ar t in theorem on approx imat ions [85]. Le t R be a field o r an excel lent d i sc re te valua- t ion r ing, A be the Hensel iza t ion of some E - a l g e b r a of finite type with r e s p e c t to a p r ime ideal, and m be a max ima l ideal of A. Let y = (Yl . . . . . YN) be a solut ion of the sy s t em of a lgebra ic equations f i (Y) = 0 with coeff ic ients in A, whose coordi lmtes belong to the n~-adic complet ion of A. Then for any in teger c > 0 a solution y = (Yi . . . . . YN) E A N ex is t s such that y~ - - ~ (r~odr~).

A spec ia l case of A r t i n ' s theorem (dimA = 1) was obtained e a r l i e r by Greenbe rg [213, 214] and E r s h o v [21].

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The close connection between Matsusaka's Q-varieties and algebraic spaces is reflected in the following result of Artin [84]. Let R-~U X U be a fiat equivalence relation and X be the factor U/R in the category of sheaves Sfppf, Then X represenSts an algebraic space. Note, however, that the class of algebraic spaces over C is already a class of Q-varieties since the latter, in general, is not representable by an analytic space (Holmann [276]).

By virtue of Artints result in [84] the giving of a descent for algebraic spaces relative to an absolutely flat morphism is always effective. In the category of algebraic S-spaces there always exist factors of group spaces by an arbi t rary flat S-subgroup.

Papers [5, 84, 454] are devoted to a survey of Artints results. We leave aside the questions connected with the birational geometry of algebraic spaces and with the theory of singularities [39, 40, 87].

2. Ptcard Schema. One of the f irst problems to which Grothendieck applied the techniques presented above was the problem of constructing the Picard schema. The latter was introduced by Grothendieck in [224] as a natural generalization within the framework of schema theory of the Picard variety ~(X) of a nonsingnlar algebraic variety X. The f irst algebraic construction of $ (X) as an Abelian variety parame- trizing the factor group X of divisors algebraically equivalent to zero by the principal divisor was given in 1951 by Matsusaka [359] and was next generalized to the case of a normal variety by Chow and Lang.

In Chevalley's 1958 seminar [466] the Picard variety ~c(X) corresponding to Cart ier divisors was studied. It was shown that for singular varieties ~r (X) possesses (in contrast to ~ (JO) good functorial properties [483]. Chevalley [155] constructed the Picard variety Co(X) for complete normal varieties X. This construction was generalized to arbi t rary projective varieties by Seslmdri [481, 482].

To determine the Picard schema for an arbi t rary S-schema X Grothendieck suggested the examina- tion of the relative Picard functor P icx/$ on the category (Sch/S). The value of this functor on the S- schema S is the group I-I 0 (S', R~pqcJ~. (GIn,X,)), where f':X' ~X>(S'~S' is the change-of-base morphism

S r

while R~pqc~ , (Gm~,) is the sheaf on the Grothendieck topology Sflxl c associated with the presheaf T -- FI t

(Tflxl c, Gin) = H t (Tet, Gin). Let Pic (T) be the group of classes of invertible sheaves on schema T. When

functor Picx/S is representable on (Sch/S), the S-schema representing it is called the relative Picard schema of the S-schema X and is denoted by Picx/S. For example, ff X is an algebraic k-scheme having a rational k-point, then Grothendieck shows that Picx/~ (S')~ Pic (X ~< S')/Pic (S') for any k-schema S' and, in

particular, PIeX/k (k) = Pic (X) is identified with the group of k-rational points Picx/, k (k) of the schema Ptcx/k .

