GALOIS DESCENT AND SEVERIBRAUER VARIETIES. 1. Introduction


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1 GALOIS DESCENT AND SEVERIBRAUER VARIETIES ERIC BRUSSEL CAL POLY MATHEMATICS 1. Introduction We say an algebraic object or property over a field k is arithmetic if it becomes trivial or vanishes after finite separable base extension. Since such objects or properties owe their existence to the presence of arithmetic gaps in k, i.e., the failure of k to be algebraically closed, we view them as responses to specific arithmetic properties of k, and we study them in order to gain insight into the arithmetic complexity of k, which consists of the features of k responsible for the existence and relative abundance of arithmetic objects and properties. Since the objects to be studied become trivial after a finite separable base extension, they become trivial over a finite Galois extension /k. Thus our goal is to characterize isomorphism classes of objects defined over k that become isomorphic when baseextended to, often to some trivial object. These are called twisted forms, a loosening of isomorphism class. In the ideal outcome each twisted form is obtained from the (trivial) object as a fixed object under a Galois action on the object. We then say that the twisted forms are obtained by Galois descent. Ideally, we obtain the following statement. MetaTheorem Let C : Fields Cat be a functor that takes a field k to a concrete category C k, and a morphism k to the extension functor E : C k C. Let /k a finite Galois extension with group G. Let C [G] be the category whose objects are pairs (W, α) for W an object of C and α a Galois Gaction on V, and whose morphisms are Gequivariant morphisms in C. Then E maps into C [G], and the Gaction defines a fixed object functor F : C [G] C k, so that we have a category equivalence C E k C [G] For each object V in C k, the set of twisted forms of [V ] admits a pointedset isomorphism F TF k (V ) = {[V ] : V C k and E(V ) E(V )} H 1 (G, Aut C (E(V )) TF k (V ) 1
2 2 ERIC BRUSSEL CAL POLY MATHEMATICS taking a class [c] to the class [F ( c E(V ))] TF k (V ), where c E(V ) is E(V ) with twisted Gaction σ c x = c σ σ x. The inverse takes a class [V ] TF k (V ) to the class [c] defined by c σ = φ 1 σφ, for any φ Isom C (E(V ), E(V )). This summary is aimed at students with a graduatelevel background in algebra, and for some topics a course in algebraic geometry, including basics on schemes and projective varieties. We prove the metatheorem in categories of vector spaces with tensor (e.g., quadratic spaces, algebras, commutative algebras, and central simple algebras), quasiprojective varieties, quasicoherent sheaves, and locally free sheaves of fixed rank. We use this background to work out the basic facts about SeveriBrauer varieties, as presented in Artin s classic paper [2]. All of this material is classical. We rely on the sources [4] and [5] (krationality), [8] (tensors), [10] (Galois descent), [15] (Galois descent and torsors), and [2] (SeveriBrauer varieties). Serre gives a polished account of Galois descent in [15] and an introduction to SeveriBrauer varieties in [14], and Artin gives the canonical indepth account of Severi Brauer varieties in [2]. Jahnel treats most of our topics in detail, and kindly specializes Grothendieck s theory of faithfully flat descent ([9]) to Galois descent. GilleSzamuely also give a thorough treatment of all of these subjects (and more) in the valuable resource [8]. Seminal founding papers include Weil ([16]), Châtelet ([6]), Amitsur ([1]), and others; see [2] and [10] for historical background and bibliography. 2. Concrete Categories with Left GAction Recall a category is concrete if it admits a faithful ( forgetful ) functor to the category Set. Examples include the categories of vector spaces, rings, algebras, central simple algebras, and quadratic forms, and these extend to schemes, sheaves, and projective varieties. If C is a category we write Hom C (V, W ), Isom C (V, W ), and Aut C (V ) for the sets of morphisms, isomorphisms, and automorphisms, respectively Left GObjects. Definition 2.1. Let G be a group, and let C be a concrete category. We say a left Gaction on an object V of C is in C if there is a group homomorphism α : G Aut C (V ) Denote by C [G] the category whose objects are pairs (V, α), where V is an object of C and α is a left Gaction on V in C, and whose morphisms are Gequivariant C morphisms. Let [(V, α)] and [V ] denote the isomorphism classes in C [G] and C, respectively. Set [V ]/G = {[(V, α )] : [V ] = [V ]}, the Gisomorphism classes in C [G] that are isomorphic in C. The set [V ]/G has distinguished element [(V, α)], making it a pointed set.
