differentially algebraic group chunks

Differentially Algebraic Group ChunksAuthor(s): Anand PillaySource: The Journal of Symbolic Logic, Vol. 55, No. 3 (Sep., 1990), pp. 1138-1142Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2274479 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 55, Number 3, Sept. 1990

DIFFERENTIALLY ALGEBRAIC GROUP CHUNKS

ANAND PILLAY

We point out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over K. This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem).

I restrict myself here to showing (Theorem 20) how one can find a large "differentially algebraic group chunk" inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B].

What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1-4 below are known.

Fact 1. The theory of differentially closed fields of characteristic 0 is complete and has quantifier elimination in the language of differential fields (+, *, 0, 1, ', d).

Fact 2. Affine n-space over a differentially closed field is a Noetherian space when equipped with the differential Zariski topology.

Fact 3. If K is a differentially closed field, k c K a differentialfield, and a and b are in k, then a is in the definable closure of k u b iff a E k (where k denotes the differentialfield generated by k and b).

Fact 4. The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, stable).

REMARKS. Fact 1 appears in Sacks [S], and implies that every definable subset of K' is a Boolean combination of "differential Zariski closed sets". Fact 2 is the Ritt-Raudenbusch basis theorem. Another proof appears in [Sr]. Fact 3 is an easy consequence of Fact 1. Fact 4 follows from Facts 1 and 2.

1. Independence: differentially algebraic groups. We fix once and for all a large differentially closed field ( of characteristic 0. k, k',... will denote differential subfields of ( (i.e. substructures), and K, K',... differentially closed subfields of ( (i.e. elementary substructures).

Affine n-space on is endowed with the differential Zariski topology: a set X c (2n is called closed if it is the set of zeros of a collection of differential polynomials in

Received February 20, 1989; revised October 19, 1989. The author was partially supported by NSF grant DMS-8601289.

C) 1990, Association for Symbolic Logic

0022-4812/90/5503-001 7/$0 1.50

1138

DIFFERENTIALLY ALGEBRAIC GROUP CHUNKS 1139

variables X1,.. ., Xn with coefficients in Q. By Fact 2, the collection can be taken to be finite. As usual, any closed X is a finite union of irreducible closed sets.

By a type p e S'(k) we mean a maximal consistent set of k-definable subsets of O' (where consistent means having the finite intersection property). We write p =

tp(d/k) (a- e on) if p is the set of k-definable sets containing a. DEFINITION 5. For p E Sn(k), V(p) = the intersection of all closed sets in p. (So

V(p) is itself closed.) DEFINITION 6. Let a E on and b E Qm. a- and b are independent over k if

V(tp(d/k)) contains an irreducible component of V(tp(d/k)). Fact 7. By Pillay and Srour [PS] the above notion of independence agrees

with the stability theoretic notion of independence, and is in particular sym- metric. Note also: a- and b are independent over K (differentially closed) iff V(tp(d/K)) = V(tp(d/K)).

LEMMA 8. Let K be differentially closed, and p E Sn(K). Then V(p) is irreducible. Conversely if X c on is closed, irreducible and k-definable (k any differentialfield), then there is a unique p e Sn(k) such that X = V(p). p is characterized by the following: a k-definable set Z is in p just if X n Z contains a (dense) open subset of X; p is denoted p(X, k).

PROOF. Pillay and Srour [PS]. DEFINITION 9. Let X c (2 be closed, irreducible and k-definable. a E Qn is said to

be a generic point of X over k if X = V(tp(d/K)). LEMMA 10. Let X c on and y c Qm be closed, irreducible, k-definable sets. Then

X x Y is a closed, irreducible, k-definable subset of Qnm, and for a e Qn and b e QOm (b, a) is a generic point of X x Y over k if and only if a- is a generic point of X over k, b is a generic point of Y over k, and a- and b are independent over k.

PROOF. Suppose X x Y = Z1 u Z2, where the Zi are closed subsets of 2n I m.

Suppose the Zi are definable over k' D k. Let a- and b be independent over k' and be generic points over k' of X and Y respectively. So (a-, b) e Zi for i = 1 or 2. Suppose without loss that (a-,b) e Z1. Now X0 = {x e X: (x,b) e Z1} is closed, k'- definable and contains a generic point (d) of X over k'. By Lemma 8, X0 = X, whereby X x {b} c Z. Similarly YO = {y e Y: X x {y} c Z1} is k'-definable, closed (why?) and contains a generic point (b) of Y over k'. Again, by Lemma 8, YO = Y. whereby X x Y ' Z1. So X x Y is irreducible. The same argument proves the second part of the lemma.

DEFINITION 1 1. The affine sets are the closed subsets of on , n < wo. A set U c on is called quasiaffine if it is an open subset of a closed X c on . A quasiaffine set U is said to be irreducible if it is an open subset of an irreducible closed X c (2n.

Note. If U is quasiaffine irreducible and k-definable, then the closure of U is k- definable and irreducible. Then we will say that a e on is a generic point of U over k if a is a generic point of the closure of U over k.

