aa r X i v : . [ m a t h . AG ] F e b ON ALGEBRAIC GROUP VARIETIES
VLADIMIR L. POPOV
Abstract.
Several results on presenting an affine algebraic groupvariety as a product of algebraic varieties are obtained.
This note explores possibility of presenting an affine algebraic groupvariety as a product of algebraic varieties. As starting point servedthe question of B. Kunyavsky [6] about the validity of the statementformulated below as Corollary of Theorem 1. For some special presen-tations, their existence is proved in Theorem 1, and, on the contrary,nonexistence in Theorems 2–5.Let G be a connected reductive algebraic group over an algebraicallyclosed field k . The derived subgroup D and the connected component Z of the identity element of the center of the group G are respectivelya connected semisimple algebraic group and a torus (see [3, Sect. 14.2,Prop. (2)]). The algebraic groups D × Z and G are not always isomor-phic; the latter is equivalent to the equality D ∩ Z = 1, which, inturn, is equivalent to the property that the isogeny of algebraic groups D × Z → G, ( d, z ) dz , is their isomorphism. Theorem 1.
There is an injective algebraic group homomorphism ι : Z ֒ → G such that ϕ : D × Z → G, ( d, z ) d ι ( z ) , is an isomorphism of algeb-raic varieties. Corollary 1.
The underlying varieties of ( generally nonisomorphic ) algebraic groups D × Z and G are isomorphic. Remark 1.
The proof of Theorem 1 contains more information thanits statement (the existence of ι is proved by an explicit construction). Example 1 ([9, Thm. 8, Proof]) . Let he group G be GL n . Then D =SL n , Z = { diag( t, . . . , t ) | t ∈ k × } , and one can take diag( t, . . . , t ) diag( t, , . . . ,
1) as ι . In this Example, G and D × Z are nonisomorphicalgebraic groups. Proof of Theorem . Let T D be a maximal torus of the group D , andlet T G be a maximal torus of the group G containing T D . The torus T D is a direct factor of the group T G : in the latter, there is a torus S such that the map T D × S → T G , ( t, s ) ts, is an isomorphism of algebraicgroups (see [3, 8.5, Cor.]). We shall show that ψ : D × S → G, ( d, s ) ds, (1)is an isomorphism of algebraic varieties.As is known (see [3, Sect. 14.2, Prop. (1), (3)]),(a) Z ⊆ T G , (b) DZ = G. (2)Let g ∈ G . In view of (2)(b), we have g = dz for some d ∈ D , z ∈ Z ,and in view of (2)(a) and the definition of S , there are t ∈ T D , s ∈ S such that z = ts . We have dt ∈ D and ψ ( dt, s ) = dts = g . Therefore,the morphism ψ is surjective.Consider in G a pair of mutually opposite Borel subgroups containing T G . The unipotent radicals U and U − of these Borel subgroups lie in D . Let N D ( T D ) and N G ( T G ) be the normalizers of tori T D and T G inthe groups D and G respectively. Then N D ( T D ) ⊆ N G ( T G ) in viewof (2)(b). The homomorphism N D /T D → N G /T G induced by thisembedding is an isomorphism of groups (see [3, IV.13]), by which weidentify them and denote by W . For each σ ∈ W , fix a representative n σ ∈ N D ( T D ). The group U ∩ n σ U − n − σ does not depend on the choiceof this representative, since T D normalizes U − ; we denote it by U ′ σ .It follows from the Bruhat decomposition that for each g ∈ G , thereare uniquely defined σ ∈ W , u ∈ U , u ′ ∈ U ′ σ and t G ∈ T G such that g = u ′ n σ ut G (see [5, Sect. 28.4, Thm.]). In view of the definition of S ,there are uniquely defined t D ∈ T D and s ∈ S such that t G = t D s , andin view of u ′ , n σ , u, t D ∈ D , the condition g ∈ D is equivalent to thecondition s = 1. It follows from this and the definition of the morphism ψ that the latter is injective.Thus ψ is a bijective morphism. Therefore, to prove that it is anisomorphism of algebraic varieties, it remains to prove its separability(see [3, Sect. 18.2, Thm.]). We have Lie G = Lie D + Lie T G (see[3, Sect. 13.18, Thm.]) and Lie T G = Lie T D + Lie S (in view of thedefinition of S ). Therefore,Lie G = Lie D + Lie S. (3)On the other hand, it is obvious from (1) that the restrictions of themorphism ψ to the subgroups D × { } and { } × S in D × S are isomor-phisms respectively with subgroups D and S in G . Since Lie ( D × S ) =Lie ( D × { } ) + Lie ( { } × S ) , it follows from (3) that the differentialof morphism ψ at the point (1 ,
1) is surjective. Therefore (see [3, Sect.17.3, Thm.]), the morphism ψ is separable. N ALGEBRAIC GROUP VARIETIES 3
Since ψ is an isomorphism, it follows from (1) that dim G = dim D +dim S . On the other hand, (2)(b) and finiteness of D ∩ Z imply thatdim G = dim D + dim Z . Therefore, Z and S and equidimensional, andhence isomorphic tori. Whence, as ι we can take the composition of anyisomorphism of tori Z → S with the identity embedding S ֒ → G . (cid:3) Theorem 2.