3 GALOIS DESCENT 3 Two actions α and β of G on an object V in C are equivalent if [(V, α)] = [(V, β)] in C [G], i.e., if there is an automorphism b Aut C (V ) such that β(σ) = b 1 α(σ) b for all σ G, i.e., a commutative diagram β(σ) V V We often suppress the action notation, writing σ(x) or σ x instead of α(σ)(x), and consider that two objects V and V in C [G] may be isomorphic but not Gisomorphic. For V, V in C [G] the set Isom C (V, V ) admits a left Gaction defined by σ φ = σ φ σ 1. b b V V α(σ) 3. Galois Cohomology We follow [15, Section I.5] and introduce the first cohomology set and principal homogeneous spaces in the category of left Gsets Cohomology Sets. Let G be a finite group and let A be a left Ggroup, with action written b σ b for b A. A (left) cocycle is a function c : G A such that c στ = c σ σ c τ. We sometimes write (c σ ) for c. Let Z 1 (G, A) denote the set of cocycles. We say cocycles (c σ ) and (c σ) are cohomologous, and write (c σ ) (c σ), if there exists an element b A such that c σ = b 1 c σ σ b for all σ. Now define the degree 1 cohomology set H 1 (G, A) df = Z 1 (G, A)/. The trivial cocycle [1] given by 1 σ = e gives H 1 (G, A) a pointedset structure Torsors. Let G be a finite group, and let A be a left Ggroup. A principal homogeneous Gset over A, or Atorsor, is a left Gset P that admits a principal transitive right Aaction, such that σ (x a) = σ x σa for all x P, σ G, and a A. We say two Atorsors are isomorphic if they admit a G and Aequivariant set bijection. Let ATors G denote the pointed set of isomorphism classes of Atorsors, with distinguished element the left Gset A, with the obvious right Aaction. Main Example. For each c Z 1 (G, A), let c A denote the set A with ctwisted Gaction σ c x df = c σ σ x and right Aaction x a = xa. Then c A is an Atorsor, an affine set for A. Proposition 3.1. There is a pointedset isomorphism g : ATors G H 1 (G, A)
4 4 ERIC BRUSSEL CAL POLY MATHEMATICS taking an Atorsor P to the class [c] of the cocycle c defined by σ x = x c σ for some x P. A different x P determines a cohomologous cocycle, inducing a surjection from elements x P to cocycles c cohomologous to c. The inverse of g is given by g 1 ([c]) = c A. Proof. This is [15, Proposition 33]. If P is an Atorsor and x P then for each σ G we have σ x = x c σ for some c σ A, and we claim c = (c σ ) is in Z 1 (G, A), i.e., c στ = c σ σ c τ. Let s check: στ x = x c στ = σ ( τ x) = σ (x c τ ) = ( σ x) σc τ = x c σ σ c τ Since P is principal, we conclude c στ = c σ σ c τ, as desired. It is easy to see that varying x varies the cocycle in the class [c], so the map g is welldefined, and if c σ = b 1 c σ σ b then c arises from x = xb. Therefore each cocycle in the class of [c] arises from some x P. Since id A is Ginvariant, g(a) = [1], so g is a map of pointed sets. We show g : H 1 (G, A) ATors G [c] c A defines an inverse. It is well defined, since if c c and c σ = b 1 c σ σ b for some b A, then left multiplication by b is an Atorsor isomorphism c A c A: b(σ c x) = bc σ σ x = c σ σ b σ x = σ c bx b(x a) = (bx) a To compute g g we must compute g( c A). Fixing x c A, we obtain a cocycle c by the formula σ c x = x c σ. Since σ c x = c σ σ x by the definition of the ctwisted action, we have c σ = x 1 c σ σ x for x A, and we conclude g( c A) = [c ] = [c], proving g g = id. To show g g = id, suppose P ATors G and x P defines the cocycle (c σ ), so that g(p ) = [c]. Define a function φ : P c A x a a We claim φ is an Atorsor isomorphism. It is Gequivariant: φ( σ (x a)) = φ(x c σ σ a) = c σ σ a = σ c x It is Aequivariant: φ(x a) = a = e a = φ(x e) a = φ(x) a. This proves the claim. Finally, since g clearly maps the distinguished class [1] H 1 (G, A) to the distinguished Atorsor A, g is an isomorphism of pointed sets, and g = g 1, as desired.
5 GALOIS DESCENT Twisted GAction. Let C and C [G] be the concrete categories of Definition 2.1. Let (V, α) be an object of C [G], and set A = Aut C (V ). Then A is a left Ggroup via the action σ a = σ a σ 1 for a A and σ G, and A acts on V on the left, so that σ (a(x)) = σ a σ(x) for a A, σ G, and x V. If (V, α), (V, α ) are in C [G], the induced left Gaction and right Aaction on Isom C (V, V ) are given by σ φ = σ φ σ 1 and φ a = φ a. Proposition 3.2. If (V, α) and (V, α ) are in C [G] and V and V are C isomorphic, then Isom C (V, V ) is an Atorsor. Proof. The Aaction is Gequivariant: σ (φ b) = σ φ b σ 1 = σ φ σ 1 σ b σ 1 = σ φ σb for σ G, b A, and φ Isom C (V, V ). The Aaction is principal: If φ Isom C (V, V ) and b A then φ b = φ if and only if b = φ φ 1 = 1. The Aaction is transitive: If φ, φ Isom C (V, V ) then φ = φ b for b = φ 1 φ, and b is in A. Observe that if (V, α) and (V, α ) are C isomorphic then for each φ Isom C (V, V ) there is a (possibly not commutative) diagram σ V φ V σ for all σ G. The failure of the diagram to commute is encoded in the cocycle V φ V (3.3) c σ = φ 1 σφ Z 1 (G, A) This cocycle is the identity if and only if φ is a Gisomorphism. Definition 3.4. Suppose (V, α) is in C [G], and A = Aut C (V ). For each c Z 1 (G, A) let (V, c α) (or c V if α is understood) denote the object V with twisted Gaction given by for all x V. σ c x = c σ σ(x) Proposition 3.5. Suppose (V, α) and (V, α ) are objects in C [G], V is C isomorphic to V, and φ Isom C (V, V ). Then [(V, α )] = [(V, c α)] for the cocycle c Z 1 (G, A) of (3.3). Proof. We check that the diagram σ cv cv φ φ V V σ
