Elena RubeiAlgebraic Geometry

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|Advances in GeometryGrundhfer/Strambach (Managing Editors)ISSN: 1615-715X, e-ISSN 1615-7168

Elena Rubei

AlgebraicGeometry

|A Concise Dictionary

Mathematics Subject Classication 201014-00, 14-01

AuthorProf. Dr. Elena RubeiUniversit degli Studi di FirenzeDipartimento di Matematica e Informatica U. DiniViale Morgagni 67/A50134 FirenzeItalyrubei@math.uni.it

ISBN 978-3-11-031622-3e-ISBN 978-3-11-031623-0Set-ISBN 978-3-11-031600-1

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Notation

We denote by the ane space of dimension over.() For any manifold , () denotes the vector space of the -forms on.,() For any almost complexmanifold (see Almost complexmanifolds, holomor-phic maps, holomorphic tangent bundles), ,() denotes the vector spaceof the (, )-forms on .(, ) For any topological space and ring , (, ) (Betti number) denotes therank of (, ); it is also denoted by (, ); see Singular homology andcohomology.

b.p.f. The abbreviation b.p.f. stands for base point free, see Bundles, bre -.() For any algebraic variety , () denotes the -th Chow group of ; seeEquivalence, algebraic, rational, linear -, Chow, NeronSeveri and Picardgroups.() For any algebraic variety , () denotes the divisor class group of ; seeDivisors.(, ) For any vector bundle on a manifold , (, ) denotes the set ofthe sections of ; see Bundles, bre -., , stands (as usual) for the Kronecker delta.() For any Cartier divisor , () denotes the line bundle associated to ; see Di-visors., , 1 See Pull-back and push-forward of cycles, Direct and inverse imagesheaves, Singular homology and cohomology.||, L() For any divisor, we denote by || the complete linear system associated to,see Linear systems; see also Linear systems for the denition of L(). For any line bundle on a variety , denotes the map associated to ; seeBundles, bre -. denotes a linear system on a Riemann surface of degree and projectivedimension ; see Linear systems.(, ), (,) For any vector space,(, ) denotes the Grassmannian of -planes in, seeGrassmannians; analogously, for any projective space ,(,) denotes theGrassmannian of projective -planes in .(, ), (, ) For any topological space and any ring,(, ) denotes the -th homologymodule ofwith coecients in (see Singular homology and cohomology);(, ) denotes its rank; (, ) is also denoted by (, ) (Betti number)(, ), (, ) For any topological space and any ring ,(, ) denotes the -th cohomol-ogy module of with coecients in (see Singular homology and cohomol-ogy); (, ) denotes its rank.

viii | Notation

(,E) For any sheaf of Abelian groups E on a topological space ,(,E) denotesthe -th cohomology group of E, see Sheaves; (,E) denotes its rank; if is a holomorphic vector bundle on a complex manifold or an algebraic vec-tor bundle on an algebraic variety , we sometimes write (, ) instead of(,O()).() For any topological space, () denotes the EulerPoincar characteristic of, i.e., the sum =1,...,(1)(,), when it is dened.(,E) For any sheaf of Abelian groups E on a topological space , (,E) denotesthe EulerPoincar characteristic of E, i.e.,=1,...,(1)(,E), when it is de-ned., For any complex manifold or smooth algebraic variety , denotes thecanonical bundle and the canonical sheaf, i.e.,O(); see Canonical bun-dle, canonical sheaf.F, F See Serre correspondence.( , ) We denote by( , ) the space of the matrices with entries in .

nef The abbreviation nef stands for numerically eective; see Bundles, bre -.() For any algebraic variety,() denotes the -th NeronSeveri group of;see Equivalence, algebraic, rational, linear -, Chow, NeronSeveri and Picardgroups.

O, O() If is a complex manifold, O (or simply O) denotes the sheaf of holomor-phic functions; if is an algebraic variety, it denotes the sheaf of the regularfunctions; more generally, it denotes the structure sheaf of a ringed space; seeSpace, ringed -; for the denition of O(), see Hyperplane bundles, twist-ing sheaves.

O(), () Let be an algebraic vector bundle on an algebraic variety or a holomorphicvector bundle on a complex manifold; then O() denotes the sheaf of the(regular, resp. holomorphic) sections of ; see Bundles, bre -. We denote O() by (). For any complexmanifold, denotes the sheaf of the holomorphic -forms;for any algebraic variety, denotes the sheaf of the regular -forms; seeZariski tangent space, dierential forms, tangent bundle, normal bundle. We denote by the projective space of dimension over.(), (), () The symbols (), () and () denote respectively the arithmetic genus,the geometric genus and the -th plurigenus of (for variety or manifold);see Genus, arithmetic, geometric, real, virtual -, Plurigenera.() For any algebraic variety , () denotes the Picard group of ; see Equiv-alence, algebraic, rational, linear -, Chow, NeronSeveri and Picard groups.

PID PID stands for principal ideal domain, i.e., for an integral domain such thatevery ideal is principal.(, ) For any topological space and any , (, ) denotes the -th funda-mental group of at the basepoint ; see Fundamental group.

Notation | ix

() For any complex manifold or algebraic variety , () denotes the irregularityof; see Irregularity., See Direct and inverse image sheaves() For any ideal , () denotes the saturation of ; see Saturation. For any , we denote by the group of the permutations on elements.(), () For any ring , () denotes its spectrum, see Schemes. See Schemesalso for the denition of () for graded ring. The symbol denotes the bred product of and over ; see Fibredproduct. The symbol denotes the intersection of cycles; see Intersection of cycles.Sometimes it is omitted. For any ideal in a ring , we denote by the radical of , i.e., = { | {0} s.t. }., + , ( : ) Let be a commutative ring and and be two ideals in . We dene tobe the ideal {=1,..., | , , } .Moreover, we dene+ ={ + | }, and ( : ) := { | }. For any vector space , we denote by its dual space. denotes the disjoint union. : (, ) (, ) If and , the notation : (, ) (, ) stands for a map : such that () .

Note. The end of the denitions, theorems, and propositions is indicated by the symbol .

AAbelian varieties. See Tori, complex - and Abelian varieties.

Adjunction formula. ([72], [93], [107], [129], [140]). Let be a complex manifoldor a smooth algebraic variety and let be a submanifold, respectively a smooth closedsubvariety. We have = | ,,where , is the normal bundle and and are the canonical bundles respec-tively of and (see Canonical bundle, canonical sheaf, Zariski tangent space,dierential forms, tangent bundle, normal bundle).If, in addition, is a hypersurface, the formula becomes = | ()|,where () is the bundle associated to the divisor , since, in this case the bundle,is the bundle given by (see Bundles, bre - for the denition of bundles associatedto divisors).

Albanese varieties. ([93], [163], [166]). The Albanese variety is a generalizationof the Jacobian of a compact Riemann surface (see Jacobians of compact Riemannsurfaces) for manifolds of higher dimension.Let be a compact Khler manifold of dimension (see Hermitian and Khlerianmetrics and Hodge theory). The Albanese variety of is the complex torus (seeTori, complex - and Abelian varieties)() := 0(, 1)(1(,)) ,where : 1(,) 0(, 1) is dened by () = for any 1(,).The Albanese map : ()is dened in the following way: we x a point 0 of (base point) and we dene() = 0for any , where0 is the integral along a path joining 0 and (thus, obviously, itdenes a linear function on0(, 1) only up to elements of (1(,))). If we choosea basis 1, . . . , of0(, 1) we can describe in the following way:() = ( 0 1, . . . , 0 ).

2 | Algebras

For any compact Khler manifold of dimension , the Albanese variety of is iso-morphic to the -th intermediate Griths Jacobian of . The Albanese variety of asmooth complex projective algebraic variety is an Abelian variety, that is, it can beembedded in a projective space. See Jacobians, Weil and Griths intermediate -,Tori, complex - and Abelian varieties.

Algebras. We say that is an algebra over a ring if it is an -module and a ringwith unity (with the same sum) and, for all , and , we have() = () = ().Algebraic groups. ([27], [126], [228], [235], and references in Tori, complex - andAbelian varieties). An algebraic group is a set that is both an algebraic variety (seeVarieties, algebraic -, Zariski topology, regular and rational functions, morphismsand rational maps) and a group and the two structures are compatible, that is, themap ,(, ) 1is a morphism between algebraic varieties.

Structure theorem for algebraic groups (Chevalleys theorem). Let be an algebraicgroup over an algebraically closed eld. Then there exists a (unique) normal anesubgroup such that / is an Abelian variety.Denition. We say that an algebraic group is reductive if all its representations arecompletely reducible (see Representations).

Almost complex manifolds, holomorphic maps, holomorphic tan-gent bundles. ([121], [147], [192], [251]). An almost complex manifold is thedata of a manifold; a section of the bundle (where is the real tangent bundle)

such that, if we see as a map : (, ) (, )(where (, ) is the vector space of the sections of ), we have that2 = ,where is the identity map; in other words, for every , the linear map : induced on the real tangent space of at is such that2 = .

Almost complex manifold | 3

We can extend to := by -linearity. We dene the holomor-phic tangent space of at to be the -eigenspace of ; we denote it by 1,0 and we denote the holomorphic tangent bundle, i.e., the bundle whose bre in is1,0 , by 1,0. We dene the antiholomorphic tangent space of at to be the-eigenspace of ; we denote it by0,1 andwe denote the antiholomorphic tangentbundle, i.e., the bundle whose bre in is 0,1 , by 0,1. Thus = 1,0 0,1 .Obviously a complex manifold (see Manifolds) is an almost complex manifold: if{ = + } are the coordinates on a coordinate open subset, we can dene by( ) = , ( ) = for any . Thus, in the case of a complex manifold,1,0 = ( ) 0,1 = ( ) ,where := , := + .We dene {, } to be the dual basis of , . Let := + and := .Observe that ( ) = 2,, where , is the Kronecker delta.Remark. There is an isomorphism 1,0 given by 12 ( ()) (observe that, through such isomorphism, goes to 12 ).Analogously, there is an isomorphism 0,1 given by 12 ( + ()).For any almost complex manifold , let ,() be the space of the (, )-formson.NewlanderNirenberg theorem. Let (, ) be an almost complex manifold of (real)dimension 2. The almost complex structure is induced by a complex structure if andonly if one of the following conditions holds:

4 | Almost complex manifold

(1) for all , (, 1,0), we have [, ] (, 1,0) (where [, ] standsfor );

(2) for all , (, 0,1), we have [, ] (, 0,1);(3) for all 0,1(), we have 1,1() 0,2() and for all 1,0(), we

have 1,1() 2,0();(4) for all ,(), we have +1,() ,+1() for every , {0, . . . , };(5) for all , (, ), we have[, ] + [, ] + [, ] [, ] = 0

(the rst member of the equality is called Nijenhuis tensor).