The f irst theorem on the representabtlity of the functor PiCx/S is due to Grothendieckandrelates tothe case when f : X -- S is a projective morphism with geometrically integral fibers. In this case the schema Picx/S is locally finitely presented separable group S-schema. If S = Spec (k), where k is a field, then the connected component of unity Pic~/S of the schema PlCX[ k is an algebraic k-schema,and the corresponding reduced k-schema (Pic~/k)re d is precisely the Picard variety ~c (JO in Chevalley's definition. The presence of nilpotsnt elements in the local rings of schema Pio2x/k yields much additional information on the Picard variety and permits us to explain various pathologdesof algebraic varieties over a field of characteristic p > 0. We remark that by virtue of Cart ier 's theorem (cf. [169, 381, 422]) the schema Ptc~/k is always reduced if p = 0, Igusa has cited an example of a nonsingular algebraic surface F with a one-dimen- siotml Picard variety and with dimtH ~ (F, a~)-----2 [286]. Noting that the space H ~ (F, ~p) is identified with a tangent space to the unity of the Picard schema Pic~/k, Grothendieck explains this fact in [225] by the irreducibility of the corresponding schema. An analogous explanation can be ~ven for the old Igusa result dirn~H Z (X, a z ) > q, where q is the irregularity of variety X, coinciding with the dimension of its Picard variety ~(X). Mumford [381], using Setters techniques from [479] of Bockstein operations on cohomotogies of sheaves of Witt rings, investigated the reducibility of the Picard schema of a smooth algebraic surface F. In particular, he proved Grothendieek*s theorem [225] that if H~(F, #z)-----0, then the schema PicF/k is reduced. If base S is a nonsingular curve while the fibers of the morphism f : X -- S are curves, the connected component of the unity Pic~~ of schema PICK/S is an analog of Igusa's "family of Jacobian varieties ~ [228] (cf. [447]).

In [389] Murre proved the representability of the Plcard functor for an arbitrary proper k-schema X. In [418, 419] Oort investigatss the connection between ~e schemata PiCx/k and PlCXred/k. This connec-

282

tiou was investigated in particular detail for multiple algebraic curves [417]. Murre's theorem was generalized in [491] to the case of a proper schema over a local Artin ring.

At the Stockholm congTess in 1962 Mumford announced a theorem on the representabilit y of the Pi- card functor, generalizing somewhat the theorem of Grothendieck (the fibers of )e are not necessarily irreducible; however, all irreducible components are geometrically irreducible). The proof of this theorem was based on the methods developed in [380]. A specialization of this theorem for the case of a nonsingular algebraic projective surface was presented in [381]. Already in the case treated by Mumford the schema PicX/S is not necessarily separable over S.

An essential improvement in the question of the representability of the Picard functor was Artin's theorem in [86] stating that for any proper flat morphism/: X -- S (finitely presented if S is non-Noetherian) for which/~ (@x') =@s" for any change of base /' : X" = X X S' ~ S', the functor Plcx/S is an algebraic space

S

over S. Note that the condition /: (&x,) == #s. (cohomological flatness) is ahvays fulfilled if the fibers of morphism f are reduced [242]. Artin's theorem reduces the question of the representability of the Picard func- for by a schema to the question of the representability of a group algebraic space by a schema. Artin him- self answered this question affirmatively for the case when the base schema is the spectrum of a local Artin ring, by the same token obtaining anew the result of ~[urre [3891 and of Sivaramakhrishna [491]. Ray- naud [447] generalized this result to the case when S is a normal locally Noetherian schema of dimension 1 and the group algebraic space G is smooth over S (the latter condition canbe excludedbyvirtue of Avantara- man's result). Furthermore, Rayrmud obtained nontrivial sufficient conditions for the cohomological flatness of the morphism/: X -'- S. A detailed investigation was carried out in [447] for the case when the fibers of f

0 are algebraic curves; in particular, the connection between the schema Picx/S and the Neron model [414, 444] of the Jacobian variety of the common fiber of f was investigated. Rayrmud's results were announced in [445, 446].

In [295] Iversen, using Raynaud's results on the descent of group schemata [449], generalized the classical Well construction of the Jacobian variety of a curve and constructed the Picard schema for a proper flat morphism with geometrically integral one-dimensional fibers.

Important theorems on the finiteness of the Picard schema, due to Grothendieck, are discussed in [225]. Nfurre's report [390] gives an account of Grothendieck's results on local representability.

In 1961 Mumford [377] defined the P i c a r d group of a normal singular point on a complex a lgebraic surface and posed the quest ion of introducing the s t ruc tu re of an a lgebraic group on it. In [227] Grothen- dieck sharpened this definition (by examining a r b i t r a r y local Hensel rings) and gave an af f i rmat ive answer to Murnford's question.