DEFINITION 12. A morphism between quasiaffine sets U c on and V c Qm is a map f: U -* V which is locally a quotient of differential polynomials (i.e. there is a (finite) covering of U by open Ui such that, on each Ui, f = fi/gi are differential polynomials and gi =A 0 on U}). Note that a morphism is continuous for the differential Zariski topology.

1140 ANAND PILLAY

DEFINITION 13. (i) A differential prevariety is a topological space V together with a covering V1,..., V, of V by open subsets, and homeomorphisms gi of V with quasiaffine Ui such that the induced transition maps between open subsets of the UL are morphisms.

(ii) A morphism between differential prevarieties and the product of prevarieties is defined in the obvious way.

(iii) A differential variety is a differential prevariety V such that the diagonal is closed in V x V.

Note. It is clear what it means for a quasiaffine set and a morphism between quasiaffine sets to be defined over a differential field k. We will say that a variety V is defined over k if the Ui in Definition 13(i) as well as the transition maps are defined over k.

DEFINITION 14. A differentially algebraic group is an object which is a group and also a differential variety such that multiplication and inversion are morphisms G x G -> G and G -+ G respectively.

(N.B. This agrees with Kolchin's definition [K].)

2. Stable groups: differential algebraic group chunks. A stable group G is a group first order definable in a model N of a stable theory T. We may refer to N as the ambient model. An co-stable (or totally transcendental) group G is a group defined in a model N of an co-stable theory T. [Note that if N = Q, our large differentially closed field, then G is a "constructible" subset of some affine space Q( and the graph of multiplication is a "constructible" subset of Q?n2. As such G has no a priori differentially algebraic-geometric structure.]

A stable group G is said to be connected if it has no definable (in the sense of the ambient model) proper subgroup of finite index.

Fact 15. If G is co-stable then G has the DCC on definable subgroups. In particular, there is a smallest definable subgroup Go of G of finite index-the connected component of G.

If T is a stable theory and N a large model of T, then we have a notion of independence: a- and b are independent over A, where a-, b, A c N. If N = Q as in ?1, this agrees with the notion mentioned in ?1, as we have already pointed out.

Fact 16. Let G be a connected stable group, G c Nn, G defined over M -< N (M a model). Then there is a type p e Sn(M) with the following (characteristic) properties: (i) G E p, and (ii) if X c G is an M-definable set then X e p iffinitely many translates of X cover G. (See Poizat [P]). p is called the generic type of G over M; p = p(G, M). a e Nn is called a generic point of G over M if p = tp(U/M).

Note. If G is connected and co-stable, the generic type of G over M can also be characterized as the (unique) type p E Sn(M) which contains G and has Morley rank equal to the Morley rank of G.

Fact 17. Let G be a connected stable group G c Nn, G defined over M. Let b E G and let a- be a generic point of G over M with a- and b independent (in the stability- theoretic sense) over M. Then a-- 1, a- * b and b a- are generic points of G over M, and each is independent from {b, b' } over M. (See Poizat [P].)

We also point out a fact about stability-theoretic independence which streamlines subsequent arguments.

DIFFERENTIALLY ALGEBRAIC GROUP CHUNKS 1141

Fact 18. Let N be a big model of a stable theory, M -< N and p E SJ(M). Let Z be an M-definable subset of N".+m Then {b E Nm:for some (any) a- with tp(d/M) = p and a and b independent over M, (d, b) E Z} is an M-definable subset of Nm.

Note what this implies in the context of ?1. Fact 19. Let V1 c (n and V2 c Qm be quasiaffine irreducible sets, both defined

over K c Q. Let a- and b be independent over K generic points of V1 and V2 over K, and suppose Z c QOlm is a K-definable (constructible) set containing (d, b). Then W = - V1: for some (any) generic point c of V2 over K <x>, (x,c) E Z} contains an open (dense) K-definable subset of V1.

Proof. By Fact 18 and ?1, W is a K-definable set. By assumption a- e W, i.e. W E tp(d/K). By Lemma 8, tp(a/K) is generated as a filter by the open K-definable subsets of V. Thus there is V' c V1, V' open and V' c W (as V1 is irreducible, V' is dense in V1).

In the statement of the following theorem "generic" has the meaning of ?1. THEOREM 20. Let G be a connected group first order definable (constructible) in

the large differentially closed field Q. Suppose G c O' and G is defined over K c Q. Then there are V c G and U c V x V, both defined over K, such that

(i) V is quasiaffine and irreducible, and finitely many translates of V cover G. (ii) U is open (dense) in V x V, (iii) the multiplication map (a, b) -+a * b restricted to U is a morphism m: U - V, (iv) the inversion map a a-1 restricted to V is a morphism i: V -> V, and (v) for any b e V, for some (any) generic point a of V over K , (a, b) e U and

a-,a b) e U. Note. V is what we call a differentially algebraic group chunk. PROOF. Note that "connected" has the stable group meaning here. First, let

p = p(G,K) as in Fact 16. By Lemma 8, X = V(p) is irreducible and K-definable, and by Lemma 8 again there is a (principal) open K-definable V1 c X such that V1 c G. (Note that p(X, K) = p(G, K).)