Denition. Let be a complex manifold. We can decompose : ,() +1,() ,+1()as = + ,where : ,() +1,(), : ,() ,+1()are the compositions of respectively with the projections+1,() ,+1() +1,(),+1,() ,+1() ,+1() .Let : be a map between two complex manifolds. By extending thedierential : by -linearity, we get a map : .We say that is holomorphic if one of the following equivalent conditions holds:(i) for every component = 1 + 2 of in local coordinates of and, we have1 = 2 and 2 = 1 ;(ii) = 0 for every = 1, . . . , (), where {} are local coordinates of;(iii) = (where and denote the operators on and re-

spectively), i.e., the dierential of is -linear for the complex structures givenby ;

Bertinis theorem | 5

(iv) (1,0) 1,0:(v) (0,1) 0,1.Ample and very ample. See Bundles, bre - (or Divisors).

Anticanonical. See Fano manifolds.

Arithmetically CohenMacaulay or arithmetically Gorenstein. SeeCohenMacaulay, Gorenstein, (arithmetically -,-).

Artinian. See Noetherian (and Artinian).

BBase point free (b.p.f.) See Bundles, bre -.

Beilinsons complex. ([5], [23], [63], [207], [209], [137]).Beilinsons theorem I. LetF be a coherent sheaf on (see Coherent sheaves). Thenthere exists a complex of sheaves0 +1 +1 2 1 1 0with = , s.t. = ()(F()) such that 1 = { F if = 0,0 if = 0.Every morphism () () induced by one of the morphisms is zero.Beilinsons theorem II. Let F be a coherent sheaf on . Then there exists a complexof sheaves 0 +1 +1 2 1 1 0with = , s.t. = O()(F()) such that 1 = { F if = 0,0 if = 0.Every morphism O() O() induced by one of the morphisms is zero.Bertinis theorem. ([93], [104], [107], [129], [136], [228]). On a smooth quasi-projective algebraic variety over an algebraically closed eld of characteristic 0,

6 | Bezouts theorem

the general element of a nite-dimensional linear system (see Linear systems) issmooth away from the base locus of the system.

Bezouts theorem. ([72], [93], [104], [107], [196]). Let be an algebraic closedeld.

Bezouts theorem. Let and be two projective algebraic varieties of respective di-mension and in . Suppose that + and that, for every irreduciblecomponent of and for general point of , and are smooth at and() = () + (), where denotes the tangent space at . Thendeg( ) = deg() deg() .(See Degree of an algebraic subset for the denition of degree).

By using intersection multiplicities (see Intersection of cycles), we can state astronger result: suppose that and are two projective algebraic varieties of di-mension and in with + and suppose they intersect properly, i.e.,the codimension of every irreducible component of is the sum of the co-dimension of and the codimension of ; then, by using an appropriate denitionof intersectionmultiplicity of and along, whichwe denote by (, ), we havethat () () = (, ) (),where the sum runs over all irreducible components of (see, e.g., [72]).Bielliptic surfaces. See Surfaces, algebraic -.

Big. See Bundles, bre - or Divisors.

Birational. See Varieties, algebraic -, Zariski topology, regular and rational func-tions, morphisms and rational maps.

Blowing-up (or -process). ([22], [93], [104], [107], [196], [228]). We followmainly [93] and [104].Roughly speaking, the blow-up of a manifold along a subvariety is a geometric trans-formation replacing the subvariety with all the directions pointing out from it. Forinstance the blow-up of amanifold in a point replaces the point with all the directionspointing out from it.

We dene the blow-up of in a point as follows. By changing coordinates wecan suppose = 0; we dene the blow-up of in 0 as the set0() := {(, ) 1 | }

Blowing-up | 7

(recall that1 = () is the set of lines of passing through 0), with the projection : 0() ,(, ) .Observe that 1() = { a point if = 0,{0} 1 if = 0.Precisely, if = 0, the set 1() is {(, )} where is the unique line through and0. Thus we can say that 0() is obtained from by replacing 0 with 1 , whichrepresents the set of the lines of passing through 0.We call 1(0) the exceptional divisor of the blow-up, and we denote it by .

Fig. 1. Blowing up.

By using the denition of blow-up of in a point, we can dene the blow-up of anymanifold in a point. Let be a complexmanifold and let . We dene the blow-up of in as follows: let be a neighborhood of ; the blow-up of in is

8 | Blowing-up

the set := () := (( {}) ())/ ,where is the relation that identies the points in {} in {} with the pointsof {} in (), with the obvious projection : () .Observe that restricted to () 1() is an isomorphism onto {}. Again, := 1() is called the exceptional divisor of the blow-up in . We say that isobtained from by blowing down .Let be an analytic subvariety of . The proper or strict transform of in , de-noted by , is dened by := 1( {}) = 1() ,that is, it is the closure of 1()minus the exceptional divisor.We can dene also the blow-up of a manifold along a submanifold. Let < ; theblow-up of along = { | +1 = = = 0} is dened to be () = {(, ) 1| [+1 : : ] }with the projection map : () ,(, ) .Observe that the map is an isomorphism from () 1() onto .Let be a complex manifold of dimension and let be a submanifold of dimen-sion . Let {} be a family of open subsets such that for every , is con-tained in := and, for every , there exist local coordinates 1, . . . , on suchthat = {+1 = = = 0}. Let : be the blow-up of along . We have that 1 ( ) and 1 ( ) areisomorphic, so we can glue the blow-ups : to get a manifold and a map : . Let := () := ( ( ))/ where is the equivalence relation given by the identication of the points of in and in . We dene the blow-up of along to be () with the map : () equal to on and equal to the identity on .Let be an algebraically closed eld. The blow-up can be dened for algebraic vari-eties over in the following way. Let be an ane algebraic variety over and be a closed subvariety. Let {0, . . . , } be a set of generators of the ideal of in. Let :

Blowing-up | 9

be the rational map [0() : . . . : ()]We dene the blow-up of along as the graph of with the obvious projection : (the graph of a rational map is dened to be the closure of the graphof |, where is any open subset on which is dened). The divisor 1() is calledthe exceptional divisor of the blow-up. If is a projective variety, we take as 0, . . . , homogeneous polynomials of the same degree generating an ideal whose saturation(see Saturation) is the ideal of in, and we dene the blow-up analogously.We can prove that the denition of blow-up doesnt depend on the choice of 0, . . . , and that, for any open ane subset in, we have that the blow-up of alongis equal to the inverse image of in the blow-up of along . The map : is birational.For example, the blow-up of in 0 is0() = {(, ) 1 | }with the projection : 0() ,(, ) .The proper transform in 0() of an ane algebraic variety containing 0,i.e., 1( {0}), is isomorphic to the blow-up of in 0.Proposition. Let be a nonsingular algebraic variety of dimension over and let ; let : be the blow-up in and call the exceptional divisor. Then wehave = + ( 1)(), () = () ,where () is the line bundle associated to (see Canonical bundle, canonicalsheaf and Equivalence, algebraic, rational, linear -, Chow, NeronSeveri and Picardgroups for the denitions of and ; see Bundles, bre - for the denition ofthe line bundle associated to a divisor; see Pull-back and push-forward of cycles forthe denition of ), the bundle induced by on is the dual of the hyperplane bundle on (see

Hyperplane bundles, twisting sheaves).Suppose now that is a projective surface; then we have 2 = 1 (which is an obvious consequence of the fact that the bundle induced by on is the dual of the hyperplane bundle on ); () () = for any , divisors of; = 0 for any divisor of.Blowing-ups are the fundamental building blocks in birational geometry (see Hiron-akas decomposition of birational maps, Varieties, algebraic -, Zariski topology, reg-

10 | Buchbergers algorithm

ular and rational functions, morphisms and rational maps). Furthermore, they areuseful in the study of singularities (see Regular rings, smoothpoints, singular pointsand Genus, arithmetic, geometric, real, virtual -).

Buchbergers algorithm. See Groebner bases.

Bundles, bre -. ([93], [110], [128], [135], [169], [228], [230], [237]).Denition.Abre bundle (bundle for short in the following) is a quadruple (, , , ),where ,, are topological spaces, : is a continuous surjective map, and there exists an open covering {} of andhomeomorphisms : 1 such that the following diagram commutes:1()

|1() ##FFFFFFFFF

// 1

||yyyy

yyyy

y

,

where 1 : is the projection onto the rst component. We call the trivializing homeomorphisms.

Observe that, for all , the bre 1() is homeomorphic to . It is generally de-noted by .We say that is the bre of the bundle, is the base and is the total space. For thesake of brevity, we sometimes say that is the bundle (on ) or that : is thebundle.We say that two bundles on , (, , , ) and (, , , ), are isomorphic if andonly if there is a homeomorphism : such that = .More generally, amorphism from a bundle on, (, , , ), to another bundle on, (, , , ), is a continuous map : such that = .We say that a bundle (, , , ) is trivial if it is isomorphic to ( , 1, , ) where1 : is the projection onto the rst component.If (, , , ) is a bundle, we say that an open subset of is trivializing if 1() istrivial, i.e., there is a homeomorphism : 1() such that 1 = .Example. One of the simplest examples of a nontrivial bundle is the Mbius strip,which is a bundle on the circle with a segment as bre.

