aa r X i v : . [ m a t h . N T ] S e p LATTICES AND COHOMOLOGY.
Luis Arenas-Carmona. ∗ Universidad de Chile,Facultad de Ciencias.
Casilla 653, Santiago,[email protected]
Abstract
We give an interpretation of the cohomology of an arithmetically de-fined group as a set of equivalence classes of lattices. We use this interpre-tation to give a simpler proof of the connection established in [11] betweengenus and cohomology.
Galois cohomology is a fundamental tool for the classification of certain algebraicstructures.To be precise, let k be a field, G a linear algebraic group acting on a space V , both defined over k . It is known (see [4]), that if G is defined as the setof automorphisms of a tensor τ on V , e.g., a quadratic form or an algebrastructure, the cohomology set H ( K/k, G K ) classifies the K/k -forms of τ , i.e.,those tensors of the same type also defined over k that become isomorphic to τ over the larger field K (see section 4).It would be reasonable to expect, therefore, that a similar theory were avail-able for structures for which the corresponding automorphism group is not analgebraic group but an arithmetically defined subgroup of an algebraic group.Such a theory is already hinted at in [11]. In this reference, two finitenessresults are proven. The first one deals with the finiteness of the local cohomologyset H ( G w , Γ w ), for an arithmetically defined group Γ. Notations are as in [11].The second one deals with the finiteness of the kernel of the map H ( G , Γ) −→ Y v place of k H ( G w ( v ) , Γ w ( v ) ) , ∗ Supported by Fondecyt, proyecto No. 3010018, and the Chilean Catedra Presidencial inNumber Theory. w ( v ) of K dividing each place v of k . It is the proofof the second result which requires expressing the given kernel in terms of theset of double cosets G k \ G A k / Y w Γ w (see corollary 3.3 in [11]). These double cosets are the same ones that classifythe classes of lattices in a genus. This is the relation we want to pursue.In this paper, we show that the results in [11] are part of a much moregeneral theory that relates cohomology sets of arithmetically defined groupswith certain equivalence classes of lattices in V .Let k be a local or number fields, K/k a Galois extension with Galois group G = G K/k . Let V K , G K denote the sets of K -points of V and G (see section 2).We establish the following result: Proposition 1.1.
There exists a correspondence between G k -orbits of G -stablelattices in V K , that are in the same G K -orbit, and elements of ker (cid:16) H ( G , G Λ K ) i ∗ −→ H ( G , G K ) (cid:17) , where G Λ K is the stabilizer of one particular lattice, and i ∗ the map induced bythe inclusion (see prop. 5.1). The cocycles described above can be thought as equivalence classes of latticesin the same space . We develop a concept of lattices in other spaces , at least forthe case that G is the stabilizer of a family of tensors (see 4). In this context,the following result is obtained: Proposition 1.2.
Assume G is the stabilizer of a family of tensors T on V .Then, the set H ( G , G Λ K ) is in one-to-one correspondence with the set of G k -orbits of G -invariant lattices in other spaces that are isomorphic over K . Theset ker (cid:16) H ( G , G Λ K ) i ∗ −→ H ( G , G K ) (cid:17) , where i is the inclusion, corresponds to those lattices that are in the same spaceas Λ k (see prop. 5.4). The relation between the set of cohomology classes and the set of latticeclasses is established by taking advantage of the long exact sequence in coho-mology that arises from a short exact sequence over K .We analyze the cohomology of the general linear group and its relation toclassification of lattices. In particular, one obtains that the set of G k -orbits offree lattices, that are isomorphic over K to a given lattice, corresponds to theset ker H ( G K/k , G Λ K ) −→ H (cid:16) G K/k , GL Λ K ( V ) (cid:17)! (see prop. 5.9). 2 lattice is defined over k , if it is generated by its k -points. This is a strongercondition that G K/k -invariance. It is necessary to look at localizations to obtaina cohomological characterization of the set of lattice classes defined over k . Weshow that this set corresponds to the kernelker H ( G K/k , G Λ K ) −→ Y v H (cid:16) G w , GL Λ K w ( V ) (cid:17)! , where G w is the local Galois group (see prop. 5.14).Section 5.1 is devoted to the study of the relation between integral cohomol-ogy and genera, ie., how cohomology can be used to study sets of lattices thatbecome isometric over some extension. This is expressed in terms of the notionof cohomological genus . In particular, we recover the results in [11]. In all of this article, k, K, E denote number or local fields of characteristic 0,or algebraic extensions of them. If k is a number field, Π( k ) denotes the set ofplaces of k . Remark 2.1.