6 6 ERIC BRUSSEL CAL POLY MATHEMATICS commutes for all σ G, proving (V, c α) and (V, α ) are Gisomorphic via φ. Theorem 3.6. Fix (V, α) C [G], put A = Aut C (V ), and let [V ]/G be the pointed set of Definition 2.1. Let [(V, α)], A, and [1] be distinguished elements of [V ]/G, ATors G, and H 1 (G, A), respectively. Then we have pointedset isomorphisms [V ]/G f g ATors G H 1 (G, A) where f, g, and h = g f and their inverses are as follows: (a) f([(v, α )]) = Isom C (V, V ); (b) g(p ) = [c] for c Z 1 (G, A) defined by σ x = x c σ for any x P, and g 1 ([c]) = c A; (c) h([(v, α )]) = [c] for c Z 1 (G, A) defined by c σ = φ 1 σφ for any φ Isom C (V, V ), and h 1 ([c]) = [(V, c α)]. Proof. The map g and its inverse were established in Proposition 3.1. We show f is welldefined. Suppose V, V are in [V ], and (V, α ) is in [(V, α )]. Then any Gisomorphism ψ : V V induces a map ψ : Isom C (V, V ) Isom C (V, V ) defined by ψ (φ) = ψ φ, which we claim is an Atorsor isomorphism. It is clearly a setisomorphism, since ψ is an isomorphism, and it is Gequivariant since ψ is Gequivariant: For if φ Isom C (V, V ) we compute ψ ( σ φ) = ψ σφ = ψ (σ φ σ 1 ) = σ (ψ φ) σ 1 = σ ψ (φ) Showing Aequivariance is trivial, and we conclude Isom C (V, V ) and Isom C (V, V ) are isomorphic Atorsors. This shows f is well defined. Since f([(v, α)]) = Isom C (V, V ) = A, f is a map of pointed sets. We show f is onto. If P ATors G then P = c A for some c Z 1 (G, A) by Proposition 3.1. It is trivial to check that c A = Isom C (V, c V ), hence P = f([(v, c α]), hence f is onto. We show f is 11. Suppose f([(v, α )]) = f([(v, α )]) via an Atorsor isomorphism θ : Isom C (V, V ) Isom C (V, V ), and φ is any element in Isom C (V, V ). We claim ψ = θ(φ) φ 1 : V V is a Gisomorphism. It is an isomorphism as a composition of isomorphisms. Since θ is Aequivariant and Isom C (V, V ) is an Atorsor, we compute for any φ Isom C (V, V ), ψ (φ ) = ψ φ = θ(φ)(φ 1 φ ) = θ(φ ) hence ψ = θ. Since θ is Gequivariant we have ( σ θ)(φ ) = θ(φ ), i.e., σ ψ σ 1 φ = ψ φ, since θ = ψ. Since φ is a bijection, we conclude σ ψ = ψ for all σ, i.e., ψ is Gequivariant. This shows f is 11, hence f is a pointedset isomorphism. We compute h = g f. Suppose [(V, α )] [V ]/G. Then f([(v, α )]) = Isom C (V, V ) by (i), and so h([(v, α )]) = [c] where c is defined by σ φ = φ c σ for any φ Isom C (V, V ). Then c σ = φ 1 σφ, as desired.
7 GALOIS DESCENT 7 It remains to compute h 1. But if h([(v, α )]) = [c] then f([(v, α )]) = c A = Isom C (V, c V ), hence [(V, α )] = [(V, c α)], so h 1 ([c]) = [(V, c α)]. This completes the proof. 4. Twisted Forms, Rationality, and Galois Descent Definition 4.1. Let C : Fields Cat be a functor that takes a field k to a concrete category C k, and a morphism k of fields to the scalar extension functor E : C k C If V is an object in C k we write V for V s scalar extension to. For each V C k the twisted forms of V relative to is the pointed set of isomorphism classes with distinguished element [V ]. TF k (V ) = {[V ] : V C k and V V in C } Conversely, we say an object W of C has a kstructure V if there exists an object V in C k such that V W. A morphism g : W W between objects with kstructures V and V has a kstructure if there is a morphism f : V V such that f = g. If W and g have kstructures we say they are defined over k, and rational over k (see [4] and [5]). Let G be a finite group. Recall a field extension /k is GGalois if G acts by field automorphisms on and k is the field of fixed points. We want to extend this notion to C, defining a functor F : C [G] C k that assigns to each object W in C [G] a Gfixed object V in C k and to each Gequivariant morphism a morphism of the Gfixed objects. If C is covariant then every Ginvariant map W W should factor through a morphism i : V W, and we think of V as a subobject of W whose elements are Gfixed points. If C is contravariant then every Ginvariant W W should factor through a morphism p : W V, and we think of V as a quotient of W, and the fibers of p as Gorbits. The main goal of this paper is to describe situations in which we have the following theorem: Theorem 4.2. The scalar extension and fixed point functors define a category equivalence C E k C [G] Suppose V is an object of C k. Then there is a pointedset isomorphism F H 1 (G, Aut C (V )) TF k (V ) [c] [( c V ) G ]
8 8 ERIC BRUSSEL CAL POLY MATHEMATICS whose inverse takes a class [V ] TF k (V ) to the class [c] defined by c σ = φ 1 σφ, for any φ Isom C (V, V ), as in Theorem 3.6. Definition 4.3. In the situation of Theorem 4.2 we say the Gaction is Galois, and an object V in C k in the image of F is obtained from C by Galois descent. For Theorem 4.2 to be realized we require that all extended objects in C be in C [G] and that all objects of C k be obtained by Galois descent. We then require that E and F be inverse, and that for each object V in C k they induce a category equivalence TF k (V ) [V ]/G. Then we invoke Theorem 3.6, which relates [V ]/G to H 1 (G, Aut C (V ), and shows that each object of C k is the fixed object V under the twisted Gactions defined by elements c Z 1 (G, Aut C (V )). 5. Vector Spaces with Tensors We will investigate twisted forms and Galois descent on the concrete categories of vector spaces, algebras, and more generally vector spaces with the additional structure given by a multilinear tensor. We now define these objects, using [5] as reference Algebras. A algebra is a vector space V equipped with a (bilinear and associative) multiplication map Φ : V V V. We prove below that there is a natural vector space isomorphism Hom (V V, V ) V V V, consequently we will write Φ V V 2 and call Φ a tensor of type (1, 2). Proposition 5.1. There is a natural vector space isomorphism Hom (V V, V ) V V V Proof. We have a canonical isomorphism V 2 Hom (V V, ) = (V 2 ) by the universal property of the tensor product. The canonical isomorphism V Hom (, V ) then defines a balanced pairing (V, (V 2 ) ) Hom (V 2, V ) by composition of functions, hence a unique linear map V (V 2 ) Hom (V 2, V ), hence a map T : V V 2 Hom (V 2, V ) v φ ψ (x y φ(x)ψ(y)v) If {e i } and {φ j } are dual bases for V and V then {e i e j } is a basis for V 2, and since T (e k φ i φ j )(e i e j ) = δ ii δ jj e k we find {T (e k φ i φ j )} is a basis for Hom (V 2, V ), which shows that T is an isomorphism. The inverse of T is given by T 1 : Hom (V 2, V ) V V 2 Φ i,j,k a ijk e k φ i φ j