Let (, , , ) be a bundle and let {} be a trivializing open covering of and : 1 for be trivializing homeomorphisms. For any , , let

Bundles, bre - | 11

us consider the composition( ) |1() 1( ) |1() ( ) ;it sends (, ) to (, ) for some ; we dene, : (),(where() is the set of the homeomorphisms of ) to be the map such that(, ) = (, ,()())for any , . The , are called the transition functions of the bundle . They satisfy() ,() ,() = ,() , , , .Conversely, given topological spaces , , an open covering {} of and func-tions , : () satisfying () and such that the maps (, ) (, ,()()) are homeomorphisms on ( ) , it is easy to construct a bundleon with bre and with the , as transition functions: dene = ( )/ where, if (, ) and (, ) , we say that (, ) (, ) if and only if = and = ,()(). The transition functions determine the bundle up to isomorphism.Let (, , , ) be a bundle. Let and and suppose there exist a trivializingopen covering {} for and trivializing homeomorphisms : 1() such that (1() ) = ;in this case we say that (, |, , ) is a subbundle of (, , , ).A section of a bundle (, , , ) is a continuous map : such that () for all .If {} is a trivializing open covering for the bundle and : 1 is atrivializing homeomorphism, we can consider( ) | : .It sends a point to a point whose rst coordinate is , let us say (, ()). Themaps :

12 | Bundles, bre -

which we have just dened have the property that() = ,()(())for any . Giving is equivalent to giving the maps .If (, , , ) is a bundle, is a topological space and : a continuous map,then the pull-back of through is (, , , ) where := := {(, ) | () = ()}and : is the projection onto the rst factor. The pull-back bundle isdenoted by . Observe that() = {(, )| () = ()} = ().We say that a bundle (, , , ) is a dierentiable, respectively holomorphic, if ,, and are dierentiable manifolds, respectively complexmanifolds, and the trivial-izing functions are dieomorphisms, respectively biholomorphisms. In this case, wehave obviously that the functions (, ) ,()() are also dierentiable, respec-tively holomorphic.By a section of a dierentiable, respectively holomorphic, bundle (, , , ), wemean (unless otherwise specied) a dierentiable, respectively holomorphic, map : such that () for any .We say that a bundle (, , , ) is a vector bundle over a eld if is a topological-vector space, the bres have a structure of-vector space and there exist {}trivializing open covering and trivializing homeomorphisms such that the maps | : {} are vector spaces isomorphisms.Thus the images of the transition functions , are in ().In this case, the dimension of is called the rank of the bundle. A vector bundle ofrank 1 is called a line bundle. A complex vector bundle is a vector bundle over theeld .Let (, , , ) and (, , , ) be two-vector bundles on and let {} be an opencovering of trivializing for both bundles. Let the transition functions be , and,respectively.Thedirect sum is the bundlewhose bre is andwhose transition functionsare ( , 00 ,, ) .The tensor product is the bundle whose bre is and whose transitionfunctions are,,,. The tensor product ( repeated times) is denotedby , or, if is a line bundle, also by or .

Bundles, bre - | 13

The wedge product is the bundle whose bre is and whose transition func-tions are ,. If = , where is the rank of , then is called determinantbundle and denoted by () (and the transition functions are obviously (,)).The dual bundle is the bundle with bre on is and whose transitionfunctions are ,()1.If has rank 1, the bundle is denoted also by 1, since is trivial.Let (, , , ) be a vector bundle. Let and be a vector subspace of andsuppose there exist a trivializing open covering {} for and trivializing homeo-morphisms : 1() such that (1() ) = .We say that (, |, , ) is a (vector) subbundle of (, , , ).Let , be the transition functions of . Then the transition functions of are(, ,0 ,)for some functions ,, ,. The quotient bundle / is the bundle whose bre on is / and whose transition functions are ,. We have that = if and only if, = 0 for all , .A morphism between vector bundles on , precisely from (, , , ) to (, ,, ), is a continuous map : such that = and such that : is linear.Let () := () and () := (). They determine subbundles if andonly if the rank of does not depend on .A frame of a vector bundle (, , , ) on an open subset of is a set of sections{1, . . . , } of on such that for all the set {1(), . . . , ()} is a basis of .Obviously, giving a frame of on is the same as giving a trivializing of on .The projectivized of a vector bundle on, whichwe denote by(), is the bundleon whose bre on is () with the obvious trivializing maps.Let be a topological group. A bundle (, , , ) is said to be a principal -bundleif there is an action of on that preserves the bres and acts freely and transitivelyon every bre.Let (, , , ) be a principal bundle with transition functions ,. Let be a topolog-ical space. A homomorphism : () determines a bundle on with bre: the bundle := / ,

14 | Bundles, bre -

where (, ) (, (1))for any .Obviously, if is a topological vector space and : (), then the determinedbundle is a vector bundle on with bre .Denition. An (algebraic) vector bundle of rank on an algebraic variety over aeld is a scheme with a surjective morphism : such that every bre hasa structure of-vector space and there is an open cover {} and trivializing maps : 1() such that the are isomorphisms, we have 1 = , where1 : is the projection onto the rst factor, and the maps induced by the from 1() to {} are vector spaces isomorphisms.By a section of an algebraic bundle : on an algebraic variety , we mean(unless otherwise specied) a morphism : such that () for any ,where is the bre 1(). In an obvious way we can dene the morphisms and theisomorphisms between algebraic vector bundles.

When we speak of a vector bundle on an algebraic variety, we always mean an alge-braic vector bundle.

Let be an algebraic variety. We can associate a sheaf to any vector bundle onin the following way: let O() be the sheaf associating to any open subset of theset of the sections of on. Analogously for holomorphic vector bundles on complexmanifolds. We call O() the sheaf of sections of .The map sending a vector bundle to the sheaf O() gives a bijection between theset of vector bundles on a smooth algebraic variety up to isomorphisms and theset of locally free sheaves of O-modules of nite rank up to isomorphisms (see [228,Chapter 6] or [107, Chapter 2, Exercise 5.18]).

We recall that the group of Cartier divisors of an algebraic variety up to linear equiv-alence is called Picard group of and denoted by () (see Divisors and Equiv-alence, algebraic, rational, linear -, Chow, NeronSeveri and Picard groups).

Denition. We can associate to any Cartier divisor of an algebraic variety aline bundlewe denote by () in the following way: if the Cartier divisor is given bylocal data (, ), we dene () to be the line bundle whose transition functions ,from to are /.Theorem. The map associating to a Cartier divisor of an algebraic variety the linebundle () induces an isomorphism from the Picard group () to the group of theisomorphism classes of line bundles on, where themultiplication is the tensor prod-uct and the inverse is the dual. The group of the isomorphism classes of line bundles

Bundles, bre - | 15

on is isomorphic also to1(,O), via the isomorphism induced by the map send-ing a bundle to its transition functions.

If is a smooth projective algebraic variety over , one can prove that, for any divisor, the rst Chern class (see Chern classes) of the line bundle associated to is thePoincar dual of the class of .Denition. Let be a projective algebraic variety over an algebraically closed eld(or a compact complex manifold) of dimension . Let be an algebraic line bundleon (resp. a holomorphic line bundle on the complex manifold ); in this case wehave that 0(,O()) is nite dimensional, see for instance [107, Chapter 2, 5] and[93, Chapter 0, 6]. Let : (0(,O())),be the rational map { 0(,O())| () = 0};dened only on the set of the points of such that there exists 0(,O())with() = 0.We say that isveryample if the associatedmap iswell denedon thewhole andis an embedding. In this case we have that (O(1)) = , where O(1) is the hyperplanebundle on (0(,O())) (see Hyperplane bundles, twisting sheaves).We say that is ample if there exists such that is very ample.We say that is numerically eective (nef) if (|) 0 for any (irreducible)curve in, that is the intersection number is nonnegative for any divisor suchthat the associated line bundle is and any (irreducible) curve (see Intersection ofcycles).Let () = { | 0(,O()) = 0};we dene the Iitaka dimension of to be if() = {0}, to bemax(){ ()}if() = {0} and is normal, to be the Iitaka dimension of on if is not normaland : is the normalization of (see Normal).The line bundle is said to be big if its Iitaka dimension is equal to .We say that a vector bundle on is ample ifO()(1) is ample on(), whereO()(1)is the line bundle that restricted to any bre of () is O(1), i.e., the dual of theuniversal bundle (see Tautological (or universal) bundle).Let be a vector bundle on. We say that 0(,O()) is base point free (b.p.f.)if there does not exist such that () = 0 for all . We say that is basepoint free if there does not exist such that () = 0 for all 0(,O()).

16 | Bundles, bre -

Let be a vector bundle on . We say that is globally generated if for all wehave that is generated by 0(,O())().Obviously, for line bundle, being globally generated is equivalent to beingbasepointfree.We say that a Cartier divisor is ample, very ample, big, nef, b.p.f., if and only if thecorresponding line bundle () is.We say that a holomorphic line bundle on a complexmanifold is positive if in therst Chern class 1() 2(,) there is a positive (1, 1)-form (see Positive).If is a nef Cartier divisor on a complex projective algebraic variety of dimension ,we can prove that is big if and only if > 0 (see Intersection of cycles for thedenition of the intersection number ); see, e.g., [169].NakaiMoishezon theorem. ([107], [142], [190], [203]). Let be a projective algebraicvariety of dimension over an algebraically closed eld. Let be a Cartier divisoron . Then is ample if and only if > 0 for all irreducible subvariety in with = and for all (see Intersection of cycles for the denition of theintersection number ).Thus ample implies nef. We have also that, under nice assumptions, for instance if is a smooth projective algebraic variety over an algebraically closed eld, the interiorpart of the cone generated by the nef divisors in the space of the divisors up to numer-ical equivalence is the cone generated by the ample divisors; see [142] (we say that twodivisors are numerically equivalent if their intersection number with any curve is thesame).