By an algebraic group, we mean a linear algebraic group. Allalgebraic groups are assumed to be subgroups of the general linear group of avector space V , of finite dimension, over a sufficiently large algebraically closedfield Ω of characteristic 0. We assume that all localizations of number fieldsinject into Ω. G denotes an algebraic group over Ω. GL ( V ) , SL ( V ) denotethe general and special linear groups over Ω. When we work over a fixed localor number field k , we say that G is defined over k if the equations defining G have coefficients in k (see section 2.1.1 in [10]). This is the case for all groupsconsidered here . For any field E , k ⊆ E ⊆ Ω, we write G E for the set of E -pointsof G , e.g., if G = GL ( V ), the set of E points is denoted GL E ( V ). The sameconventions apply to spaces and algebras. All spaces and algebras are assumedto be finite dimensional.Exceptions to this rule are the multiplicative and additive groups. We denotethe multiplicative group Ω ∗ by G m , and the additive group Ω by G a , whenconsidered as algebraic groups. For the set of k -points we write k ∗ , k . Insteadof ( G m ) k , ( G a ) k .The orthogonal group of a quadratic form Q on V is written O n ( Q ) or O n ( Q, V ), where n = dim Ω ( V ). The set of E -points is denoted O n,E ( Q ).The field on which a particular lattice is defined is always written as asubindex. If K/k an extension of local or number fields and Λ k is a lattice in V k , Λ K denotes the O K -lattice in V K generated by Λ k .If G is an algebraic group acting on a space V , both defined over k , and Λ k isa O k -lattice on V k , the stabilizer of Λ k in G k is denoted G Λ k . If G = GL ( V ), thisset is denoted GL Λ k ( V ). Similar conventions apply to special linear or orthogonalgroups. 3 emark 2.2. Whenever
K/k is a Galois extension of a number field k , and v a place of k , w denotes a place of K dividing v . We assume that one fixedsuch w has been chosen for every v . This convention is also applied for infiniteextension, e.g., K = ¯ k . Remark 2.3. G K/k denotes the Galois group of the extension
K/k . If there isno risk of confusion, we write simply G . If K is not specified, we assume K = ¯ k .If k is a number field and v ∈ Π( k ), we also use the notation G w = G K w /k v .If Γ is a group acting on a set S , S/ Γ denotes the set of orbits and S Γ theset of invariant points. The action of γ ∈ Γ is denoted s s γ , for s ∈ S . In this section we recall some of the cohomology results that we need. Theresults in this section are found in chapter 1 in [5], and p. 13-26 in [10].
Definition 3.1.