3. Hi lber t Schema and Other Construct ions. The Hi lber t schema was introduced by Grothendieck in [223] as a natural general izat ion within the f ramework of schemata theory of the c lass ica l Chow var ie ty of a lgebraic cycles . F o r any coheren t sheaf ; on an S-schema X, Grothendieck defined the functor Quot~r~x,s: (Sch/S)~(Ens#. The value of this functor on an a r b i t r a r y S-schema Z is the set Quot~-lxls(Z) of c lasses by i somorphism of the fac tor sheaves of the sheaf S'z=;| z on XXsZ which a re flat on Z, finitely presented ,

@s and have a support which is p roper over Z. For the case ~ '=@x the functor Quot@x:ms is denoted by H i lbx / s and is called the Hilbertufunctor assoc ia ted with the S-schema X. For any polynomial P 6Q ix] there is defined a subfunctor Hilb~r/S.~. whose value on an S-schema Z is a set of closed subschemata TcXXsZ, flat and p roper over Z, such that for any point z 6 Z the f iber T2=T~z/~(z ) has P as the Hi lber t

polynomial. The functor Hi lbx /S is r ep resen tab le if and only if the functors HilbP/s.. a r e represen tab le , and in this case, for the represen t ing object (the Hi lber t schema) we have Hllbx/s= 11 HllbxP/s, where HtlbxP/s

P

is the r ep resen t ing schema for Hi lh~/S.

Grothendieck proved~n [2231 that w h e n / : X - - S is a quas_.iprojective morphism with a Noetherian schema S, the functor Hilb~/S is r epresen tab le by schema H i l t ~ / S of finite type and quasiproject ive o v e r

S. More g, enera l ly , in this case the functor Quot~r~x~a is r epreseh tab le by a quasiproject ive schema of locally finite type over S. When the m o r p h i s m / : X - - S is not quasiproject ive, the schema Hi lbx /S does not exisz, in general . However, by vir tue of Ar t in ' s r e su l t [86] the funetor Quot~-~x,s is r epresen tab le by an algebraic space locally finitely presented over S for any schema X locally finitely presenmd over S. By the same token the question of the representability of the functor Quot~ixls is fully answered in the category of algebraic spaces.

283

Y ) The construction of the schema H i l l ~ S ~ , ~ for the case when X is a s m o o t h algebraic s u r f a c e over field

k and P is a polynomial of degree 1 is contained in Mumford's book [381]. In this case the functor Hilb~/S~ coincides with the functor Div~x/k, also studied by Grothendieck [223] for an a rb i t ra ry S-schema X. Fur- thermore, Mumford showed the connection between the Hilbert schema and the classical constructions of Chow and van der Waerden of the Chow variety (also see [187, 380]).

Fogarty [188] investigated the connection between the Hilbert schema of an algebraic surface and its symmetric product.

In [257] tIartshorne proved the connectedness (and even the linear connectedness)of the Hilbert schema Hilb~/ , generalizing the classical asser t ion on the connectedness of "the variety of modules of space

curves." An algebraic proof of this resul t (in the language of ideals) is due to Car t ier [150]. The I-Iilbert schemata Hllbp~j~ play an important role in Mumford's construction of the variety of modules of space

curves [380]. Mumford [378] showed that the tIilbert schema of nonsingular curves of degree 14 and genus 24 in a three-dimensional projective space over the complex number field has nilpotent elements in local rings. The presence of the latter explains certain pathologies in classical algebraic geometry.

In [221] Grothendieck investigated the representabil i ty of other important functors in algebraic geometry. His approach to the study of the automorphism group of an algebraic variety proved to be part icular- ly interesting. Grothendieck defined the automorphism schema Autx/S of an S-schema X as the represent - ing object for the functor Autx/s:S'-+ Auts, (XX S') from the category (Sch/S) into the category of groups.

s in particular, if S = Spec (k) is the spectrum of a field, then the group of rational k-points of the group schema Autx/k coincides with the k-automorphism group of the k-schema X. Grothendieck proved the existence of the schema Autx/S for any projective S-schema X. The structure of an algebraic group on the automorphism set of a projective algebraic variety had been introduced earI ier by Matsusaka and Matsu- mura (unpublished). In [358] Matsumura and Oort generalized Grothendieck's resul t to an a rb i t ra ry proper S-schema X. The structure of an algebraic group on the automorphism set of a complete variety was introduced ear l i e r by Ramanujam [437]. The schema Autx/S is a group S-schema locally finitely presented and, in general, unreduced [358].