By Kodaira embedding theorem (see Kodaira embedding theorem) a holomorphicline bundle on a compact complex manifold is positive if and only if it is ample.Observe that, if is a line bundle associated to a divisor , then the positivity of isnot equivalent to the eectivity of (and no implication is true: to show that the im-plication is false one can take a noneective divisor of positive degree on aRiemannsurface; to show that that the implication is false one can take the exceptional di-visor of the blow-up of a surface in a point: this is eective, but the associated linebundle () is not positive, because, by NakaiMoishezon criterion, it is not ample,since 2 is equal to 1.)Finally one can easily prove that b.p.f. implies nef.

Canonical bundle, canonical sheaf | 17

Summarizing, for a line bundle on a projective algebraic variety over an algebraicallyclosed eld, the following implications hold:

very ample b.p.f. ample

nef big

See also Sheaves, Chern classes, Equivalence, algebraic, rational, linear -, Chow,NeronSeveri and Picard groups, CartanSerre theorem.

CCalabiYau manifolds. ([94], [252]). There are several possible (also nonequiva-lent) denitions. One of themost common denitions is the following (see Canonicalbundle, canonical sheaf and Hermitian and Khlerian metrics for the denitionsof canonical bundle and Khler manifold).

Denition.A CalabiYaumanifold is a compact Khler manifold with trivial canonicalbundle.

Another (nonequivalent, precisely weaker) denition is: a CalabiYau manifold is acompact Khler manifold such that there exists a nite (holomorphically) coveringspace (see Covering projections) of with trivial canonical bundle.Examples. Elliptic Riemann surfaces and K3 surfaces satisfy both denitions, whileEnriques surfaces satisfy the second denition, but not the rst (see Riemann sur-faces (compact -) and algebraic curves, Surfaces, algebraic - for the denitions).

Canonical bundle, canonical sheaf. ([93], [107], [228]). Let be a complexmanifold. The canonical bundle is the determinant bundle of the dual of the holo-morphic tangent bundle 1,0 (see Almost complex manifolds, holomorphic maps,holomorphic tangent bundles). The canonical sheaf, denoted by , is the sheaf ofthe holomorphic sections of the canonical bundle, i.e., O().If is a smooth algebraic variety of dimension over an algebraically closed eld, wedene the canonical bundle to be the determinant of dual of the tangent bundle ,i.e., to be the bundle determined by the sheaf (see Zariski tangent space, dieren-tial forms, tangent bundle, normal bundle).The associated sheaf of sections, i.e., O(), coincides obviouslywith and is calledcanonical sheaf; it is denoted also by .

18 | Cap product

Example. = ( 1),where is the hyperplane bundle on (see Hyperplane bundles, twistingsheaves).

See also Dualizing sheaf.

Cap product. See Singular homology and cohomology.

CartanSerre theorems. ([42], [93], [103], [107], [223]).Serres theorem. Let F be a coherent sheaf (see Coherent sheaves) on a projectivealgebraic variety over a eld. For any , letF() beFO() (see Hyperplanebundles, twisting sheaves for the denition of O()). Then(i) the-vector space(,F)has nite dimension for any andhas dimension0 for every > ();(ii) there exists such that, for any and for any , the stalk F() is

spanned, as O,-module, by the elements of0(,F());(iii) there exists such that(,F()) = 0 for all , > 0.Statements (ii) and (iii) are often called Serres theorem A and B, respectively.

Furthermore, in [42], Cartan and Serre proved that, for any coherent sheaf F on a com-pact complex manifold, the complex vector space(,F) is nite dimensional forany .CastelnuovoEnriques Criterion. See Surfaces, algebraic -.

CastelnuovoEnriques theorem. See Surfaces, algebraic -.

CastelnuovoDe Franchis theorem. See Surfaces, algebraic -.

Categories. ([26], [79], [116], [168], [179]). We follow mainly the exposition in [79].A category C consists of(i) a set of objects;(ii) for every ordered pair of objects, , , a set denoted C(, ) (or simply(, )), whose elements are calledmorphisms or arrows;(iii) for each triple of objects , , , a map(, ) (, ) (, )

(the image of (, ) (, ) (, ) by this map is called the composi-tion of and and is denoted by or by ) such that

Categories | 19

(a) (1, 1) and(2, 2) are disjoint unless 1 = 2 and 1 = 2;(b) the associativity of the composition holds;(c) for each object , there exists an arrow 1 (,) such that 1 =

and 1 = for any , arrows.We write an arrow (, ) as : .Denition. Let C, D be two categories; a (covariant) functor from C to D (we willwrite : C D) is given by a map, which we denote again by , from the set of theobjects of C to the set of objects of D and, for each ordered pair of objects , of C, amap, which we denote again by , fromC(, ) toD((), ()) such that(i) () = ()(),(ii) (1) = 1().We can dene the composition of two functors in the obvious way.

Denition. Let be a functor from a category C to a category D.We say that is faithful if, for any , objects of C, we have that : C(, ) D((), ())is injective.We say that is full if, for any , objects of C, we have that : C(, ) D((), ())is surjective.

Denition. Let C be a category. The dual category C is the category dened in thefollowing way: the objects are the objects of C (an object in C will be denoted by as an object of C); we dene C (, ) := C(, ) (an arrow : inC(, ) will be denoted : as an element of C (, )) and wedene = () and = ().A controvariant functor from C D is a covariant functor C D, thus it is givenby amap from the set of the objects of C to the set of objects ofD and, for each orderedpair of objects , of C, a mapC(, ) D((), ()) .Denition.We say that a category C is a subcategory of a category C if the set of the objects of C is a subset of the set of the objects of C; for any , objects of C, we have thatC (, ) is a subset ofC(, ): the compositionof themorphisms in C coincideswith the compositionof themor-

phisms in C and, for any object of C, the identity morphism in C coincideswith the identity morphism in C.

20 | Categories

Denition.We say that two objects , in a category C are isomorphic if and only ifthere are twoarrows : and : such that = 1 and = 1.Denition. Let C, D be two categories and and be two functors from C to D. Amorphism of functors (called also a natural transformation) from to : is a family of arrows in D,() : () (), one for each object of C, such that,for any arrow : in C, the diagram() () //()

()()() () // ()

is commutative, i.e., () () = () ().Taking the natural transformations as morphisms, we get that the functors from C toD form a new category, which we denote by (C,D).In particular two functors , from C toD are isomorphic if there exist a natural trans-formation from to and a natural transformation from to such that = and = . Equivalently, and are isomorphic if there exists a natural trans-formation from to such that () : () () is an isomorphism for any object of C.

Denition.We say that two categories C and D are isomorphic if and only if there arefunctors : C D and : D C such that = D and = C.Denition.We say that two categories C and D are equivalent if and only if there arefunctors : C D and : D C such that is isomorphic to D and isisomorphic to C.Proposition. Two categories C andD are equivalent if and only if there exists a functor : C D such that : C(, ) D((), ())is a bijection for any, objects of C (that is, is full and faithful) and, for any object of D, there exists an object of C such that and () are isomorphic.LetC be a category and S be the category of the sets (that is, the categorywhose objectsare the sets and, for any sets 1, 2,(1, 2) is the set of the functions from 1 to 2).Let be an object of C. Let : C S be the covariant functor dened by() = C(, )

Categories | 21

for any object of C and sending an arrow : to the arrowC(, ) C(, )given by the composition with . Let : C S be the controvariant functor denedby () = C(, )for any object of C and sending an arrow : to the arrowC(, ) C(, )given by the composition with . Sometimes and are denoted respectively by(, ) and( , ).Denition.We say that a covariant functor : C S is representable if it is isomor-phic to for some object of C (in this case, we say that represents ).We say that a controvariant functor : C S is representable if it is isomorphic to for some object of C (in this case, we say that represents ).We can prove that an object representing a functor is unique up to isomorphism.

Denition.Wesay that a categoryC isadditive if it satises the following three axioms: C(, ) is an Abelian group for any objects and of C and the composition

of arrows is bi-additive, i.e.,( + ) = + , ( + ) = + for all morphisms , , , .

Zero object. There exists an object 0 of C such thatC(0, 0) is the zero group. Direct sum. For any two objects 1, 2 in C there is an object in C and arrows : , : for = 1, 2

such that = for = 1, 2, 2 1 = 1 2 = 0 and1 1 + 2 2 = .Denition. Let C andD be two additive categories. We say that a functor from C toDis additive if, for any , , objects of C, the map : C(, ) D((), ())is a homomorphism of Abelian groups.

Let be the category of Abelian groups, that is, the category whose objects are theAbelian groups and, for any Abelian groups 1, 2, (1, 2) is the set of the ho-momorphisms from 1 to 2. Let C be an additive category and let : be anarrow in C. We dene a controvariant functor : C ,

22 | Chern classes

in the following way: for any object of C, we dene( )() = (C(, ) C(, )),where C(, ) C(, ) is the map given by the composition with , and,for any : arrow inC, let ( )()be themorphism ( )() ( )(),i.e., (C(, ) C(, )) (C(, ) C(, )),given by the composition with . If the functor is represented by an object ,one can prove easily that there is a morphism : such that = 0. Themorphism , or the pair (, ), or , is called the kernel of .The cokernel of a morphism : is a morphism : such that, for all object in C, the following sequence (where the maps are induced by and ) is anexact sequence of groups:0 (, ) (, ) (, ).Again, sometimes we call a cokernel or .Denition. We say that a category C is Abelian if it is additive and it satises the fol-lowing axiom:Ker and Coker. For any arrow : in C, there exists a sequence such that(i) = ;(ii) is the kernel of , is the cokernel of ;(iii) is the kernel of and the cokernel of .Chern classes. ([45], [93], [96], [107], [135], [146], [147], [181],[188]). Let be a holo-morphic vector bundle of rank on a complex compactmanifold. The Chern classes() 2(,)for = 1, . . . , and the total Chern class() = 1 + 1() + + () =0,..., 2(,)(where 1 0(,) ) are dened by the following three axioms:Axiom 1: Normalization. if = 1, 1 is the map 1(,O) 2(,) induced by theexponential sequence on , 0 O O 0

Chern classes | 23

(the rst map is given by the inclusion and the second by 2; see Exponentialsequence).