9 GALOIS DESCENT 9 where the constants a ijk are determined by the expression Φ(e i e j ) = k a ijke k. If A and B are algebras represented by (V, Φ) and (W, Ψ), then a algebra homomorphism ϕ : A B is map f : (V, Φ) (W, Ψ), where f : V W is a vector space map such that f Φ = Ψ (f f). Various algebra properties amount to conditions on Φ. To say that A = (V, Φ) is associative is to say that Φ (id V Φ) = Φ (Φ id V ). To say that A is commutative is to say that Φ = Φ τ, where τ : V V V V is the canonical transposition isomorphism. To say that A has a unit element, see [5, p.432]. To say that A is a central simple  algebra of degree n is to say Φ V V 2, and then that (V, Φ) becomes isomorphic to (M n (L), Ψ L ) over some finite extension L/, where Ψ L is the multilinear map describing matrix multiplication More General Tensors. More generally, a tensor of type (n, m), with n, m 0, is an element of V n V m. We consider the set of pairs (V, Φ), with V a vector space and Φ a tensor of type (n, m). A tensor of type (n, m) on V gives an element of Hom (V m, V n ): If Φ = IJ a IJ (e j1 e jn ) (φ i1 φ im ), then we define Φ Hom (V m, V n ) by Φ(v 1 v m ) = IJ a IJ φ i1 (v 1 ) φ im (v m )e j1 e jn This correspondence is bijective. We have the following familiar structures. 1. If Φ has type (0, 0), then (V, Φ) is a vector space. 2. If Φ has type (0, 2), then (V, Φ) is a quadratic space. 3. If Φ has type (1, 2), then (V, Φ) is a algebra. Definition 5.2. The category of vector spaces with tensor of type (n, m) is the category whose objects are pairs (V, Φ), where V is a vector space and Φ is a tensor of type (n, m) (which we will view as an element of Hom (V m, V n )), and whose morphisms are maps f : (V, Φ) (W, Ψ) given by a morphism f : V W of vector spaces such that f Φ = Ψ f, where f acts on V m and V n by acting on each tensor factor. 6. Galois Descent for Vector Spaces. Definition 6.1. Let C : Fields Cat be the functor that assigns to each field k the category of kvector spaces with klinear maps, and to each morphism k the scalar
10 10 ERIC BRUSSEL CAL POLY MATHEMATICS extension functor E : C k C V V := V k φ φ := φ k id Let /k be a finite Galois extension with group G. Let W be a vector space, and let α : G Aut Ck (W ) be an action in C k, where W is viewed as a kvector space via the map k. We say the action α is Galois (or semilinear) if σ(aw) = σ(a)σ(w) for each a and w W. As in Definition 2.1, let C [G] denote the category whose objects are pairs (W, α) where W is a vector space and α is a Galois Gaction, and whose morphisms are Gequivariant morphisms. Every vector space admits at least one Galois Gaction, given by fixing a basis for W and letting G act on coefficients. In particular the action of G on the right tensor factor of V k is Galois, since for all a, a, σ G, and v V, σ(a (v a)) = v σ(a a) = v σ(a )σ(a) = σ(a )(v σ(a)) = σ(a )σ(v a). Moreover, if φ : V V is in C k, then φ id : V k V k is a Gequivariant map of vector spaces, since for all σ G, v V, and a we compute (φ id σ)(v a) = φ id (v σ(a)) = φ(v) σ(a) = σ(φ(v) a) = (σ φ id )(v a) Thus the extension functor actually maps into C [G]: E : C k C [G] Definition 6.2. Let C [G] be the category of vector spaces with Galois Gaction, as in Definition 4.3. The fixed point functor F : C [G] C k is defined by W W G and ψ ψ W G We show F is welldefined. If W is an object of C [G] then since the Gaction on W is Galois, multiplication by k stabilizes W G, so W G is a kvector space. Then, since any morphism ψ : W W in C [G] is Gequivariant, the restriction of ψ to W G has image in (W ) G. Since F clearly takes the identity morphism to the identity and preserves composition of morphisms, it is a welldefined functor. Theorem 6.3. Let /k be a finite Galois extension with group G, and let C k, C, and C [G] be the categories of vector spaces in Definition 6.1. Then:
11 GALOIS DESCENT 11 (i) The scalar extension and fixed point functors define a category equivalence C E k C [G] F (ii) Suppose V is an object of C k. Then there is an isomorphism of trivial pointed sets H 1 (G, GL(V )) TF k (V ) [c] [( c V ) G ] where c V C [G] is V with twisted Galois Gaction σ v = c σ σ(v). The inverse takes a class [V ] TF k (V ) to the class [c] defined by c σ = φ 1 σφ, for any φ Isom C (V, V ). In particular, H1 (G, GL(V )) = {[1]}, TF k (V ) = {[V ]}, and any Galois Gaction on a vector space is equivalent to the standard one on the coefficients of some basis. Proof. For (i), it is obvious that F E equals the identity, since the Gaction on V k is via the right tensor factor. Thus every object of C k is obtained by Galois descent from its own extension. It remains to show that E F is isomorphic to the identity in C [G], i.e., that each object of C [G] has a kstructure, equal to its fixed point set. We state this as a lemma. Lemma 6.4. Let /k be a finite Galois extension with group G, W is an object of C [G], and f : W W is a morphism in C [G]. Then W is Gisomorphic to W G k, and f (f W G) k id. Thus E F is isomorphic to the identity in C [G]. Proof. Let W G k be vector space with (left) Gaction on the right factor. We will show the canonical map φ : W G k W v a av is a Gisomorphism. Let {x 1,..., x n } be a kbasis for. Write G = {σ 1,..., σ n }, with σ 1 = e. Fix w W, and consider the n elements n n v j = σ i (x j w) = σ i (x j )σ i (w) W G i=1 i=1 By linear independence of characters [11, VI.5.4], the matrix (σ i (x j )) M n () is invertible, so that we may invert this system, and in particular express w = σ 1 (w) as a linear combination of the v j. Thus φ is onto. We show φ is Gequivariant: φ(σ(v x)) = φ(v σ(x)) = σ(x)v = σ(xv) = σ(φ(v x)). We show φ is 11: Since φ is Gequivariant, ker(φ) is stable under G: If φ( v i x i ) = 0 then φ(σ( v i x i )) = σ(φ( v i x i )) = 0. Let {e i } I be a kbasis for W G, then {e i 1} I is
12 12 ERIC BRUSSEL CAL POLY MATHEMATICS a basis for W G k. Since φ(v 1) = v, ker(φ) (W G k 1) = (0). Suppose ker(φ) (0), and w ker(φ) is a nonzero element whose expression w = m i=1 a ie i 1 involves the fewest number of nonzero a i, of all nonzero elements of ker(φ). We may assume that a 1 = 1 is nonzero, after reordering and rescaling if necessary. Since w W G k 1, we may assume a 2 k. Then there exists a σ G such that σ(a 2 ) a 2, hence σ(w) w 0, and since ker(φ) is stable under G we have σ(w) w ker(φ). But since the basis e i is in k, σ(w) is supported on the same e i as is w, so σ(w) w has a shorter expression than w, contradiction. This proves ker(φ) = (0), hence that E F is isomorphic to the identity on objects in C [G]. If f : W W is a Gequivariant morphism, and φ and φ are the maps defined above, then it is immediate that f W G id : W G k (W ) G k is a Gequivariant morphism that equals (φ ) 1 f φ : W G k (W ) G k. Thus E F is isomorphic to the identity in C [G]. It remains to prove (ii). The set TF k (V ) consists of the one class [V ], since if V is any kvector space such that V V then the kdimension of V equals the dimension of V, since the scalar extension functor preserves dimension. Since vector spaces are classified up to isomorphism by dimension, we conclude all twisted forms of V are isomorphic to V, hence TF k (V ) = {[V ]}. Since GL(V ) is linear, we see immediately that the Gaction on V is Galois if and only if the Gaction on c V is Galois for each c Z 1 (G, GL(V ). Since each W isomorphic to V is Gisomorphic to some c V by Theorem 3.6, the set [V ]/G is in C [G], and by Theorem 3.6 again, we have an isomorphism [V ]/G H 1 (G, GL(V )). On the other hand, by (i) the categories C k and C [G] are equivalent, and since an equivalence preserves isomorphism classes, we have a pointedset isomorphism TF k (V ) [V ]/G given by [V ] [V ] G, whose inverse is the fixed point functor. Composing these isomorphisms yields the induced pointedset isomorphisms in the statement of Theorem 6.3. In particular, H 1 (G, GL(V )) = {[1]}. This completes the proof. 7. Galois Descent for Vector Spaces with Tensor We aim to apply descent to vector spaces with the additional structure provided by a tensor. By Theorem 6.3 there is only one kstructure on a given vector space, since vector spaces are classified (up to isomorphism) by dimension. Adding structure to the objects and morphisms restricts the automorphism groups and subdivides the isomorphism classes, creating the prospect of a nontrivial theory of twisted forms. Definition 7.1. Let C : Fields Cat be the functor of Definition 5.2, taking a field k to the concrete category C k of kvector spaces with tensor of type (n, m), and a morphism
13 GALOIS DESCENT 13 k of fields to the extension functor E : C k C defined by E(V, Φ) = (V k, Φ k id ) and E(f) = f k id. Suppose /k is a finite Galois extension with group G, and (W, Ψ) is an object of C. A left Gaction on (W, Ψ) in C k is Galois if it is Galois on W, and Ψ is Gequivariant: if Ψ Hom (W m, W n ) then σ Ψ = Ψ σ for each σ G, where σ acts on W m and W n by acting on each tensor factor. Let C [G] denote the category whose objects are pairs ((W, Ψ), α) with W a vector spaces with tensor Ψ of type (n, m) and Galois Gaction α, and whose morphisms are Gequivariant morphisms. (If (W, Ψ) is an algebra, i.e., Ψ Hom (W 2, W ), then this means each σ is a ring automorphism.) We will omit explicit reference to the action α, and just write (W, Ψ) for a vector space with tensor Ψ and some Galois action. If (V, Φ) is in C k let (V, Φ ) denote the object (V, Φ ) with standard Gaction on scalars. This action is Galois, so under this convention E takes C k into C [G]: Let E : C k C [G] F : C [G] C k denote the fixed point functor, which takes an object (W, Ψ) to (W, Ψ) G := (W G, Ψ (W G ) m) and a morphism f : (W, Ψ) (W, Ψ ) to f (W,Ψ) G. Fix an object (V, Φ) of C k. As in Definition 4.1 and Definition 5.2, the set of twisted forms of (V, Φ) is the pointed set TF k (V, Φ) df = {[(V, Φ )] : (V, Φ ) C k, (V, Φ ) (V, Φ )} Theorem 7.2. Let /k be a finite Galois extension with group G, and let C k, C, and C [G] be the categories of vector spaces with tensor in Definition 7.1. Then: (i) The scalar extension and fixed point functors define a category equivalence C E k C [G] (ii) Suppose (V, Φ) is an object of C k. Then there is a pointedset isomorphism F H 1 (G, Aut (Φ )) TF k (V, Φ) [c] [( c (V, Φ )) G ] where ( c V, c Φ ) C [G] is (V, Φ ) with twisted Galois Gaction σ v = c σ σ(v). The inverse takes a class [(V, Φ )] in TF k (V, Φ) to the class [c] defined by c σ = g 1 σg for all σ G, for any g Isom (Φ, Φ ).