(a) Note that V1 is quasiaffine and irreducible, and moreover a generic point of V1 over K in the sense of ?1 is precisely a generic point of G over K in the sense of Fact 16, the same being also true if we replace V1 by an open K-definable subset of V1.

(b) Note also that if Z is a K-definable subset of V1, then Z e p iff Z contains a (dense) open K-definable subset of V1.

Now let a be a generic point of V1 over K. So a-' is definvable over K<a> and moreover, by Fact 17 and (a), a- 1 e V1. By Fact 3, a-' = f (a)/g(a), where f and g are differential polynomials over K and g(a) =A 0. By (b) above, {x e V1: g(x) =A 0 and x-' = f (x)/g(x) e V1 } contains an open K-definable set V2 . Thus the inversion i: V2 -+ V1 is a morphism.

Now, let a and b be independent generic points of V2 over K. By Lemma 10 and the note following Definition 11, (a, b) is a generic point over K of the irreducible quasiaffine set V2 x V2 . As a * b is definable over K<a, b>, we can as above find an open K-definable U1 C V2 x V2 such that multiplication m: U1 -+ V1 is a morphism.

Now, note that if a and b are independent (over K) generic points of V2 over K, then, by (a) above and the previous paragraph, both (a, b) and (a-', a * b) are generic points of V2 x V2 over K and thus are contained in U1. Thus by Fact 19 there is an

1142 ANAND PILLAY

open K-definable subset V3 of V2 such that for all b E V3 and for generic a of V2 over K , (a,b) E U1 and (a-', a* b) E U1.

Now let V = {a E V3: a-' E V3}, and U = {(a, b) E U1,n (V x V): a * b E V}. By the continuity of i and m, V is open dense in V1 and U is open dense in V x V. Easily (i), (ii), (iii), and (iv) of the theorem hold for U and V (finitely many translates of V cover G because V E p(G, K)). Now for (v), let b E V and let a be a generic point of V over K; so by the choice of V3 we have (a,b) E U1 n (V x V). By Fact 17, a * b is a generic point of V1 over K and thus a * b E V. So (a-', a * b) E U1 n (V x V), and clearly (a,b), (a-', a * b) E U.

This completes the proof of Theorem 20. Now let G1 be a group definable in ( and G2 a differentially algebraic group

covered by open sets homeomorphic to quasiaffine sets U1,..., Uk. A map f: G1 -+

G2 is said to be a definable isomorphism if, first, f is a group isomorphism and, second, for each i the restricted map f: f -'(Ui) -+ U is definable in Q.

THEOREM 21. Let G be a groupfirst order definable in Q, and defined over K c Q (K differentially closed). Then there is a differentially algebraic group G' defined also over K and a K-definable isomorphism of G with G'.

PROOF. As G is wo-stable we may easily assume G to be connected. Let V be as in Theorem 20. G can be covered by finitely many translates of V. As in Weil [W] or [B], Theorem 20 can be used to show that this covering induces a well-defined structure of a variety on G with respect to which multiplication and inversion are morphisms. The covering of G by, say, V, a1, V,..., a, * V is a covering by sets homeomorphic to the quasiaffine set V. As K is a model, the ai may be chosen in K, and thus this differentially algebraic group is defined over K. The rest follows easily.

Note. It seems to be unknown whether in Theorem 21 we can replace K by a differential field k throughout.

REFERENCES

[B] E. BOUSCAREN, Model theoretic versions of Weil's theorem on pregroups, The model theory of groups (stable group seminar, Notre Dame, Indiana, 1985-1987; A. Nesin and A. Pillay, editors), Notre Dame Mathematical Lectures, vol. 11, Notre Dame University Press, Notre Dame, Indiana, 1989, pp. 177-185.

[K] E. R. KOLCHIN, Differentially algebraic groups, Academic Press, Orlando, Florida, 1985.

[PS] A. PILLAY and G. SROUR, Closed sets and chain conditions in stable theories, this JOURNAL, vol. 49 (1984), pp. 1350-1362.

[P] B. POIZAT, Groupes stables, avec types generiques reguliers, this JOURNAL, vol. 48 (1983), pp. 339-355.

[S] G. SACKS, Saturated model theory, Benjamin, Reading, Massachusetts, 1972.

[Sr] G. SROUR, The notion of independence in categories of algebraic structures. Part I: Basic properties, Annals of Pure and Applied Logic, vol. 38 (1988), pp. 185 - 211.

[W] A. WEIL, On algebraic groups of transformations, American Journal of Mathematics, vol. 77

(1955), pp. 355-391.

DEPARTMENT OF MATHEMATICS

UNIVERSITY OF NOTRE DAME

NOTRE DAME, INDIANA 46556

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