Proof: Giving a k-subalgebra isomorphic to kn is equivalent to specifying n elements e1, . . . en that form a system of orthogonal idempotents, i.e. sat-isfy e2i = 1 for all i and eiej = 0 for i 6= j. Identifying Mn(k) with the endomorphism algebra of ann-dimensional k-vector spaceV, we may regard the ei as projections to 1-dimensional subspaces Vi in a direct product de-composition V =V1 ⊕ · · · ⊕Vn of V. Choosing a vector space isomorphism V ∼= kn sending Vi to the i-th component of kn = k⊕ · · · ⊕k gives rise to the required conjugation.
Proof of Proposition 2.2.8: Of courseMn(k) contains subalgebras isomor-phic tokn, whence the necessity of the condition. Conversely, assume given a k-algebra embedding i : kn → A, and let e1, ..., en be the images of the standard basis elements ofkn. By Rieffel’s Lemma it will be enough to show thatAe1 is a simple leftA-module and that the natural mapk →EndA(Ae1) is an isomorphism. Ifk is algebraically closed, thenA ∼=Mn(k) by Corollary 2.1.7. Lemma 2.2.9 then enables us to assume thatiis the standard diagonal embedding kn ⊂ Mn(k), for which the claim is straightforward. If k is not algebraically closed, we pass to an algebraic closure and deduce the result by dimension reasons.
Corollary 2.2.10 Let A be a central simplek-algebra containing a commu-tativek-subalgebraK which is a Galois field extension ofk of degreen. Then K is a splitting field for A.
Proof: By Galois theory, theK-algebra K⊗kK is isomorphic to Kn (see the discussion after the statement of Lemma 2.3.8 below), and thus is a K-subalgebra of A⊗kK to which the proposition applies.
2.3 Galois Descent 37 Examples 2.3.1 The following special cases will be the most important for us:
• The trivial case Φ = 0 (with anyp, q). This is justV with no additional structure.
• p= 1, q= 1. In this case Φ is given by a k-linear endomorphism of V.
• p = 0, q = 2. Then Φ is the tensor product of two k-linear functions, i.e. a k-bilinear form V ⊗kV →k.
• p= 1, q= 2. This case corresponds to a k-bilinear map V ⊗kV →V. Note that the theory of associative algebras is contained in the last ex-ample, for the multiplication in such an algebra A is given by a k-bilinear map A⊗kA→A satisfying the associativity condition.
So consider pairs (V,Φ) ofk-vector spaces equipped with a tensor of fixed type (p, q) as above. A k-isomorphism between two such objects (V,Φ) and (W,Ψ) is given by a k-isomorphism f : V→∼W of k-vector spaces such that f⊗q ⊗(f∗−1)⊗q : V⊗p ⊗k (V∗)⊗q → W⊗p ⊗k (W∗)⊗q maps Φ to Ψ. Here f∗ : W∗ ∼→V∗ is thek-isomorphism induced by f.
Now fix a finite Galois extension K|k with Galois group G= Gal (K|k).
Denote by VK the K-vector space V ⊗k K and by ΦK the tensor induced onVK by Φ. In this way we associate with (V,Φ) a K-object (VK,ΦK). We say that (V,Φ) and (W,Ψ) become isomorphic over K if there exists a K-isomorphism between (VK,ΦK) and (WK,ΨK). In this situation, (W,Ψ) is also called a (K|k)-twisted formof (V,Φ) or a twisted form for short.
Now Galois theory enables one to classifyk-isomorphism classes of twisted forms as follows. Given ak-automorphism σ: K →K, tensoring byV gives ak-automorphismVK →VKwhich we again denote byσ. EachK-linear map f : VK →WK induces a mapσ(f) : VK →WK defined byσ(f) =σ◦f◦σ−1. If f is a K-isomorphism from (VK,ΦK) to (WK,ΨK), then so is σ(f). The map f → σ(f) preserves composition of automorphisms, hence we get a left action of G = Gal (K|k) on the group AutK(Φ) of K-automorphisms of (VK,ΦK). Moreover, given two k-objects (V,Φ) and (W,Ψ) as well as a K-isomorphism g : (VK,ΦK) →∼ (WK,ΨK), one gets a map G → AutK(Φ) associating aσ =g−1◦σ(g) to σ ∈G. The map aσ satisfies the fundamental relation
aστ =aσ·σ(aτ) for all σ, τ ∈G. (1) Indeed, we compute
aστ =g−1◦σ(τ(g)) =g−1◦σ(g)◦σ(g−1)◦σ(τ(g)) =aσ ·σ(aτ).
Next, let h : (VK,ΦK) →∼ (WK,ΨK) be another K-isomorphism, defining bσ :=h−1◦σ(h) forσ ∈G. Then aσ and bσ are related by
aσ =c−1bσσ(c), (2) where c is the K-automorphism h−1◦g. We abstract this in a general defi-nition:
Definition 2.3.2 LetGbe a group andA another (not necessarily commu-tative) group on which G acts on the left, i.e. there is a map (σ, a)→ σ(a) satisfyingσ(ab) =σ(a)σ(b) andστ(a) =σ(τ(a)) for allσ, τ ∈Ganda, b∈A.