14 14 ERIC BRUSSEL CAL POLY MATHEMATICS Proof. We start with (i). Because of Theorem 6.3 we only have to check that E and F are inverse on the tensors in each category. But if (V, Φ) C k it is obvious that Φ V = Φ, hence that F E equals the identity in C k. On the other hand if (W, Ψ) C [G] then it is easy to see the Gisomorphism (W G ) k W of Theorem 6.3, which is E F on the underlying vector space, induces (Ψ W G) id Ψ. Thus (W, Ψ) G k (W, Ψ) in C [G], hence E F is isomorphic to the identity. This proves (i). For (ii), we note that the set of Gisomorphism classes [(V, Φ )]/G is indeed in C [G]. For since each c σ is an automorphism in C, c σ Φ = Φ c σ by definition of Aut (Φ ). Therefore c σ σ Φ = Φ c σ σ, since the (untwisted) Gaction is Galois. Furthermore since c σ is linear we have the semilinear property c σ σ(av) = σ(a)c σ σ(v) for all a, v V. Therefore every twisted Gaction on (V, Φ ) is Galois, so [(V, Φ )]/G is in C [G]. The functorial equivalence [(V, Φ )]/G TF k (V, Φ) given by F and E now gives the desired pointedset isomorphism exactly as in Theorem 6.3, and we omit the details Example: Algebras of Finite Type. Let C : Fields Cat be the functor that assigns to each field k the category C k of commutative kalgebras of finite type. We check that the additional hypothesis (finite type) is compatible with Theorem 7.2. Suppose /k is a finite Galois field extension with group G. If A is an object of C k then B = A is a algebra, and since /k is finite, B/A is finite, hence B is a algebra of finite type. Therefore E : C k C [G] is well defined. Conversely if B is a algebra of finite type and B admits a Galois Gaction, then B k is a kalgebra of finite type and B is integral over A = B G, hence A is a kalgebra of finite type by [3, Proposition 7.8]. Therefore F : C [G] C k is well defined, and Theorem 7.2 applies to this situation Example: Étale kalgebras. A finitedimensional commutative algebra A over a field k is a pair (V, Φ) where V is a finite dimensional kvector space and Φ is a tensor of type (1, 2) that commutes with the transposition of tensor factors in the domain. Let C : Fields Cat be the functor that assigns to each field k the category C k of finitedimensional commutative kalgebras, and kalgebra morphisms. If /k is a finite Galois field extension with group G, a left Galois Gaction on an object A of C is a group action by ring automorphisms, such that σ(ca) = σ(c)σ(a) for all c and a A. Let C [G] be the corresponding category of pairs (A, α). Since the action is Galois, the fixed point object A G is a finitedimensional commutative kalgebra, so the fixed point functor takes C [G] to C k, and we are in the situation of Theorem 7.2. Let n denote the split étale algebra of degree n. Let ( n, α) C [G] be defined by the standard action σ (a 1,..., a n ) = ( σ a 1,..., σ a n ), which is easily checked to be Galois. Then ( n ) G = k n.