Axiom 2: Multiplicativity. if 0 0 is an exact sequence of vectorbundles on , then () = ()(),where the product at the second member is the cup product, see Singular homologyand cohomology.

Axiom 3: Functoriality. if : is a continuous map and a vector bundle on ,then () = ().We dene the Chern polynomial of to be()() = 1 + 1() + + ().LerayHirsch theorem. Let : () be the projectivized bundle of . Let bethe tautological bundle on (), i.e., the subbundle of whose bre on a point of() is the line represented by this point (see Tautological (or universal) bundle) ()

() // .

We have that((),) is a free(,)-module generated by 1, , . . . , 1, where = 1(1).Theorem. If : () is the projectivized bundle of and = 1(1), where isthe tautological bundle on (), we have + (()) = 0 .The theorem above is sometimes used (once 0 and 1 are dened) to dene the Chernclasses (to be the unique elements satisfying the equation above).

Some useful formulas about Chern classes are:(i) For any () = (1)().(ii) Let and be holomorphic vector bundles on . Let()() = (1 + ), ()() = (1 + )

24 | Chern classes

be the formal factorizations of the Chern polynomials of and . Then( )() =, (1 + ( + )).From (ii) we easily get that1( ) = ()1() + ()1().Moreover, if () = and () = 1,( ) = =0,..., ( )()1().In particular, if is a bundle on the projective space and () = , then1(()) = 1() + ,where () = , where is the hyperplane bundle (see Hyperplane bundles,twisting sheaves).

The denition of Chern classes can be given, more generally, for complex bundles oncompact manifolds in a way analogous to the denition above (the only dierencebeing in the normalization axiom). We skip it, but we mention other ways to deneChern classes.If is a complex vector bundle on a compact -manifold and is a connection on (see Connections), we can dene for any () = ( ( 2)) ,where is the curvature of , is the trace, and is the imaginary unit.Another way is the following: if the rank of is , we dene () to be the Poincar dual of the zero locus of a general section of ; 1() to be the Poincar dual of the zero locus of 1 2 where 1, 2 are general section of ; more generally, () to be the Poincar dual of the zero locus of 1 +1,

where 1, . . . , +1 are general sections of .Finally we want to mention the denition of Chern character.Let ()() = (1 + )be the formal factorization of the Chern polynomial ()() of a complex bundle ;the are called the Chern roots. Observe that, if splits, i.e., = 1

CohenMacaulay, Gorenstein, (arithmetically -,-) | 25

with the line bundles, the are the 1( ). We dene the Chern character () = () of by() = =1,..., exp() = =1,...,(1 + + 22! + 33! + ) .One can easily see that() = + 1() + 12 [1()2 22()] + 16 [1()3 31()2() + 33()] + .We have that ( ) = () + (),( ) = ()().Chows group. See Equivalence, algebraic, rational, linear -, Chow,NeronSeveriand Picard groups.

Chows theorem. ([93], [103], [196]). Any analytic subvariety in is algebraic,i.e., it is the zero locus of a nite number of homogeneous polynomials (see Varietiesand subvarieties, analytic -, Varieties, algebraic -, Zariski topology, regular and ra-tional functions, morphisms and rational maps).See G.A.G.A.

Class group, divisor -. See Equivalence, algebraic, rational, linear -, Chow,NeronSeveri and Picard groups.

Cliords Index and Cliords theorem. See Riemann surfaces (compact -)and algebraic curves.

CohenMacaulay, Gorenstein, (arithmetically -,-). ([20], [39], [62], [159],[186], [187]). Let (, ) be a local Noetherian ring. We say that a nitely generated -module is CohenMacaulay if() = ()(see Local, Noetherian, Artinian, Depth and Dimension).A ring is Cohen-Macaulay if it is Cohen-Macaulay as module on itself.Let be an algebraically closed eld.(1) We say that an algebraic set over is CohenMacaulay if, for any , thelocal ring O, (the stalk in of the sheaf of the regular functions on ) is CohenMacaulay.

26 | CohenMacaulay, Gorenstein, (arithmetically -,-)

One can prove that a CohenMacaulay algebraic set is equidimensional: for instance,the union of a line and a plane meeting in a point is not CohenMacaulay.Hartshornes connectedness theorem states that a CohenMacaulay algebraic setmust be locally connected in codimension 1, i.e., removing a subvariety of codimen-sion 2 cannot disconnect it.For instance two surfaces meeting in a point in a space of dimension 4 cannot beCohenMacaulay.

Example of CohenMacaulay algebraic sets: (locally) complete intersections.

(2)An equidimensional projective algebraic set of codimension in is said arith-metically CohenMacaulay if the minimal free resolution (see Minimal free resolu-tions) of the sheaf O has length (which is the minimal possible length):0 F F1 F1 O O 0.Equivalently, if we dene = [0, . . . , ] and = () is the ideal associated to ,theminimal free resolution of the projective coordinate ring /, asmodule over , haslength : 0 1 1 / 0.This is equivalent to the condition (/) = ,where the stands for projective dimension and the projective dimension of an -module is theminimal length of a projective resolution of (a projective resolutionof is an exact sequence 0 0 0with projective -modules; see Injective and projective modules and Injectiveand projective resolutions; is the length of the resolution).The condition (/) = is obviously equivalent to the condition(/) = () = () (/),that is, () (/) = (/),which is equivalent, by the AuslanderBuchsbaum theorem (see Depth), to(/) = (/).Thus is arithmetically CohenMacaulay if and only if the coordinate ring /, isCohenMacaulay as -module.The rank of F is called the CohenMacaulay type of .

Coherent sheaves | 27

One can show that, for of dimension 1, being arithmetically CohenMacaulayis also equivalent to the condition (I()) = 0for 1 and for all , where I is the ideal sheaf of .Proposition. Arithmetically CohenMacaulay CohenMacaulay.(3) An equidimensional projective algebraic set of codimension in is said to bearithmetically Gorenstein if it is arithmetically CohenMacaulay of type 1, i.e., theminimal free resolution of the sheaf O has length 0 F F1 F1 O O 0and the rank of F is 1.Remark. (See [186].) If is arithmetically CohenMacaulay of codimension in and 0 1 1 / 0is a minimal free resolution of /, then0 1 (/, ) 0is a minimal free resolution of (/, ).The module := (/, )( 1) is called the canonical module.Proposition. Let be an a.C.M. algebraic subset of . We have: is arithmeticallyGorenstein / = () for some the minimal free resolution of / is selfdualup to twist by + 1.(4) We say that an equidimensional projective algebraic subset of of codi-mension is Gorestein if it is CohenMacaulay and its dualizing sheaf, that isEXT

(O,O )( 1) (see Dualizing sheaf, , EXT) is locally free of rank 1.Remark. Strict complete intersection arithmetically Gorenstein arithmeticallyCohenMacaulay (the rst implication is due to the fact that, if is a strict completeintersection, then its minimal resolution is the Koszul complex).

Coherent sheaves. ([93], [107], [129], [146], [223]).The most common denition of coherent sheaf is the following.

Denition. Let (,O) be a ringed space (see Space , ringed -). We say that a sheaf F of O-modules (see Sheaves) is of nite type if for every

point there is an open neighborhood of and a surjective morphism ofsheaves of O-modules O| F| for some .

28 | Coherent sheaves

We say that a sheaf F of O-modules is coherent if the following two propertieshold:(i) F is of nite type,(ii) for any open subset of , any , and any morphism of sheaves of O-

modules : O| F|, we have that () is of nite type. We say that a sheaf of O-modules F is quasi-coherent if, for every point ,

there is an open neighborhood of and an exact sequenceO| O| F| 0,

where O and O are direct sum of (possibly innite) copies of O.Theorem. Let (,O) be a ringed space.(i) Let 0 F1 F2 F3 0 be an exact sequence of sheaves of O-modules. If

two of the F are coherent, so is the third.(ii) The kernel, the image and the cokernel of a morphism of sheaves of O-modules

between two coherent sheaves are coherent.(iii) Let F and G be two coherent sheaves on . Then F O G and HOMO (F,G) are

coherent.

In some texts (for instance [107]), the denition of coherent sheaf on a scheme (seeSchemes) is the following.

Denition. Let be a commutative ring with unity and an -module. We dene Fto be the following sheaf on (): for any open subset of (), let F() bethe set of the functions : (where is the localization of in , see Localization, quotient ring, quotienteld) such that () for every and is locally a fraction, that is, forall , there exists a neighborhood of in and , such that for all , we have and () = . Consider the obvious restriction maps.Let (,O) be a scheme and F a sheaf ofO-modules. We say F is a coherent (respec-tively quasi-coherent) sheaf if there is an open ane covering of , { = ()},and nitely generated (respectively not necessarily nitely generated) -modulessuch that F| = F .The two denitions of coherent sheaf agree on Noetherian schemes (see, e.g., [129,Theorem 1.13] for a proof).

By Serres G.A.G.A. theorem (see G.A.G.A.) there is a functor from the categoryof schemes of nite type over to the category of analytic spaces (see Spacesanalytic -) and, given a projective scheme over , the functor induces an equiv-alence of categories from the category of coherent sheaves on to the category of co-

Completion | 29

herent sheaves on the analytic space associated to and this equivalence maintainsthe cohomology.

See also CartanSerre theorems.

Cohomology of a complex. See Complexes.

Cohomology, singular -. See Singular homology and cohomology.

Complete intersections. ([104], [107], [228]). We say that a projective algebraicvariety of dimension in is a strict complete intersection if there exist el-ements of the ideal of that generate the ideal of; we say that is set-theoreticallycomplete intersection if it is the intersections of hypersurfaces.Complete varieties. ([107], [129], [140], [197], [201], [202], [228]).Denition. An algebraic variety is said to be complete if, for all algebraic varieties, the projection morphism : is a closed map, i.e., the image through of any closed subset (closed in the Zariski topology) is a closed subset.