An algebraic variety on which there is a nonconstantinvertible regular function, cannot be a direct factor of a connectedsemisimple algebraic group variety.Proof of Theorem . If the statement of Theorem 2 were not true, thenthe existence the nonconstant invertible function specified in it wouldimply the existence of such a function f on a connected semisimplealgebraic group. Then, according to [10, Thm. 3], the function f /f (1)would be a nontrivial character of this group, despite the fact thatconnected semisimple groups have no nontrivial characters. (cid:3) In Theorems 3, 5 below we assume that k = C ; according to theLefschetz principle, then they are valid for fields k of characteristic zero.Below, topological terms refer to the Hausdorff C -topology, homologyand cohomology are singular, and the notation P ≃ Q means that thegroups P and Q are isomorphic. Theorem 3.
If a d -dimensional algebraic variety X is a direct factorof a connected reductive algebraic group variety, then H d ( X, Z ) ≃ Z and H i ( X, Z ) = 0 for i > d .Proof. Suppose that there are a connected algebraic reductive group R and an algebraic variety Y such that the algebraic variety R is isomor-phic to X × Y . Let n := dim R ; then dim Y = n − d . The algebraicvarieties X and Y are irreducible, smooth, and affine. Therefore (see[7, Thm. 7.1]), H i ( X, Z ) = 0 for i > d, H j ( Y, Z ) = 0 for j > n − d. (4)By the universal coefficient theorem, for any algebraic variety V andevery i , we have H i ( V, Q ) ≃ H i ( V, Z ) ⊗ Q , (5)and by the K¨unneth formula, H n ( R, Q ) ≃ H n ( X × Y, Q ) ≃ L i + j = n H i ( X, Q ) ⊗ H j ( Y, Q ) . (6)Therefore, it follows from (4) that H n ( R, Q ) ≃ H d ( X, Q ) ⊗ H n − d ( Y, Q ) . (7)On the other hand, if K is a maximal compact subgroup of thereal Lie group R , then the Iwasawa decomposition shows that R , as VLADIMIR L. POPOV a topological manifold, is a product of K and a Euclidean space, andtherefore, the manifolds R and K have the same homology. Since thealgebraic group R is the complexification of the real Lie group K , thedimension of the latter is n . Therefore, H n ( K, Q ) ≃ Q because K is aclosed connected orientable topological manifold. Whence, H n ( R, Q ) ≃ Q . This and (7) imply that H d ( X, Q ) ≃ Q . In turn, in view of (5), thisimplies that H d ( X, Z ) ≃ Z because H d ( X, Z ) is a finitely generated(see [4, Sect. 1.3]), torsion free (see [1, Thm. 1]) Abelian group. (cid:3) Corollary 2.
A contractible algebraic variety ( in particular, A d ) ofpositive dimension cannot be a direct factor of a connected reductivealgebraic group variety. Theorem 4.
An algebraic curve cannot be a direct factor of a connectedsemisimple algebraic group variety.Proof.
Suppose an algebraic curve X is a direct factor a connected semi-simple algebraic group R variety. Then X is irreducible, smooth, affine,and there is a surjective morphism π : R → X . In view of rationality ofthe algebraic variety R (see [2, 14.14]), the existence of π implies unira-tionality, and therefore, by L¨uroth’s theorem, rationality X . Hence X isisomorphic to an open subset U of A . The case U = A is impossibledue to Theorem 3. If U = A , then on X there is a nonconstantinvertible regular function, which is impossible due to Theorem 2. (cid:3) Theorem 5.
An algebraic surface cannot be a direct factor of a con-nected semisimple algebraic group variety.Proof.
Suppose there are a connected semisimple algebraic group R andthe algebraic varieties X and Y such that X is a surface and X × Y is isomorphic to the algebraic variety R . We keep the notation of theproof of Theorem 3. Since R is semisimple, K is semisimple as well.Therefore, H ( K, Q ) = H ( K, Q ) = 0 (see [8, §
9, Thm. 4, Cor. 1]).Since R and K have the same homology, and Q -vector spaces H i ( K, Q )and H i ( K, Q ) are dual to each other, this yealds H ( R, Q ) = H ( R, Q ) = 0 . (8)Since R is connected, X and Y are also connected. Therefore, H ( X, Q ) = H ( Y, Q ) = Q . (9)It follows from (6), (8), and (9) that H ( X, Q ) = 0. In view of (5), thiscontradicts Theorem 3, which completes the proof. (cid:3) Remark 2.
It seems plausible that, using, in the spirit of [2], ´etalecohomology in place of singular homology and cohomology, one can
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