15 GALOIS DESCENT 15 Proposition 7.3. Let /k be a finite GGalois extension, and let n be the algebra. Then the pointed set of twisted forms TF k (k n ) consists of the étale ksubalgebras of n of degree n, which are precisely the kalgebras of the form A = m i=1 J i for fields J i /k contained in, such that m i=1 [J i : k] = n. Proof. Let A be a ksubalgebra of n. Since k is a field, the ideals of A are finitedimensional kvector spaces, hence they satisfy the descending chain condition, hence A is artinian. Since A is artinian it is a direct product of local artinian kalgebras, and since n is reduced, each of these is a field extension of k. If J/k is one of A s direct factors, then J s image under one of the standard projections of n to is nonzero, hence J is isomorphic to a subfield of. Thus A m i=1 J i for fields J i /k contained in. By Theorem 7.2 the twisted forms of k n are the kalgebras of fixed points of n under Galois actions, hence they are isomorphic to ksubalgebras of n, hence they are of the form A i J i, with J i /k field extensions contained in, such that i [J i : k] = n. Conversely, suppose A is a ksubalgebra of n of degree n, so A m i=1 J i with J i /k field extensions contained in, such that i [J i : k] = n. Since each J i is separable we may write J i = k[t ]/(p i ) for a monic irreducible p i k[t ], and since /k is Galois we have (J i ) [T ]/(p i ) ni by the Chinese Remainder Theorem. Therefore A i (J i) i ni n. Since A (k n ), A is a twisted form of k n. We compute Aut ( n ) = S n in C the group of linear ring automorphisms of n, which are completely determined by their action on the n orthogonal idempotents. The induced left action σ b = σ b σ 1 is evidently trivial, since S n permutes the components while G acts componentwise. Since S n is a trivial Ggroup, the cocycle condition reads c στ = c σ c τ, hence Z 1 (G, S n ) = Hom(G, S n ). Two homomorphisms c, c : G S n are cohomologous if c σ = b 1 c σ b for some fixed b S n. Thus H 1 (G, S n ) is the set of homomorphisms up to conjugacy, with distinguished element represented by the trivial homomorphism. 8. Galois Descent for Modules Definition 8.1. Let A be a commutative kalgebra of finite type, and let Mod : Fields/k Cat be the functor that assigns to each field containing k the category Mod A of A  modules and A module homomorphisms, and to each morphism k the scalar extension functor E : Mod A Mod A defined by E(M) = M A A = M k and φ φ k id. Suppose /k is a finite Galois extension with group G, and A admits a left Galois Gaction, as in Definition 7.1. Every A module N may be viewed as an Amodule N A via the map A A. We say a left Gaction on N A in Mod A is Galois (or semilinear) if
16 16 ERIC BRUSSEL CAL POLY MATHEMATICS σ(an) = σ(a)σ(n) for all a A and n N. Let Mod A [G] denote the category whose objects are pairs (N, α) where N is an A module and α is a Galois Gaction, and whose morphisms are Gequivariant A module morphisms. If M is an object in Mod A then M admits a Galois Gaction via the standard action on the scalars, so that E takes Mod A into Mod A [G]: Let E : Mod A Mod A [G] F : Mod A [G] Mod A denote the fixed point functor, which takes N to N G. Note that a Galois action on an A module N is also a Galois action on the vector space N as in Definition 6.1. We show F is welldefined. If N is an object of Mod A [G] then since the Gaction is Galois, multiplication by A stabilizes N G, so N G is an object of mod A. Since any morphism ψ : N N in Mod A [G] is Gequivariant, we have ψ N G : N G (N ) G, hence ψ N G is in Mod A. The remaining requirements are easy to check. Theorem 8.2. Suppose /k is a finite Galois extension with group G, and A is a kalgebra of finite type. Let Mod A and Mod A [G] be the categories of A and A moduleswithgaloisaction defined above. Then: (i) The scalar extension and fixed point functors define a category equivalence Mod E A Mod A [G] F (ii) Suppose M is an object of Mod A. Then there is a pointedset isomorphism H 1 (G, Aut A (M ) TF k (M) [c] [( c M ) G ] whose inverse takes a class [M ] TF k (M) to the class [c] defined by c σ = φ 1 σφ, for any φ Isom A (M, M ). Proof. Since (a) A k and A modules are k and vector spaces, (b) the Gaction on A  modules is Galois with respect to the vector space structure, and (c) E and F are welldefined on Mod A and Mod A [G], all that remains for (i) is to check that E F is isomorphic to the identity in Mod A [G]. The latter amounts to showing the vector space isomorphism µ : (N G ) A A N given by n a an is a Gequivariant A module isomorphism for any object N in Mod A, which is immediate, and that for each Gequivariant morphism φ Hom A (N, N ) we have a commutative diagram φ µ = µ φ N G in Mod A [G], also easy to check.
17 GALOIS DESCENT 17 For (ii) we show that the twisted actions are Galois in Mod A. For each c σ Aut A (M ) we have c σ (am) = ac σ (m), hence σ c am = σ(a)c σ σ(m), as required. Thus if N admits a Galois Gaction, so it is an object of Mod A [G], then the A modules c N are also in Mod A [G], hence [N]/G is in Mod A [G]. The rest of the proof now proceeds as before, and we omit the details. 9. Galois Descent for Affine Schemes 9.1. Affine Schemes. Theorem 7.2 (and Example 7.1) is a theory of descent and twisted forms for algebras of finite type over fields, and their modules. Let C : Fields Cat be the functor that assigns to each field the category C of finitely generated commutative algebras. The prime spectrum functor defines a contravariant category isomorphism Spec : C C B Y = Spec B where C : Fields Cat is the functor that assigns to each field the category C of affine schemes of finite type. We will use this equivalence to extend Theorem 7.2 to the category of affine kschemes. For each morphism k in Fields the scalar extension functor E : C k C X X = X k Spec φ φ = φ k id Spec is given by the fiber product, so that (Spec A) k Spec = Spec (A k ). The functor Spec takes the scalar extension map A A to the projection morphism X X, given by intersecting prime ideals of A with A. We say an object Y of C is rational over k or defined over k if Y X k Spec for some kscheme X, which is then a kstructure for Y, and a morphism ψ : Y Y of objects with kstructures in C is defined over k if the map arises from a morphism of kstructures via E Opposite Actions. Since Spec is contravariant, it reverses all morphisms. Let Aut C (Y ), Hom C (Y, Y ), and Isom C (Y, Y ) denote the opposite groups, acting on the left, so that if a, b Aut C (Y ) equal Spec of a, b Aut C (B), then (ab) = a b = b a, where the operation is the product in the opposite group, and the operation is composition of functions. A left Gaction α : G Aut Ck (B) on B in C [G] induces a left G action α : G Aut C k (Y k ) σ (α(σ))