We remark that even in the classical case when X =Spec (K), S = Spec (k), where K/'k is the finite extension of a field, the schema AutK/k yields a new point of view on Galois theory. Namely, the algebraic group AutK/k plays the role of the "present-day s Galois group, while its group of k-rat ional points is precisely the classical Galois group. For example, if tC/k is purely inseparable, AUtK/k (k) = {e}, and dimAutK/k > 0 (!). This classical case was examined in detail in [110]. See [505], for example, for the applications of this point of view to the theory of automorphisms of Lie algebras.

Analogously to the functor Autx/S, Grothendieck defined and investigated the functors Horn S (X, X'), Isoms (X, X'), and others.

In [368, 369] Miyanishi investigated the representabfl i ty of the functor PH (G; X/S) associating with each S-schema S' the set of classes by isomorphism of principal homogeneous spaces on Xs. = X >< S" re l -

s ative to the group S-schema G. This functor was introduced by Grothendieck in [14] as a natural generalization of the Picard functor (the localization of the functor Pl{ (Gin; X/S) relative to the flxlC-topology is the Picard functor P icx / s ) . When S = Spec (k) and G is a linear commutative algebraic group, this functor is representable, and the k-schema representing it has a close relation with the Picard variety of divisors of type G, constructed and studied by Bertin [124-127]. Miyanishi proved the representabil i ty of the functor PH (G; X/S) for the case when the morphism is cohomologically fiat, the Picard ~nc tor P icx /S is representable by a sum of quasiprojective S-schemata, and G is a finite locally free group S-schema.

In the present survey we do not touch upon the very interesting questions connected with the r ep re - sentability of "global schemata of modules ~ [166, 192, 285, 379, 380].

w F o r m a l G e o m e t r 7

1. Formal Schemata. The definition and the foundations of the theory of such schemata are contained in [219, 240, 248]. Formal schemata play an important role in the infinitesimal study of schemata and also yield a convenient context for the theory of formal groups (cL [169]). in the affine case they correspond to the formal spectra of topological algebras, i.e., ~o the set of closed prime ideals equipped with the Zariskt

284

topology and with a sheaf of local r ings . Genera l ly f o r m a l schemata a r e obtained by pas t ing together affine f o r m a l schemata . The m o s t impor tan t spec ia l case (and, it s e e m s , as yet the only one usable in app l i ca - tions) cons is t s in the examinat ion of a commuta t ive r ing A with identity and of an I-adic topology on A. In this case the fo rm a l spec t rum S p / ( A ) is a r inged space whose suppor t is a c losed subse t Spec (A) defined by the ideal I, while the s t ruc tu re sheaf is the r e s t r i c t i o n of the sheaf (A), where ~ is the comple t ion of A in the i -ad ic topology. The m o s t impor tan t cons t ruc t ion leading to a f o r m a l s chema is the cons t ruc t ion of _the fo rma l comple t ion of the s c h e m a X along a c losed subschema Y. The la t te r is a f o rma l schema with a topological space Y and a s t ruc tu re sheaf e } = tim @x/I~, where I is the sheaf of ideals of subschema Y.

The schema V~=,(Y, ~x//"+9 is ca l led the n- th inf in i tes imal neighborhood of subschema Y, and X=l imF=.

The canonical m o r p h i s m of r inged spaces i: ~ - - X defines, for any coheren t sheaf ~ on X, i ts f o r m a l c o m - pletio n ~ = l i_m,~/I"§ = i* (5).

In Chap. IF/of his monumental treatise [242] Grothendieck developed the cohomology theory of formal schemaza. The fundamental results of this theory are the theorems of comparison and of existence for p r o p e r m o r p h i s m s . Let):: X - - S be a p rope r m o r p h i s m of Noether ian schemata , SocS be a c losed sub- s chema of schema S, X o ~ / - 1 (So) , ~ ( repsec t ive ly , S) be the fo rma l comple t ion o f X ~ e s l ~ c t i v e l y , S) along X 0 ( respec t ive ly , So), i: X - X and j: S - - S be cor responding canonic morph i sms , ) : : X - - S be the r e s t r i c - tion of f on X. Gro thendieck ' s c o m p a r i s o n theorem a s s e r t s that for any coheren t sheaf F on X the canonic

homomorphism of sheaves W A is an isomorphism. In particul=, if s S, then the comple t ion of the f iber (Rq):.(F)) s a t point s is i somorphic with lim/-/r F| For q = 0 this