Let G be a finite group, A a group provided with a G -action. H ( G , A ) is defined as the quotient H ( G , A ) = n α : G 7→ A (cid:12)(cid:12)(cid:12) α ( hg ) = α ( h ) α ( g ) h o / ≡ , where α ≡ β if and only if there exists a ∈ A such that α ( g ) = a − β ( g ) a g forall g ∈ G . If G acts trivially on A , then H ( G , A ) ∼ = Hom( G , A ) /A , where A actsby conjugation. In what follows we write α g instead of α ( g ).In case that A ⊆ B is a subgroup, there is a long exact sequence0 −→ A G −→ B G −→ ( B/A ) G −→ H ( G , A ) −→ H ( G , B ) , and furthermore, under the natural action of B G on ( B/A ) G ,( B/A ) G /B G ∼ = ker (cid:16) H ( G , A ) −→ H ( G , B ) (cid:17) . (1)To simplify notations, in all that follows we assume that whenever a sequenceof pointed sets . . . −→ U −→ V −→ W −→ X −→ Y −→ Z is written, X, Y, Z denote pointed sets,
W, V, U, . . . denote groups, and W actson X with X/W ∼ = ker( Y −→ Z ) . If A is normal in B , we have in the sense just described a long exact sequence0 / / A G / / B G / / ( B/A ) G EDBCGF@A / / H ( G , A ) / / H ( G , B ) / / H ( G , B/A ) . (2) This result is not found in [5], but can be found in [10] p.22.
4n case A is central in B , the higher order cohomology groups for A are alsodefined, and we have a long exact sequence0 / / A G / / B G / / ( B/A ) G EDBCGF@A / / H ( G , A ) / / H ( G , B ) / / H ( G , B/A ) EDBCGF@A / / H ( G , A ) . Finally, if A and B are both Abelian this sequence extends to cohomology ofall orders (see [13] or [6]). All results of this section can be extended via directlimits to profinite groups acting continuously on discrete groups (see [13], p. 9and p. 42).We are specially interested in the case in which G is the Galois group G K/k of a possibly infinite Galois extension
K/k , where k is a local or number field.In what follows, the subgroups A, B, . . . are groups of algebraic or arithmeticalnature.The following is a known fact, (see [5], ex. 1, p. 16).
Proposition 3.2.
For any finite dimensional algebra A , defined over k , andany algebraic extension K/k , it holds that H ( G K/k , A ∗ K ) = { } . Example 3.3 (Hilbert’s theorem 90) . GL K ( V ) ∼ = (End K ( V )) ∗ . Therefore, H (cid:16) G , GL K ( V ) (cid:17) = { } . K/k -forms.
By a tensor of type ( l, m ) on V , we mean an Ω-linear map τ : V ⊗ l −→ V ⊗ m ,where V ⊗ r = r O i =1 V for r ≥ , V ⊗ = Ω .τ is said to be defined over k , if τ ( V ⊗ lk ) ⊆ V ⊗ mk . All tensors mentioned in thiswork are assumed to be defined over k . GL ( V ) acts on the set of tensors of type( l, m ) by g ( τ ) = g ⊗ m ◦ τ ◦ ( g ⊗ l ) − . It makes sense, therefore, to speak aboutthe stabilizer of a tensor.Let I be any set. By an I -family of tensors, we mean a map that associates,to each element i ∈ I , a tensor t i of type ( n i , m i ). GL ( V ) acts on the set ofall I -families by acting in each coordinate. In all that follows, we say a familyinstead of an I -family unless the set of indices needs to be made explicit. Let T be a family of tensors and H = Stab GL ( V ) ( T ). Then, H is a linear algebraicgroup. 5f K/k is a Galois extension with Galois group G , we get an exact sequence { } −→ H K −→ GL K ( V ) −→ X K −→ { } , where X K is the GL K ( V )-orbit of T . It follows from (1), and example 3.3,that X G K /GL k ( V ) ∼ = H ( G , H K ). The elements of X G K /GL k ( V ) can be thoughtof as isomorphism classes of pairs ( V ′ k , T ′ ) that become isomorphic to ( V k , T )when extended to K . These classes are usually called K/k -forms of ( V, T ), orjust k -forms if K = ¯ k . Observe that two vector spaces of the same dimensionare isomorphic, so we can always assume that the vector space V , in which alltensors are defined, is fixed. Definition 4.1.
We call a pair ( V, T ), where T is a family of tensors on V , a space with tensors , or simply a space . By abuse of language, we identify ( V, T )and ( V, T ′ ) whenever T , T ′ are in the same GL k ( V )-orbit, i.e., if they correspondto the same K/k -form. We say that ( V, T ′ ) is a K/k -form of ( V, T ), if T and T ′ are in the same GL K ( V ) orbit. Example 4.2.