Then a1-cocycle of G with values in A is a mapσ 7→aσ from G toA satis-fying the relation (1) above. Two 1-cocycles aσ and bσ are called equivalent orcohomologousif there exists c∈A such that the relation (2) holds.
One defines the first cohomology set H1(G, A) of G with values in A as the quotient of the set of 1-cocycles by the equivalence relation (2). It is a pointed set, i.e. a set equipped with a distinguished element coming from the trivial cocycle σ 7→1, where 1 is the identity element of A. We call this element the base point.
In our concrete situation, we see that the class [aσ] in H1(G,AutK(Φ)) of the 1-cocycle aσ associated with the K-isomorphism g : (VK,ΦK) →∼ (WK,ΨK) depends only on (W,Ψ) but not on the mapg. This enables us to state the main theorem of this section.
Theorem 2.3.3 For a k-object (V,Φ) consider the pointed setT FK(V,Φ) of twisted(K|k)-forms of(V,Φ), the base point being given by(V,Φ). Then the map (W,Ψ)→[aσ] defined above yields a base point preserving bijection
T FK(V,Φ)↔H1(G,AutK(Φ)).
Before proving the theorem, we give some immediate examples, leaving the main application (that to central simple algebras) to the next section.
Example 2.3.4 (Hilbert’s Theorem 90) Consider first the case when V has dimension n over k and Φ is the trivial tensor. Then AutK(Φ) is just the group GLn(K) of invertible n ×n matrices. On the other hand, two n-dimensional vector k-spaces that are isomorphic over K are isomorphic already over k, so we get:
H1(G,GLn(K)) ={1}. (3) This statement is due to Speiser. The case n = 1 is usually called Hilbert’s Theorem 90 in the literature, though Hilbert only considered the case when
2.3 Galois Descent 39 K|kis a cyclic extension of degreen. In this case, denoting byσa generator of G= Gal (K|k), every 1-cocycle is determined by its valueaσ onσ. Applying the cocycle relation (1) inductively we get aσi = aσσ(aσ). . . σi−1(aσ) for all 1 ≤ i ≤ n. In particular, for i = n we get aσσ(aσ). . . σn−1(aσ) = a1 = 1 (here the second equality again follows from the cocycle relation applied with σ =τ = 1). But aσσ(aσ). . . σn−1(aσ) is by definition the norm of aσ for the extension K|k. Now formula (3) together with the coboundary relation (2) imply the original form of Hilbert’s Theorem 90:
In a cyclic field extensionK|k withGal (K|k) =< σ >each element of norm 1 is of the form σ(c)c−1 with somec∈K.
Example 2.3.5 (Quadratic forms) As another example, assume k is of characteristic different from 2, and takeV to ben-dimensional and Φ a tensor of type (0,2) coming from a nondegenerate symmetric bilinear form< , > on V. Then AutK(Φ) is the group On(K) of orthogonal matrices with respect to < , > and we get from the theorem that there is a base point preserving bijection
T FK(V, < , >)↔H1(G,On(K)).
This bijection is important for the classification of quadratic forms.
To prove the theorem, we construct an inverse to the map (W,Ψ)7→[aσ].
This is based on the following general construction.
Construction 2.3.6 Let A be a group equipped with a left action by an-other groupG. Suppose further thatX is a set on which bothGandAact in a compatible way, i.e. we have σ(a(x)) = (σ(a))(σ(x)) for all x∈ X, a ∈A and σ ∈ G. Assume finally given a 1-cocyle σ 7→aσ of G with values in A.
Then we define thetwisted action of G on X by the cocycle aσ via the rule (σ, x)7→aσ(σ(x)).
This is indeed a G-action, for the cocycle relation yields aστ(στ(x)) =aσσ(aτ)(στ(x)) = aσσ(aττ(x)).
IfX is equipped with some algebraic structure (e.g. it is a group or a vector space), and G and A act on it by automorphisms, then the twisted action is also by automorphisms. The notation aX will mean X equipped with the twisted G-action by the cocycle aσ.
Remark 2.3.7 Readers should be warned that the above construction can only be carried out on the level of cocycles and not on that of cohomology
classes: equivalent cocycles give rise to different twisted actions in general.
For instance, take G = Gal (K|k), A = X = GLn(K), acting on itself by inner automorphisms. Then twisting the usual G-action on GLn(K) by the trivial cocycle σ 7→ 1 does not change anything, whereas if σ 7→ aσ is a 1-cocycle withaσa noncentral element for someσ, thena−σ1σ(x)aσ 6=σ(x) for a noncentralx, so the twisted action is different. But a 1-cocycleG→GLn(K) is equivalent to the trivial cocycle by Example 2.3.4.