Proposition. An algebraic variety over is complete if and only if it is compact withthe usual topology.

Proposition. A projective algebraic variety over an algebraically closed eld is com-plete.

There exist complete nonprojective algebraic varieties; see [201].

Chows lemma. For any complete variety over an algebraically closed eld, thereexists a projective algebraic variety and a surjective birational morphism from to.Completion. ([12], [62], [107], [185], [256]). Let (, +) be a topological Abelian groupwith a sequence of subgroups, {}, such that = 0 1 and {} is a fundamental system of neighborhoods of 0.We say that a sequence () in is a Cauchy sequence if, for any neighborhoodof 0, there exists such that , .We say that two Cauchy sequences, () and (), are equivalent iflim( ) = 0.

30 | Complexes

The completion of , denoted by , is dened to be the set of the equivalence classesof the Cauchy sequences in .The map sending to the constant sequence = for all is injectiveif and only if is Hausdor. If the map is an isomorphism, we say that iscomplete. We can prove that a completion is complete.The completion of can be also dened by using the inverse limit (see Limits, Directand inverse): we can dene := lim /.Example. Let = and = () for every , where is a prime number. Take{} as fundamental system of neighborhoods of 0; the topology we get is called-adic and the ring is called ring of the -adic integers.More generally, let be a ring, be an ideal and dene = for any (dene0 = ) . The topology we get on by taking {} as a fundamental system ofneighborhoods of 0 is called -adic.In algebraic geometry one often meets completions of local rings (, ) (see Local);in this case we consider {} as fundamental system of neighborhoods of 0. Pre-cisely, for any algebraic variety and any , the completion of the local ring(O,, ) gives information about the local behavior of around and the study ofthe completions of the local rings (O,, ) for is linked to the study of singu-larities of .If and are two algebraic varieties over a eld and and , we say that in and in are analytically isomorphic if and only if the completions of O,and O, are isomorphic as -algebras.Cohens theorem. Let (, ) be a complete regular local Noetherian ring of dimension (see Noetherian, Artinian and Dimension) containing some eld and let be itsresidue eld, i.e., /. Then is isomorphic to the ring of the formal powers series in variables over , usually denoted by [[1, . . . , ]].More generally, if (, ) is a complete local Noetherian ring containing some eld and is its residue eld, then is isomorphic to [[1, . . . , ]]/ for some and someideal .In particular, if is a smooth point of an algebraic variety of dimension over aeld , then the completion of O, is isomorphic to [[1, . . . , ]].See Regular rings, smooth points, singular points.

Complexes. Let be a ring. A complex of -modules, which is usually written 2 // 1 1 // // +1 +1 // ,is the datum of a sequence of -modules and -homomorphisms : +1such that +1 = 0 for any .

Cone, tangent - | 31

The cohomology of the complex above in degree , usually denoted by (),is dened as () := ( : +1) (1 : 1 ) .We call it homology, instead of cohomology, if the indices of the -modules aredecreasing (instead of increasing).

Amorphism from a complex of -modules 2 // 1 1 // // +1 +1 // to another complex of -modules 2 // 1 1 // // +1 +1 // is the datum of a sequence of -homomorphism : such that the followingdiagram commutes 2 // 1 1 //1

//

+1 +1 //+1

2 // 1 1 // // +1 +1 // .

We can see easily that it induces homomorphisms from () to() for any .We say that two morphisms = () and = () from to are homotopicallyequivalent if there are -homomorphisms : 1 such that = +1 + 1 . 2 // 1 1 //

~~||||

||||

||||

//

+1+1

~~||||

||||

||||

+1 //

2 // 1 1 // // +1 +1 // .

Homotopically equivalent morphisms induce the same homomorphisms in cohomol-ogy.In an analogous way we can dene complexes of sheaves, etc.See also Exact sequences.

Cone, tangent -. ([104], [228]). Let be an ane algebraic variety over an alge-braically closed eld and let = () be the ideal of (see Varieties, algebraic -,Zariski topology, regular and rational functions, morphisms and rational maps). Let

32 | Connections

. Choose coordinates such that = 0. For every , let () be the sum of themonomials of of the lowest degree. Let() := {()| }.The tangent cone to at = 0 is dened to be the zero locus of ().Example. Let be generated by 2 2 +3. The ideal () is generated by 2 2; seeFigure 2.

Fig. 2. The tangent cone.

Connections. ([46], [93], [135], [146], [188], [251]). Let be a vector bundle ona manifold . A connection on is a map : () ( )(where, for any bundle , () denotes the set of sections of and denotesthe dual of the tangent bundle) such that(1 + 2) = 1 + 2 1, 2 (),() = + (), : .Remark. Let 1, . . . , be a frame for on an open subset of , i.e., let 1, . . . , besections of | such that, for all , {1(), . . . , ()} is a basis of the bre . Wedene the connection matrix of with respect to 1, . . . , in the following way: = =1,..., ,(the entries of are 1-forms). Then( ) = + = + , = ( + ,),which can be written for short as + ,

Connections | 33

where = ( 1... ) .Let() := (). Given a connection on , : 0() 1(), we can denea map, which is usually called again , : () +1()in the following way: ( ) = + for any (), (). By composing : 0() 1() with : 1() 2() we get a map : 0() 2(),called curvature of the connection. It satises() = ()for all () and for all maps : . If is the connection matrix of withrespect to a frame on an open subset of , we have that is given by the matrix of2-forms = + ,where denotes the usual matrix product where the entries are multiplied by thewedge product.

Remark. Suppose {1, . . . , } and {1, . . . , } are two frames on an open subset of for the vector bundle and, if = , then

( 1... ) = (1...) .

Then, if and are the connection matrices of with respect the two frames and and are the matrices of the curvatures, we have that = 1 + 1, = 1 .Proposition. Let be a connection on a vector bundle on and be its curvature.Let , . Then , = [,],

34 | Connections

where the subscript in an operator means that, after applying the operator, we evalu-ate the result in the vector indicated in the subscript and [, ]means .

Connections on tensor products and dual bundles

Remark. If and are two vector bundles and and are two connections respec-tively on and , we can dene a connection on the tensor product inthe following way: ( )( ) = + for any (), ().Remark. If is a connection on a vector bundle , there exists a connection on thedual bundle such that (, ) = (, ) + (, )for any (), () (where (, ) means that we are applying the element of() to the element of ()).Bianchis identity. Let be a connection on a vector bundle . Let be its curvature;it can be seen as an element of 2( ). Then = 0(by the remarks above, denes a connection on and thus also on = ()and thus we have a map, we call again , from 2( ) to 3( )).

Compatibility with holomorphic structures and metrics

Denition. Let (, (, )) be a complex vector bundle with a Hermitianmetric. A connec-tion on is said to be compatible with the metric if(1, 2) = (1, 2) + (1, 2)for all 1, 2 ().Denition. Let be a complex manifold and be a holomorphic vector bundle. Aconnection on is said to be compatiblewith the holomorphic structure if = where is the projection( 1,0) ( 0,1) ( (0,1 ))(i.e., the entries of the connection matrix are of type (1, 0)).Proposition. Let be a complexmanifold and be a holomorphic vector bundle witha Hermitianmetric. There is a unique connection on compatible both with the holo-morphic structure and with the metric.

Covering projections | 35

Correspondences. ([72], [93], [196]). Let and be two algebraic varieties; acorrespondence from to is an algebraic cycle in (see Cycles).Let , , be three smooth algebraic varieties and let , be the projection from onto and analogously , and ,. If 1 is a correspondence from to and 2 is a correspondence from to , we dene2 1 = ,(,1 ,2),where is the intersection of cycles (see Pull-back and push-forward of cycles,Intersection of cycles).One can prove that, if is smooth, then the set of the correspondences from to with the product and the usual sum of cycles is an associative ring.Covering projections. ([33], [91], [112], [158], [184], [215], [234], [247]).Denition. Let and be two topological spaces. We say that a map : is a topological covering projection (covering projection for short) of if, for all , there exists an open subset of such that 1() is a disjoint union of opensubsets of such that | : is a homeomorphism for all . The space issaid to be covering space.

Denitions. We say that a map between two topological spaces, : , is a local home-

omorphism if, for all , there exists an open subset of containing such that () is an open subset of and : () is a homeomorphism.

Let , , be topological spaces. Let : and : be continuousmaps. A lifting of (for ) is a continuous map : such that = .

??~~~~~~~ //

We say that a continuous map : is complete with respect to a topo-logical space , if, for every continuous map : [0, 1] , every lifting of|{0} can be extended to a lifting of .

We say that a continuous map : has unique path lifting if, given paths and in (that is two continuous maps from [0, 1] to) such that (0) = (0)and = , then = .

We say that a topological space is locally path-connected if the path-connectedcomponents of open subsets are open.

36 | Covering projections

We say that a topological space is semi-locally simply connected if every has a neighborhood such that all loops in are homotopically trivialin.

21

Fig. 3. A covering projection.

Theorem. Let and be topological spaces. A covering projection : is alocal homeomorphism and is complete with respect to any topological space.

Theorem. Let be a locally path-connected and semi-local simply connected topolog-ical space and let be a locally path-connected topological space. Let : be a continuous map complete with respect to any topological space and with uniquepath lifting. Then : is a covering projection.Denition. Let and be two topological spaces and let and be points respec-tively of and . A pointed covering projection : (, ) (, ) is a coveringprojection : such that () = .Proposition . Let : (, ) (, ) be a pointed covering projection. Let be aconnected topological space, and : (, ) (, ) be a continuous map.(a) If there exists a lifting : (, ) (, ) of , it is unique.(b) If is locally path-connected, there exists a lifting : (, ) (, ) of if and

only if (1(, )) (1(, )),where we denote by 1 the rst fundamental group (see Fundamental group,also for the denition of and ).