Let Q be a non-singular quadratic form on the space V . Then, O n ( Q ) = Stab GL ( V ) ( Q ). Equivalence classes of non-singular quadratic forms on V k are classified by H (cid:16) G , O n, ¯ k ( Q ) (cid:17) . A space, in this case, is what is usuallycalled a quadratic space. Let k be a local or number field, K/k a Galois extension, G ⊆ GL ( V ) analgebraic group defined over k , Λ k a lattice on V k , L K a G -invariant lattice on V K . Let G = G K/k . Proposition 5.1.
If there is an element ϕ ∈ G K such that ϕ ( L K ) = Λ K ,then a σ = ϕ σ ϕ − is a cocycle, and its class in H ( G , G Λ K ) is independent of thechoice of ϕ , depending only on the orbit of L K under G k . The correspondenceassigning, to every such G k -orbit of O K -lattices, an equivalence class of cocycles,is an injection. The image of this map is ker (cid:16) H ( G , G Λ K ) i ∗ −→ H ( G , G K ) (cid:17) , where i is the inclusion. Proof. G K acts on the set of O K -lattices in V K . Let X be the orbit of Λ K .We have an exact sequence { } −→ G Λ K −→ G K −→ X −→ { } . Hence, by (1), we get X G /G k ∼ = ker (cid:16) H ( G , G Λ K ) −→ H ( G , G K ) (cid:17) . (cid:3) xample 5.2. Using the fact that H (cid:16) G , GL K ( V ) (cid:17) = { } , we obtain thatthe set of GL k ( V )-orbits of G -invariant O K -lattices isomorphic to Λ K is incorrespondence with H (cid:16) G , GL Λ K ( V ) (cid:17) .If G is defined as the stabilizer of a family of tensors, e.g., the unitary groupof a hermitian form or the automorphism group of an algebra, we get a moreprecise result.Recall that in section 4 we identified K/k -forms of ( V, T ) with the corre-sponding GL k ( V )-orbits of families of tensors. Definition 5.3.
Let ( V, T ) be a space. A lattice in ( V K , T ) is a pair (Λ K , T ),where Λ K is a lattice in V K . GL K ( V ) acts on the set of pairs (Λ K , T ′ ), forall families of tensors T ′ , by acting on each component. Two lattices (Λ K , T ),( L K , T ′ ) are said to be in the same space if T , T ′ are in the same GL k ( V )-orbit. Proposition 5.4.
Assume that G is the stabilizer of a family of tensors T on V .The set H ( G , G Λ K ) is in one-to-one correspondence with the set of G k -orbits of G -invariant O K -lattices in the same G K -orbit, in all spaces that are K/k -formsof ( V, T ) . The kernel of the map H ( G , G Λ K ) i ∗ −→ H ( G , G K ) , where i is the inclusion, corresponds to the subset of orbits of lattices that arein the same space as Λ K . Proof.
We have an action of GL K ( V ) on the set of all pairs ( L K , T ′ ), where L K is a lattice and T ′ an I -family of tensors with a fixed index set I . If T isthe orbit of (Λ K , T ), we have a sequence { } −→ G Λ K −→ GL K ( V ) −→ T −→ { } , and the same argument as before applies. Last statement follows from the factthat spaces ( V K , T ′ ) are classified by H ( G K/k , G K ), (see section 4 or [5], p. 15). (cid:3) Remark 5.5.