Now the idea is to take a cocycle aσ representing some cohomology class inH1(G,AutK(Φ)) and to apply the above construction withG= Gal (K|k), A = AutK(Φ) and X = VK. The main point is then to prove that taking the invariant subspace (aVK)G under the twisted action ofGyields a twisted form of (V,Φ).
We show this first when Φ is trivial (i.e. we in fact prove Hilbert’s Theo-rem 90). The statement to be checked then boils down to:
Lemma 2.3.8 (Speiser) LetK|k be a finite Galois extension with group G andV aK-vector space equipped with asemi-linearG-action, i.e. aG-action satisfying
σ(λv) =σ(λ)σ(v) for all σ ∈G,v ∈V and λ∈K.
Then the natural map
λ: VG⊗kK →V
is an isomorphism, where the superscript G denotes invariants under G.
Before proving the lemma, let us recall a consequence of Galois theory.
Let K|k be a Galois extension as in the lemma, and consider two copies of K, the first one equipped with trivial G-action, and the second one with the action of Gas the Galois group. Then the tensor productK⊗kK (endowed with theG-action given by σ(a⊗b)∼=a⊗σ(b)) decomposes as a direct sum of copies ofK:
K⊗kK ∼=M
σ∈G
Keσ,
where G acts on the right hand side by permuting the basis elements eσ. To see this, write K = k[x]/(f) with f some monic irreducible polynomial f ∈k[x], and choose a root α of f in K. As K|k is Galois, f splits in K[x]
as a product of linear terms of the form (x−σ(α)) for σ ∈ G. Thus using a special case of the Chinese Remainder Theorem for rings (which is easy to prove directly) we get
K⊗kK ∼=K[x]/(f)∼=K[x]/(Y
σ∈G
(x−σ(α))∼=M
σ∈G
K[x]/(x−σ(α)), whence a decomposition of the required form.
2.3 Galois Descent 41 Proof: Consider the tensor product V ⊗k K, where the second factor K carries trivial G-action andV the G-action of the lemma. It will be enough to prove that the mapλK : (V⊗kK)G⊗kK →V⊗kK is an isomorphism. In-deed, by our assumption about theG-actions we have (V ⊗kK)G∼=VG⊗kK, and hence we may identifyλK with the map (VG⊗kK)⊗kK →V ⊗kK ob-tained by tensoring with K. Therefore ifλhad a nontrivial kernelA(resp. a nontrivial cokernel B), thenλK would have a nontrivial kernelA⊗kK (resp.
a nontrivial cokernel B⊗kK).
Now by the Galois-theoretic fact recalled above, the K ⊗k K-module V ⊗kK decomposes as a direct sum V ⊗kK ∼=L
W eσ, withσ(e1) =eσ for σ ∈ G. It follows that (V ⊗k K)G = W e1, whence we derive the required isomorphism (V ⊗kK)G⊗kK ∼=L
W eσ ∼=V ⊗kK.
Remark 2.3.9 We could have argued directly for λ, by remarking that K decomposes as a product K = L
keσ with σ(e1) = eσ according to the normal basis theorem of Galois theory. The above proof, inspired by flat descent theory, avoids the use of this nontrivial theorem.
Proof of Theorem 2.3.3: As indicated above, we take a 1-cocycle aσ
representing some cohomology class in H1(G,AutK(Φ)) and consider the invariant subspace W := (aVK)G. Next observe that σ(ΦK) = ΦK for all σ ∈ G (as ΦK comes from the k-tensor Φ) and also aσ(ΦK) = ΦK for all σ ∈ G (as aσ ∈ AutK(Φ)). Hence aσσ(ΦK) = ΦK for all σ ∈ G, which means that ΦK comes from a k-tensor on W. Denoting this tensor by Ψ, we have defined a k-object (W,Ψ). Speiser’s lemma yields an isomorphism W ⊗k K ∼= VK, and by construction this isomorphism identifies ΨK with ΦK. Thus (W,Ψ) is indeed a twisted form of (V,Φ). If aσ = c−1bσσ(c) with some 1-cocycle σ 7→ bσ and c ∈ AutK(Φ), we get from the definitions (bVK)G =c(W), which is a k-vector space isomorphic to W. To sum up, we have a well-defined map H1(G,AutK(Φ)) → T FK(V,Φ). The kind reader will check that this map is the inverse of the map (W,Ψ) 7→ [aσ] of the theorem.
Remark 2.3.10 There is an obvious variant of the above theory, where instead of a single tensor Φ one considers a whole family of tensors on V. TheK-automorphisms to be considered are then those preserving all tensors in the family, and twisted forms are vector spaces W isomorphic to V over K such that the family of tensors on WK goes over to that on VK via the K-isomorphism. The descent theorem in this context is stated and proven in the same way as Theorem 2.3.3.