7 theory should be cons ide red as a na tura l genera l iza t ion of Z a r i s k i ' s " theory of holomorphic func to rs" [513]. In pa r t i cu la r , Z a r i s k i ' s connectedness theorem follows eas i ly f rom it. Grothendieck a lso inves t iga ted in which case is the l imi t group i somorph ic with the group ~/q ( / - ' (s), ~'| The r e s u l t s obtained by h im a re v e r y profound and a r e connected with the quest ion of the cohomological f la tness of m o r p h i s m f . Gro- thendieck ' s ex is tence theo rem a s s e r t s , under the s a m e assumpt ions as in the c o m p a r i s o n t heo rem, that for any coheren t sheaf ; on schema X there ex i s t s a coheren t sheaf F on X for which p_--~r. The t h e o r e m s ob- mined a r e fo rma l ana logs of S e r r e ' s c o m p a r i s o n and a lgebra iza t ion theor ies f rom [478]. A su rvey of Gro- thendieck ' s r e su l t s a r e contained in his r e p o r t a t the Bourbaki s e m i n a r [219].

Grothendieck [227] gave a genera l i za t ion of the compar i son and exis tence theory to the case of not n e c e s s a r i l y p ro pe r m o r p h i s m s .

T h e o r e m s , impor tan t for appl icat ions , on the a lgebra izab i l i ty of a f o r m a l s chema X over a comple te local r ing A, whose suppor t is p ro jec t ive ove r the res idue field of A, a r e contained in [242].

In [261] H a r m h o r n e proved a f o rm a l duality theorem H ~ (X, ~ = Hom~ (H~ -~ (X, Hom~x (F, ~,x)), k), where is the cornpl.etion of a smooth p rope r schema X re la t ive to a c losed subschema Y, ~X is a canonic sheaf

on X, and H ~ -~ a r e local cohomologies r e l a t i ve to Y.

Har t sho rue [260, 261], Ma t sumura [273], and Hironaka [270, 273] studied the fo rma l complet ion of a subschema Y in a smooth p rope r k - s c h e m a X. In pa r t i cu la r , they invest igated the quest ion as to when the f ield of " m e r o m o r p h i c " functions KCXD (i.e. , global sec t ions of the sheaf of comple te r ings of pa r t i a l loca l r ings of schema X) coincides with the field of ra t ional functions K (X). This coincidence holds when Y is a comple te in te rsec t ion [260, 261] o r an effect ive C a r t i e r d iv isor with an ample no rma l sheaf [270] or , f inally, a subschema of a pro jec t ive space [261, 273].

In [261] Ha r t sho rne d i scovered an in te res t ing re la t ion of these quest ions with those on the cohomo- l o ~ c a l d imension of the complemen t X\Y and on the Lefsche tz -Gro thendieek theory. To him a lso a r e due the impor tan t t h e o r e m s on the finite d imensional i ty of the cohomotogies H i (X, F) [258, 260, 261, 264]. Hi ronaka invest igated the imbeddings of a f o rma l schema Z in a smooth k - s c h e m a T so that T ~" Z and that T would p o s s e s s the p r o p e r t i e s of un iversa l i ty re la t ive to ra t iona l mappings [270]. Here too an example was p resen ted of a nonalgebra izable fo rma l schema; another such example was given by Ha r t sho rne [261].

2. F o r m a l Deformat ions . The techniques of f o rma l geome t ry proved to be pa r t i cu la r ly useful in the theory of deformat ion of schemata , designed to genera l ize to the case of s chema ta the fundamental r e su l t s of the Koda i r a -Spence r theory of de format ions of complex va r i e t i e s . The bas ic quest ions of this theory a r e the following: a) (The lifting problem.) We a re given a k - s c h e m a X 0 of finite type and a ce r t a in m o r - phism u: Spec (k) ~ S whose image is a point so E S. Does there ex i s t a f lat S - s c h e m a X of finite type for which X 0 = Xs0? b) (The module problem.) Does there ex i s t a un iversa l fami ly ~Z'-~.W for al l t ift ings of