Recall that Λ K = Λ k ⊗ O k O K . If L K is in the same G k -orbitas Λ K , L K = L k ⊗ O k O K , since G k also acts on V k . Recall that we definedthe cocycle corresponding to L by the formula a σ = ϕ σ ϕ − (see prop. 5.1).This definition does not depend on G , as long as ϕ ∈ G . It follows that theset of G k -orbits of lattices in V k that are isomorphic as O k -modules, and whoseextensions to K are in the same G K orbit, corresponds toker H ( G , G Λ K ) −→ H ( G , G K ) × H (cid:16) G , GL Λ K ( V ) (cid:17)! . (3)In the case that G is the stabilizer of a family of tensors,ker H ( G , G Λ K ) −→ H (cid:16) G , GL Λ K ( V ) (cid:17)! G k -orbits of such lattices in all spaces that are K/k -forms of ( V, T ). Example 5.6.
If Λ k is free, (3) corresponds to the set of G k -orbits of freelattices on V k , whose extensions to K are in the same G K -orbit. Definition 5.7.
We say that an O K -lattice Λ K is defined over k , if Λ K ∼ = O K ⊗ O k Λ k for some Λ k . We say that Λ K is a k -free lattice , if Λ k is free.Assume first that G is the stabilizer of a family of tensors. Definition 5.8.
Let a ∈ H ( G , G Λ K ). We say that a is defined over k , k -freeor in ( V, T ) if some (hence any), lattice in the class corresponding to a has thisproperty. Define L def ( G, K/k,
Λ) = { a ∈ H ( G , G Λ K ) | a is defined over k } , L fr ( G, K/k,
Λ) = { a ∈ L def ( G, K/k, Λ) | a is k -free } , L V ( G, K/k,
Λ) = { a ∈ H ( G , G Λ K ) | a is in ( V K , T ) } , L V def ( G, K/k,
Λ) = L V ( G, K/k, Λ) ∩ L def ( G, K/k, Λ) , L V fr ( G, K/k,
Λ) = L V ( G, K/k, Λ) ∩ L fr ( G, K/k, Λ) . Let F : H ( G , G Λ K ) −→ H ( G , G K ) , (4) F : H ( G , G Λ K ) −→ H ( G , GL Λ K ( V )) , (5)be the maps defined by the inclusions. Then, we have the following proposition: Proposition 5.9.
Assume that Λ k is free. The following identities hold: L V ( G, K/k,
Λ) = ker F , L fr ( G, K/k,
Λ) = ker F , L V fr ( G, K/k,
Λ) = ker F ∩ ker F . (cid:3) Later, we give a similar interpretation to L def . Example 5.10. L fr ( O n ( Q ) , ¯ k/k, Λ) = ker H ( G , O Λ n, ¯ k ( Q )) −→ H (cid:16) G , GL Λ¯ k ( V ) (cid:17)! is in correspondence with the set of isometry classes of free quadratic latticesthat become isometric to Λ k over some extension. Remark 5.11.
Notice that L V , L V def , L V fr can be defined, even if G is not thestabilizer of a family of tensors, as follows: L V ( G, K/k,
Λ) = ker (cid:16) H ( G , G Λ K ) −→ H ( G , G K ) (cid:17) , L V def ( G, K/k,
Λ) = n a ∈ L V ( G, K/k, Λ) (cid:12)(cid:12)(cid:12) a is defined over k o , L V fr ( G, K/k,
Λ) = n a ∈ L V ( G, K/k, Λ) (cid:12)(cid:12)(cid:12) a is free o . In this case, the first and last identities of proposition 5.9 still hold. Notice thatwe can still interpret L V as a set of equivalence classes of lattices, because ofproposition 5.1. 