Proposition. Let : (, ) (, ) be a pointed covering projection. The maps : (, ) (, )are injective for every and are isomorphisms for 2.

Cremona transformations | 37

Theorem. Let be a path-connected, locally path-connected, semi-locally simply con-nected topological space. Let . There is a bijection between the following sets:{pointed covering maps on (, )}/ pointed covering homeomorphismsand {subgroups of 1(, )},where a pointed covering homeomorphism between two pointed covering projectionson (, ), : (, ) (, ) and : (, ) (, ), is a homeomorphism : such that () = and = . The bijection is given by associ-ating to a pointed covering projection on (, ), : (, ) (, ), the subgroup(1(, )) of 1(, ).Denition. We say that a covering projection : with path-connected, isuniversal if the rst fundamental group of is trivial.Let : (, ) (, ) be a pointed covering projection with path-connected. Let = (1(, )).Obviously the bre 1() is in bijection with the set of lateral classes of in 1(, ):{| 1(, )}.In fact, if 1(), we can send to, where is the image through of a pathfrom to .We have an action, calledmonodromy action, of 1(, ) on 1(): if is a point of1() and 1(, ), let be the lifting of such that (0) = ; we dene to be(1).Observe that, if is a point of such that () = and is the image through of a path from to , then (1(, )) = 1. Thus, in the assumptions of thetheorem above, there exists a pointed covering homeomorphism between (, ) and(, ) if and only if 1 = .In particular, if is normal, the group of covering homeomorphisms of on istransitive on 1() and is isomorphic to 1(, )/.Note. In the case of two complex manifolds, (ramied) covering projections some-times stands for holomorphic surjectivemaps between complexmanifolds of the samedimension. When it is a true topological covering projection, i.e., it is not ramied, wesay that it is an tale covering projection.

See also Riemanns existence theorem

Cremona transformations. See Quadratic transformations, Cremona transfor-mations.

38 | Cross ratio

Cross ratio. Let be a eld and let 1 = 1. The cross ratio of an ordered set offour points of 1, (1, 2, 3, 4), with 1, 2, 3 distinct, is the element [ 12 ] 1 suchthat there exists an automorphism : 1 1 such that((1), (2), (3), (4)) = ([ 01 ] , [ 10 ] , [ 11 ] , [ 12 ]) ,or, using not homogeneous coordinates,((1), (2), (3), (4)) = (, 0, 1, ),where = 2/1. It is equal to

simple ratio(1, 2, 3)simple ratio(1, 2, 4) ,

where, if = [ ] and := /, we dene simple ratio(1, 2, 3) = 3 132 . Moreprecisely, the cross ratio is equal todet( 1 31 3 ) det( 2 42 4 )det( 1 41 4 ) det( 2 32 3 ) .Since the cross ratio of (1, 2, 3, 4) and the cross ratio of ((1), (2), (3), (4)) arethe same for any composition of two disjoint transpositions, we have that the possi-ble values of the cross ratio of ((1), (2), (3), (4)) for 4 are at most 6, precisely,if is the cross ratio of (1, 2, 3, 4), the 6 possible values are, 1 , 1 , 11 , 1 1 , 1 11 .Cup product. See Singular homology and cohomology.

Curves. See Riemann surfaces (compact -) and algebraic curves.

Cusps. See Regular rings, smooth points, singular points.

Cycles. Let be an algebraic variety. An algebraic cycle of codimension in is anelement of the free Abelian group generated by the closed (for the Zariski topology)irreducible subsets of of codimension (see Varieties, algebraic -, Zariski topology,regular and rational functions, morphisms and rational maps); in other words analgebraic cycle of codimension in is an element of the form

Deformations | 39

with irreducible algebraic subset of of codimension and .Analogously for analytic cycles.See Equivalence, algebraic, rational, linear -, Chow, NeronSeveri and Picardgroups, Pull-back and push-forward of cycles.

DDeformations. ([43], [151], [153], [154], [160], [161], [218], [221]).Deformation theory is strictly connected with the theory of moduli spaces (see Mod-uli spaces), i.e., varieties parametrizing geometric objects of a certain kind. It wascreated to parametrize the possible complex structures on a xed dierentiable man-ifold. We recall that if : is a holomorphic surjective map between complexmanifolds such that the dierential of at every point has maximal rank and the -bres of are compact complex manifolds, then the bres are dieomorphic by Ehres-manns theorem (see [58] or [149, Theorem. 2.3]), and so parametrizes some complexstructures on the same dierentiable manifold.

Denition. Let be a compact complex manifold. A deformation of is the datumof a proper at morphism : of complex analytic spaces, a point and anisomorphism 1() . We denote this by : (, ).(See Proper, Flat (module, morphism), Spaces, analytic - for the denitions ofthese terms.)Let : (, ) be a deformation and let : (, ) (, ) amorphism of analyticspaces. The pull-back : (, ) of the deformation : (, ) through is dened in the following way: let = = {(, ) | () = ()}and let : be the projection onto the second factor.We say that two deformations of , : (, ) and : (, ), are isomor-phic if there exist isomorphisms of analytic spaces : and : (, ) (, )such that = and the composition of from 1() to 1() with the iso-morphisms with is the identity.Denition. We say that a deformation : (, ) of is complete if any otherdeformation of , : (, ), is locally the pull-back of : (, ), i.e.,there exist neighborhood of in and : such that the deformation |(in the obvious sense) is isomorphic to the pull-back of : (, ) through .We say that : (, ) isuniversal if it is complete and themap is locally unique.We say that : (, ) is semiuniversal if it is complete and the map (thedierential of in ) is unique.

40 | Deformations

Observe that all theuniversal deformations of a complexmanifold are locally canon-ically isomorphic, and all the semiuniversal deformations of are locally isomorphic.Obviously, from the moduli problem viewpoint, the best situation is the one of uni-versal deformations.

Denition. Let : (, ) be a deformation of . Let 1,0 be the holomorphictangent space of at and let = O(1,0), where1,0 is the holomorphic tangentbundle of . The KodairaSpencer map associated to : (, ) is the map : 1,0 1()dened as follows: let be a neighborhood of and let {} be a nite open coveringof 1 such that, dened = , there are isomorphisms : suchthat the composition of with the projection onto the second factor is ; let 1,0 ;for all there exists a unique () 1,0 orthogonal (through ) to 1,0 and such that (()) = . The family, := denes an element of1(), which we call ().Observe that the trivial deformation (i.e., ) has zero KodairaSpencer map.Remark. If (, ) and (, ) are two deformations of with KodairaSpencer maps respectively and and the second deformation is the pull-back ofthe rst through a map : (, ) (, ), then = .Theorem. For every complex compact manifold , there exists a deformation (, ), called Kuranishi family, with the following properties:(1) its KodairaSpencer map is bijective;(2) it is a semiuniversal deformation of ; furthermore, and it is complete in every

point of , if we consider it as a deformation of the bre on ;(3) if0() = 0, then it is a universal deformation of ;(4) is the zero locus of a holomorphic map from a neighborhood of 0 in 1() to2(); in particular () 1() 2() and, if 2() = 0, then is

smooth.

From the remark and from statement (1) of the theorem we have that a complete de-formation must have surjective KodairaSpencer map. Furthermore, if a complete de-formation has bijective KodairaSpencer map, then it is semiuniversal.

We considered deformations of compact complexmanifolds, but in an analogous waywe can consider deformations of other objects: schemes, bundles . . . . In [218], Schles-singer developed a theory to which many deformation theories can be related.

Degree of an algebraic subset | 41

Let be a local Noetherian complete ring with residue eld and let be the cate-gory of the local Artinian -algebraswith residue eld . Schlessingers theory studiesthe functors : (see Categories for the denition of functor) such that () is a set with only oneelement.The link between this theory and deformation theory is the following: consider forinstance deformations of schemes; consider the functor : such that() is the set of isomorphisms classes of deformations on () of a scheme and, if : is a homomorphism, () is the map from the set of isomorphismsclasses of deformations on () of to the set of isomorphisms classes of defor-mations on () of associating to a deformation on () its pull-back through : () (). If we x and (), the pull-back gives amorphismof functors(, ) ; obviously its surjectivity, that is the surjectivity of themaps(, ) () for , corresponds to the completeness of the deformation () and the bijectivity to the universality; so the universality is related withrepresentability of the functor .Degeneracy locus of amorphism of vector bundles. See Determinantalvarieties.

Degree of an algebraic subset. ([93], [104], [107], [196]). Let be an alge-braically closed eld.Let be an algebraic subset of dimension in . The degree of is dened to be !times the leading coecient of the Hilbert polynomial (see Hilbert function andHilbert polynomial).Equivalently, one can dene the degree of an algebraic variety of dimension into be the number of intersection points of with a generic ( )-dimensional sub-space (i.e., a subspace of complementary dimension) and the degree of an algebraicsubset of dimension in to be the sumof the degrees of its irreducible componentsof dimension .If is the zero locus of a homogeneous polynomial , we have that the degree of isthe degree of .If = , the degree of a -dimensional smooth algebraic variety in is its fun-damental class in 2( ,) (see Singular homology and cohomology). Moreprecisely2( ,) is, where is the fundamental class of a -subspace of ; sothe fundamental class of is for some ; is the degree of .The degree of is also , where is the Fubini-Study form on (see Fubini-Study metric).See also Bezouts Theorem and Minimal degree.

42 | Depth

Depth. ([62], [159], [185]). Let be aNoetherian ring and let be a nitely generated-module. If is an ideal of such that = , we dene the -depth (sometimescalled -grade) of to be the number of the elements of a maximal -regular se-quence in (see Regular sequences). If is also local and is its maximal ideal, wecall depth of the-depth of.AuslanderBuchsbaum theorem. Let be aNoetherian local ring. For any nitely gen-erated -module with nite projective dimension, we have that() + () = (),where is the projective dimension (see Dimension).See also CohenMacaulay, Gorenstein, (arithmetically -,-).