285

schemata X0? As a matter of fact the answers to the questions posed are in the negative in the general case. Let us assume tha~ the answer to question a) is in the affirmative for some schema X 0. In this case the formal completion X along fiber X 0 is a formal schema over Sper ~#s. ~,) whose topological space is isomorphic with schema X0, while for any integer n >- 0 the schema X| s/m" is the n-th infinitesimal neighbor-

s hood of schema X 0. Thus, we arrive at a "formal lifting" of schema X 0. Weakened forms of questions a) and b) are the following formal analogs of them: a') We are given a complete local ring A with a residue field k. Does there exist a formal schema X over A whose topological space is isomorphic with X0? b') Does there exist a universal family, i.e., a complete local ring # with residue field k and a formal schema ~r-- Spec t@) such that for any formal deformation X -- Spec (A) there exists the homomorphism #-- A of rings for which X = ~| A? We remark that question b') is a formal analog of Kuranishi's construction

[327] of a local variety of modules in analytic geometry.

Grothendieck reformulated question b') as a question on the prorepresentability of the following functor on the category of Artinian k-algebras with residue field k, F: A -- {set of A-schemata X of finite type with X| = Xo}. The prorepresenting object for this functor is the formal module schema ~Spec(~). Ap-

A

plying Schlessinger's [58] and Grothendieck's [219] criteria (see w we obtain an affirmative answer to question b') in the following case: dimkHt (X0, | ) = m < ~ and H ~ (X0, @~ ) = 0 (here @~ is the sheaf of

�9 0 2 z~0 ' z~ 0 germs of sections of the tangent bundle to X0). Moreover, if I-I CX0, ~0 ) -- 0 (for example, X 0 is a curve), then the ring r ~/~ [it, ..... t~]l. This result is precisely analogous to Kuranishi's result cited above. In the case when, in addition, schema X 0 is projective, Grothendieckts algebraization theorem mentioned in Para. 1 allows us to prove the effective prorepresentability of functor F, while Artinrs theorem yields the algebraization of the formal module schema for X 0 (see w Para. 1).

Grothendieck [467] investigated question a9 for a smooth schema X0. The formal lifting of schema X 0 to a formal A-schema X is effected in the form of a sequential construction of infinitesimal neighborhoods X n over S, = Spec (A/m ~) such that X~| ~ X~_~. In view of the smoothness of X 0 such a construction can

always be made locally. Grothendieck showed that an obstacle to the pasting together of such infinitesimal "or �9 , 2 i ne1~,hborhoods hes m the group H (X0, @X0). Moreover, the continuation of schema X0 uptoa formal schema

�9 �9 . ~__ �9 -- �9 o

XonsS2rCe (dAb);%r:2e~ l[f13H I. (Xn0,p@ laX0i)cul0r, We rff~araktivflveataC:mePle ~ qu~It~io c ~,~Ifogl;oOwfs ~S2G:e~e~d~::k,s

theory if H 2 (X0, | ) = 0 (for example, X 0 is an affine or smooth curve). If, furthermore, the schema is projective over k, ~en by virtue of Grothendieck's algebraization theorem we obtain an affirmative answer also to question a) for the schema S = Spec (A).

Particularly important applications of the preceding theory are the questions connected with the lifting of k-schema X 0 to characteristic zero. In this we are interested in question a),where schema S is a schema of characteristic zero (for example, S = Spec (W (k)) is the spectrum of the Witt vector ring over k). By virtue of Serre's example [475] the answer to this question is not always in the affirmative eve n for smooth projective surfaces. From Grothendieck's theory immediately follows an affirmative answer to this question for smooth projective curves. When X 0 is an Abelian polarized variety, this question was examined by Grothendieck and Mumford (cf. [383, 424]). For finite group schemata it was considered by Mumford and Oort [424, 425].

In 1968, with the aid of the concept of a cotangent complex of a morphism of schemata, Grothendieck succeeded in making a significant generalization of deformation theory of smooth curves to arbitrary relative schemata [235]. The theory of a cotangent complex and its application to deformation theory was con- siderably developed by Illusie [290, 292], which in its own turn gtobalized the local theory of Andree [72] and Quillen [436] (also see [341]).

1. 2.

3.

4.

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