8 he set H ( G , U K ) and the ideal group. Let k be a local or number field, K/k a finite Galois extension. Let G = G K/k , and let U K = O ∗ K denote thegroup of units of O K .For any local or number field E , let I E be its group of fractional ideals, P E the subgroup of principal fractional ideals. There is a natural map α : I k → I K defined by α ( A ) = A N O k O K . Clearly α ( P k ) ⊆ P K , so we get a map α ′ : I k /P k → I K /P K .We apply the general theory to Λ k = O k , G = G m . Any λ ∈ K ∗ acts by A 7→ λ A , for A ∈ I K , whence G Λ K = U K . It follows that, H ( G , U K ) ∼ = ( P K ) G /α ( P k ) . Non-zero prime ideals of O K form a set of free generators for I K (see [7], p.18). Let A ∈ I K . We can write A = Y ℘ ∈ Π( k ) ( Y P| ℘ P β ( P ) ) . If A is G -invariant, all the powers β ( P ) corresponding to prime divisors of thesame prime of k must be equal. In other words: A = Y ℘ ∈ Π( k ) ( Y P| ℘ P ) β ( ℘ ) , (6)where β ( ℘ ) is the common value of β ( P ) for all P dividing ℘ . This ideal is in α ( I k ) if and only if the ramification degree e ℘ divides β ( ℘ ) for all ℘ . Hence, wehave an exact sequence0 −→ ker α ′ −→ ( P K ) G /α ( P k ) −→ Y ℘ ∈ Π( k ) ( Z /e ℘ ) , where the image of the last map corresponds to those ideals of the form (6)that are principal in K . The image of ker α ′ is what we call L def ( G, K/k,
Λ).In particular, since all ideals become principal in some extension, we can takea direct limit, to obtain the long exact sequence:0 −→ I k /P k −→ H ( G ¯ k/k , U ¯ k ) −→ Y ℘ ∈ Π( k ) ( Q / Z ) −→ . A refinement of this argument gives H ( G ¯ k/k , U ¯ k ) ∼ = ( I k ⊗ Z Q ) / ( P k ⊗ Z Z ) , L def ( G, K/k,
Λ) = I k /P k . Recall remarks 2.2 and 2.3. Assume k is a number field. There exist naturallocalization maps F v : H ( G , G Λ K ) → H ( G w , G Λ K w ) , defined by inclusion and restriction. We define G Λ K w = G K w if w is Archimedean.9 emma 5.12. Let F : H ( G , G Λ K ) → H ( G , G K ) be the map induced by theinclusion. If the natural map τ : H ( G , G K ) → Y v ∈ Π( k ) H ( G w , G K w ) is injective, then ker F ⊇ T v ker F v . Proof of lemma.
Immediate from the following commutative diagram: H ( G , G Λ K ) F / / Q v F v (cid:15) (cid:15) H ( G , G K ) τ (cid:15) (cid:15) Q v H ( G w , G Λ K w ) / / Q v H ( G w , G K w ) . (cid:3) Remark 5.13.
If the hypothesis of this lemma is satisfied, one says that G satisfies the Hasse principle over k . Characterisation of L def . L def ( G, K/k,
Λ) is the set of equivalence classesof lattices defined over k that become isomorphic over K . A lattice L K is definedover k if and only if it is generated by its k -points, i.e., L K = O K ( L K ∩ V k ) . This is a local property, being an equality of lattices. On the other hand, forany local place v , all lattices defined over k v are k v -free, i.e., L def ( GL ( V ) , K w /k v , Λ) = L fr ( GL ( V ) , K w /k v , Λ) . The following result is immediate from this observation.
Proposition 5.14. L def ( G, K/k,
Λ) = ker H ( G , G Λ K ) −→ Y v H (cid:16) G w , GL Λ K w ( V ) (cid:17)! . (cid:3) Assume that In all of section 5.1, k is a number field. Definition 5.15.
Let F v be the localization map. Define C gen ( G, K/k,
Λ) = ker Y v F v ! . We call this set the cohomological genus of Λ with respect to G . Proposition 5.16.
For any linear algebraic group G , it holds that C gen ( G, K/k, Λ) ⊆ L def ( G, K/k, Λ) . roof. This follows from proposition 5.14 and the commutative diagram H ( G , G Λ K ) (cid:15) (cid:15) * * UUUUUUUUUUUUUUUUU Q v ∈ Π( k ) H ( G w , G Λ K w ) / / Q v ∈ Π( k ) H (cid:16) G w , GL Λ K w ( V ) (cid:17) . (cid:3) Remark 5.17.