Del Pezzo surfaces. See Surfaces, algebraic -.

De Rhams theorem. ([85], [93], [251]).De Rhams theorem. Let be a real manifold. Let () denote the so-calledde Rham cohomology, which is dened to be the quotient{closed -forms on}{exact -forms on} ,where a -form is said to be closed if = 0 and is said to be exact if there existsa ( 1)-form such that = . Let(,) denote the singular cohomology (seeSingular homology and cohomology). Then we have(,) () .For de Rhams abstract theorem, see Sheaves.

Derived categories and derived functors. ([5], [79], [108], [242]).Denition. Let A be an Abelian category (see Categories). The homotopy categoryof A, denoted by (A), is the following category: the objects are the complexes of objects of A; the morphisms are homotopy equivalence classes of morphisms of complexes

(See Complexes for the denition of homotopically equivalent morphisms ofcomplexes).

We denote by (A) the subcategory of (A) whose objects are the boundedcomplexes, by +(A) the subcategory whose objects are the complexes boundedbelow, and by (A) the subcategory whose objects are the complexes boundedabove.

Derived categories and derived functors | 43

We say that a morphism in(A), where is one among 0, , +, , is a quasi-isomor-phism if it induces an isomorphism in cohomology.Let be a complex 1 +1 ;we denote by () the complex such that () = +1 for any and () = .We call shift operator.Let : be a morphism in (A), where is one among 0, , +, . The cone of, denoted by (), is the complex () with the dierential( ) ( () 0 )( ) .Proposition. Let A be an Abelian category. If : and : are twomorphisms in (A) and is a quasi-isomorphism, then there exist a complex in Aand two morphisms : and : with quasi-isomorphism such that thefollowing diagram commutes:

?? .

``AAAAAAAA

__??????? >>}}}}}}}}

Denition. LetA be anAbelian category. The derived category ofA, denoted by(A)is the following category: the objects of (A) are the complexes of objects of A; a morphism : in (A) is a triplet (, , ), where is a third complex, : and : are homotopy equivalence classes of morphisms of

complexes and is a quasi-isomorphism. We write it in the following way:

__@@@@@@@ ??.

By the proposition above one can dene the composition of two morphisms

__@@@@@@@ ??

44 | Derived categories and derived functors

and

__??????? >>}}}}}}}in (A) in the following way: by the proposition above there exist , , with quasi-isomorphism such that the following diagram is commutative: ;

__??????? ??

__??????? >>||||||||

__??????? ??let the composition be given by the triplet , , . We denote by (A) thesubcategory of(A)whose objects are the bounded complexes, and analogously+(A) and(A).

Let : be the following morphism in (A) (where is one among 0, , +, and so throughout the item): .

__??????? >>~~~~~~~~

We dene the cone () to be the cone of .Denition. We dene the localizing functor from the homotopy category to thederived category A : (A) (A)in the followingway: it is the identity on the set of the objects and, for any morphismin(A), we dene A() to be

__@@@@@@@ ?? .

Observe that A() is an isomorphism in (A) for every quasi-isomorphism in(A).In fact, the idea of derived category is to identify an object of an Abelian category Awith all its resolutions; to do this we consider a category, (A), whose objects are

Derived categories and derived functors | 45

all the complexes of objects in A and the morphisms are dened in such way thattwo quasi-isomorphic complexes are isomorphic in (A). In such a way, any object of A, considered as an element of (A) (that is, 0 0) is isomorphic to all itsresolutions.

Denition. Let C be an additive category and let be an additive automorphism of C(we call shift operator). A triangle in C is a sextuple (, , , , , ) of objects, , in C and morphisms : , : , : (). It is often denoted ().A morphism of triangles is a commutative diagram //

//

//

()() // // // () .

Denition. We say that an additive category C equipped with an additive auto-morphism and with a family of triangles, called distinguished triangles, is a trian-gulated category if the following axioms hold:(1) Every triangle isomorphic to a distinguished triangle is a distinguished triangle.

For every morphism : , there is a distinguished triangle (, , , , , ).The triangle (, , 0, , 0, 0) is a distinguished triangle.

(2) A triangle (, , , , , ) is distinguished if and only if the triangle(, , (), , , ())is distinguished.

(3) Given two distinguished triangles (, , , , , ) and (, , , , , ) andmorphism : and : commuting with and , there existsa morphism : such that (, , ) is a morphism from the rst triangle tothe second.

(4) Given distinguished triangles (, , , , , ),(, , , , , ),(, , , , , ),there exist morphisms : , : , such that(, , , , , ())is a distinguished triangle and (, , ) and (, , ) are morphisms of triangles(i.e., = , = , = , = ).

46 | Derived categories and derived functors

Denition. We say that a functor between two triangulated categories is a -functorif it is additive, if it commutes with the shift operators, and if it takes distinguishedtriangles to distinguished triangles.

A distinguished triangle in(A) is dened to be a triangle isomorphic to a triangleof the form () (),where and are the natural maps () and () ().Analogously we dene the distinguished triangles in (A). With these families ofdistinguished triangles, (A) and(A) are triangulated categories.Denition. Let A and B be Abelian categories and let : (A) (B)be a -functor. A right derived functor of is a -functor : (A) (B)together with a morphism of functors from (A) to(B) : B Awith the following universal property: if : (A) (B)is a -functor and : B Ais a morphism of functors, then there exists a unique morphism : suchthat = ( A) .Analogously, a left derived functor of is a -functor : (A) (B)together with a morphism of functors from (A) to(B) : A B with an analogous universal property.

If exists, it is unique up to isomorphism of functors.Theorem. Let A and B be two Abelian categories and : (A) (B) a -functor.Suppose there exists a triangulated subcategory of(A) such that

Determinantal varieties | 47

(i) for every object of (A), there exists a quasi-isomorphism from it to an objectof ;

(ii) for every exact object of (i.e., () = 0 for any ), we have that () is alsoexact.

Then there exists a right derived functor of and, if is an object of(A) andan object of and they are quasi-isomorphic, then () and () are isomorphicin(B).Let be an additive functor between two Abelian categories. If it is a left exact functor(that is, it takes any exact sequence 0 to an exact sequence 0 () () ()), and exists, then we dene := ()andwe call it the classical -th right derived functor for . Analogously, if is a rightexact functor, we can dene the classical -th left derived functor for , .For instance, letA be anAbelian category such that every bounded complex of objectsin A admits a quasi-isomorphism to a bounded complex of injective objects (we saythat an object in an Abelian category is injective if, for any morphism : and any monomorhism : , there exists a morphism : such that = ), e.g., we can takeA equal to the category of -modules for some commutativering with unity . Let be an object of A. Let be the subcategory of (A) given bythe complexes of of injective objects. Consider the left exact functor = (, )from (A) to () (where is the category of Abelian groups). Then there existsthe derived functor (, ) and(, ) (, ).Let A be the category of -modules for some commutative ring with unity and letbe an -module. The classical -th left derived functor of the right exact functor is (, ).See Ext, EXT and Tor, TOR.

Determinantal varieties. ([15], [77], [104], [106], [209]). Let be an algebraicvariety (or a manifold) and and be two vector bundles on and let : bea morphism of vector bundles. For any , the set() = { | ( : ) }is said to be a determinantal variety (or the -degeneracy locus of ).Example. Take = and = =1,...,O(), = =1,...,O(). Then is given by amatrix whose entry , is a polynomial of degree if and 0 if < and() is the zero locus in of the minors ( + 1) ( + 1) of .

48 | Dimension

For instance, take = 1 for some and . Let = O and = O(1). Call thecoordinates in , for = 1, . . . , , = 1, . . . , . Letbe thematrix such that , = ,.We have that 1() is the image of the Segre embedding (see Segre embedding)1 1 ,(. . . , , . . .), (. . . , , . . .) (. . . , , . . .).Furthermore, () is the -secant variety to (see [104]).Dimension. ([12], [104], [107], [159], [164], [185], [228]). We dene the dimension ofa topological space to be the supremum of the set of all integers such that thereexists a chain 0 1 of distinct irreducible closed subsets of .The (Krull) dimension of a ring is dened to be the supremum of the set of allintegers such that there exists a chain 0 1 of distinct prime idealsof .The dimension of an algebraic variety is its dimension as topological space (withthe Zariski topology).

Let be an algebraically closed eld. By usingHilberts Nullstellensatz (see HilbertsNullstellensatz), we can prove easily that the dimension of an ane algebraic varietyover is the dimension of the ane coordinate ring and the dimension of a projectivealgebraic variety over is the dimensionof thehomogeneous coordinate ringminus 1.Theorem. The dimension of an integral domain that is a nitely generated algebraover a eld is the transcendence degree over of the quotient eld of (see Tran-scendence degree and Localization, quotient ring, quotient eld).

In particular, the dimension of the ane coordinate ring of an irreducible ane alge-braic variety over a eld is equal to the transcendence degree over of its quotienteld.

Proposition. The degree of the Hilbert polynomial (see Hilbert function and Hilbertpolynomial) of a projective algebraic variety is the dimension of the projectivevariety.

The (Krull) dimension of a nonzero -module is dened to be the Krull dimen-sion of /(),where () = { | = 0}.We say that an -module has nite projective dimension if there exists a projec-tive resolution of (see Injective and projective modules,Injective and projectiveresolutions) of the form 0 0 0

Discrete valuation rings | 49

(the number is said to be the length of the resolution). In this case, the projectivedimension of the -module, denoted by (), is the minimum of the lengths ofsuch resolutions.

If = [0, . . . , ] for some eld , then () is the minimum of the length of afree resolution (which is the length of a minimal free resolution), since any nitelygenerated projective[0, . . . , ]-module is free (see [159, Chapter IV, Theorem 3.15]).See also Length of a module.

Direct and inverse image sheaves. ([93], [107]). Let : be a continu-ous map between two topological spaces. Let F be a sheaf on (see Sheaves).The direct image sheaf (push-forward) F is the sheaf on F(1()),for any open subset of .The -