Assume G is the stabilizer of a family of tensors. This result tellsus that the cohomological genus corresponds to a set of equivalence classes ollattices defined over k . In fact, a ∈ C gen ( G, K/k,
Λ) if and only if a correspondsto a lattice, in some K/k -form of ( V, T ), that is in the same G k v -orbit, at everyplace v . Definition 5.18.
We define the
V C -genus of Λ k by the formula V C gen ( G, K/k,
Λ) = C gen ( G, K/k, Λ) ∩ L V ( G, K/k, Λ) . In other words, it is the kernel of the map H ( G , G Λ K ) −→ H ( G .G K ) × Y v ∈ Π( k ) H ( G w , G Λ K w ) . (7)Let G be an arbitrary linear algebraic group. The V C -genus corresponds toa set of G k -orbits of lattices in V k . In fact, it corresponds to a subset of theset of double cosets G k \ G A k /G Λ A k , i.e., the genus of G (see [10], p. 440). Inparticular, the following proposition holds. Proposition 5.19. If G has class number with respect to a lattice Λ k , then(7) has trivial kernel for every Galois extension K/k (compare with corollary 4on p. 491 of [10]). (cid:3)
This, in particular, applies to a group having absolute strong approximation(see [10]). However, we have a stronger result.
Proposition 5.20. If G has absolute strong approximation over k , the map (7)is injective. Proof.
Recall remark 2.2. Let M K , L K be two G -invariant O K -latticesin V K , that are locally in the same G k v -orbit for all v . Then, we can chooseelements σ v ∈ G k v , such that σ v M K w = L K w for every place v , and σ v = 1 atall but a finite number of places. Now, any global element σ , close enough to σ v at all finite places where σ v = 1, and stabilizing M K w = L K w at the remainingfinite places, satisfies σM K = L K , as claimed. (cid:3) The following result is just a restatement of lemma 5.12.
Proposition 5.21. If G satisfies the Hasse principle over k , then V C gen ( G, K/k,
Λ) = C gen ( G, K/k, Λ) . (cid:3) G ⊆ GL ( V ) be a semi-simple group with universal cover e G and funda-mental group µ n . Let K = ¯ k . The short exact sequence { } −→ µ n −→ e G K −→ G K −→ { } , defines a map θ : G k −→ H ( G , F ) = k ∗ / ( k ∗ ) n .Let Λ k be any lattice in V k . The following proposition holds. Proposition 5.22.
With the above notations,
V C gen ( G, K/k, Λ) is in one-to-one correspondence with the genus of G (compare with theorem 8.13 in [10], p.490) . Proof.
It suffices to show that any two G k -orbits in the same genus areidentified over some extension. Without loss of generality, we assume k is non-real. It suffices to check that they are in the same spinor genus (see [2]). Spinorgenera are classified by J k /J nk k ∗ Θ A ( G Λ A k ) , where Θ A ( G Λ A k ) is the image of the local spinor norm (see [1] or [2]). This is afinite set, and the representing adeles can be chosen to have trivial coordinatesat almost all places. Therefore, it suffices to take an extension that contains the n -roots of unity, and n roots of a finite set of local elements. (cid:3) This result allows us to use cohomology to study the genus of any Semisimplegroup.
Let [ A ] be the k ∗ -orbit of the O K -ideal A . Assume thatΛ k = O k ⊕ . . . ⊕ O k | {z } n times . The map det ∗ : H (cid:16) G , GL Λ K ( V ) (cid:17) −→ H ( G , U K ) is the map induced incohomology by the determinant. It is surjective, since det has a right inverse.However, in general it is not injective, as the example below shows. Definition 6.1.
Let L K be a G -invariant lattice in V K , and let a be the cocycleclass corresponding to the GL k ( V )-orbit of L K . We define the determinant classof L K , which we denote det ∗ ( L K ), by:det ∗ ( L K ) = det ∗ ( a ) ∈ H ( G , U K ) ∼ = P G K /α ( P k ) , and we identify it with the corresponding ideal class. The case of an orthogonal group is already considered in [3]. xample 6.2. Using the standard embedding GL ( V ) × GL ( W ) −→ GL ( V ⊕ W ), it is easy to prove that det ∗ (Λ K ⊕ L K ) = det ∗ (Λ K )det ∗ ( L K ). In particular,we obtain that det ∗ ( A ⊕ . . . ⊕ A n ) = [ A . . . A n ].Assume k ⊆ K are local fields with maximal ideals ℘ , P . Assume that ℘ O K = P e . Then,det ∗ ( P ⊕ . . . ⊕ P | {z } e ) = [ P e ] = 1 = det ∗ ( O K ⊕ . . . ⊕ O K | {z } e ) , but the latter lattice is defined over k and the first one is not.Let L def = L def ( GL ( V ) , K/k, Λ). We have the following result:
Lemma 6.3. L def ∩ ker(det ∗ ) = { } . Proof of lemma.
This follows from the fact that all k -defined lattices are ofthe form A k ⊕ O k ⊕ . . . ⊕ O k (see [8], (81:5)). It can also be proved by a diagramchasing argument. (cid:3) Now observe that, for any algebraic group G ⊆ GL ( V ), we have L def ( G, K/k,
Λ) = i − ∗ ( L def ) , where i ∗ is the cohomology map induced by the inclusion. Proposition 6.4. If G ⊆ SL ( V ) , then i − ∗ ( L def ) = ker( i ∗ ) . Proof of proposition.
It is immediate from the commutative diagram H ( G , G Λ K ) / / i ∗ ' ' NNNNNNNNNNNN (cid:8) H (cid:16) G , SL Λ K ( V ) (cid:17) v v nnnnnnnnnnnn H (cid:16) G , GL Λ K ( V ) (cid:17) det ∗ (cid:15) (cid:15) H ( G , U K )that im( i ∗ ) ⊆ ker(det ∗ ). Now recall lemma 6.3. (cid:3) In particular, such a group cannot identify a free lattice to a non-free k -defined lattice over any extension, although it can identify a free lattice to anon- k -defined lattice.In this case, a description of L fr is equivalent to a description of L def , hence L def ( G, K/k,
Λ) can be described without resorting to localization.13
Example:Commutative algebras
Let A k = k n as a k -algebra. This is a space with a tensor of type (2 , A K is the symmetric group S n for any algebraicextension K/k . Asume henceforth that K is an algebraic closure of k .The k -forms of A are all semisimple commutative k -algebras. This set isclassified by H ( G , S n ) = Hom ( G , S n ) / ≡ , where ≡ denotes the conjugation (asan equivalence relation). Recall that any semisimple commutative algebra is asum of fields. If ψ is the map that corresponds to an algebra L k = L i L i,k , asimple computation shows that there is correspondence between the fields L i,k and the orbits of im( ψ ).Let R k denote an O k -subalgebra of A k (not necessarily with 1) of maximalrank as a lattice. Let Γ = S Rn . Then, the image of the map H ( G , Γ) −→ H ( G , S n ) corresponds to the set of isomorphisms classes of algebras whoseextensions to K contain a G -invariant algebra isomorphic to R K . In particular,this includes all lattices defined over k , i.e., all O k -algebras whose extensions to K are isomorphic to R K . Example 7.1.
If Γ is not transitive, then no field can contain an O k -algebrawhose extension to K is isomorphic to R K . This is the case for example of thealgebra R k = O n − k ⊕ A or R k = { a ∈ O nk | a n − − a n ∈ A} if A is an idealdifferent from (1), and n ≥ Remark 7.2.
All result in this paper apply also to lattices over rings of S -integers. Absolute strong approximation must be replaced by strong approxi-mation with respect to S . References [1] L.E. Arenas-Carmona. “Spinor genera under field extensions for skew-hermitian forms and cohomology.” Ph. D. thesis. Ohio-State University.Columbus. 2000.[2]
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