18 18 ERIC BRUSSEL CAL POLY MATHEMATICS on Y k in C [G], where G is the opposite group, and Y k is the kscheme structure on Y obtained via the morphism Spec Spec k. The left action of G on Aut C (B) induces a left action of G on Aut C (Y ), given by σ φ = σ φ (σ ) 1. The isomorphism G G determined by Spec takes σ to σ = σ 1, so in terms of composition of functions we have a left action of G on Aut C (Y ), (9.1) σ (φ ) = σ φ σ 1 Thus we have a welldefined set H 1 (G, Aut C (Y )). The map Aut C (B) Aut C (Y ) induced by Spec takes an automorphism φ to φ, and we compute ( σ φ) = σ φ (σ 1 ) = σ φ σ 1 = σ (φ ) Thus φ φ is Gequivariant under (9.1). It now follows that Spec induces a pointedset isomorphism H 1 (G, Aut C (B)) H 1 (G, Aut C (Y )) that preserves the cocycle condition: if d = c then d στ = d σ σ d τ. Moreover, if c σ = φ 1 σφ for some φ Isom C (B, B ), as in the derivation of cocycles in Theorem 7.2, then d σ = (φ ) 1 σ (φ ) = σ (φ ) (φ ) 1. Thus the twisted action on Y induced by Spec is σ d y = d σ σ = σ d σ Galois Action. Suppose /k is a finite Galois extension with group G, B is an object of C [G], and Y = Spec B. By Definition 7.1, a Galois Gaction is given by a homomorphism α : G Aut Ck (B) such that the structure map B is Gequivariant, yielding for all τ G a commutative diagram B τ B Thus the corresponding left action α : G Aut C k (Y k ) is Galois if the structure map Y Spec is Gequivariant, i.e., for all τ G we have a commutative diagram τ Y τ Y Spec τ Spec If X is an object in C k then the induced left action of G on E(X) = X is Galois, and if φ : X X is a morphism then E(φ) = φ k id is Gequivariant. Therefore E takes C k into C [G]: E : C k C [G]
19 GALOIS DESCENT Scheme Quotient. Let Y be a scheme, and let Y k denote its kscheme structure. The scheme quotient Y/G of Y by a group G is a kscheme X together with a Ginvariant k morphism p : Y k X such that X represents the functor Hom C k (Y k, ) G on the category of affine kschemes with trivial Gaction, so that for each such scheme Z there is a bijection Hom C k (X, Z) Hom C k (Y k, Z) G given by composition with p. Suppose Y = Spec B admits a Galois Gaction, X = Spec B G, p : Y X is the map induced by B G B, and f : Y k Z = Spec C is a Ginvariant morphism of affine kschemes. Then f is induced by a morphism C B G, hence f factors through p. Thus Let Y/G = Spec (B G ) F : C [G] C k denote the functor sending an object Y in C to the object Y/G in C k and a morphism φ : Y Y to the induced morphism φ/g : Y/G Y /G. We call it the quotient functor. Theorem 9.2. Let /k be a finite Galois extension with group G, let G denote the opposite group, and let Ck and C [G] denote the categories of affine kschemes and affine schemes with Galois Gaction. (i) The scalar extension and quotient functors define a category equivalence C k E F C [G] (ii) Suppose X is an object of Ck. There is a pointedset isomorphism H 1 (G, Aut C (X )) TF k (X) [d] [( d X )/G] where the (left) Gaction on d X is given by σ d y = σ d σ (y). The inverse takes a class [X ] to the cocycle d σ = σ ψ ψ 1 for any ψ Isom C (X, X ). If X = Spec B then dx = Spec c B and ( d X )/G = Spec(( c B) G ), where c = (c σ ) Z 1 (G, Aut C (B)) is defined by c σ = d σ. Proof. Both (i) and (ii) are immediate by Theorem 7.2 and the translations into the opposite category given by the discussion in Section Galois Action on Structure Sheaves Over Affine Schemes We discuss the locallyringedspacetheoretic aspects of a Galois action on an affine scheme. We first prove a lemma that applies in a more general setting. Lemma Suppose /k is finite, and X is a kscheme. Then the projection p : X X is finite, faithfully flat, open, and closed.
20 20 ERIC BRUSSEL CAL POLY MATHEMATICS Proof. Since k is finite and flat, p is finite and flat by base change. Thus p is faithfully flat, open, and closed by [13, I.2.11,14] ktopology. Let /k be a finite field extension, let X = Spec A be an affine k scheme of finite type, let Y = X = Spec A, and let p : Y X be the projection. If U X is an open set then p 1 (U) = U is an open subset of X, since p is continuous. We call U a kopen set. Similarly if Z X is closed, we call Z a kclosed set. We call the collection of kopen sets of X the ktopology on X. It is generated by the basic kopen sets D(f) := Spec A[1/f] = Spec A [1/f], for f A. Since p is closed, by Lemma 10.1, any open subset of X contains a kopen set, hence a basic kopen set. However, the kopen sets do not in general generate the topology of X, because not every open set of X is a union of kopen sets. For example if a prime ideal p of A factors into prime ideals P 1,..., P r in A then every kopen set either contains all or none of the set {P 1,..., P r }; the kopen sets are unions of fibers Galois Action. Suppose /k is a finite Galois extension with group G, Y is an affine scheme of finite type admitting a left Galois G action α : G Aut C k (Y ), X = Y/G, and p : Y X is the projection. From the basic theory of locally ringed spaces each automorphism σ : Y Y of schemes is equivalent to a homeomorphism on the underlying topological space Y together with a structure sheaf isomorphism σ : O Y σ O Y. Proposition The Galois action on Y is equivalent to a morphism G Aut OX (p O Y ) under which the map p O Y is Gequivariant. Proof. Since Y is affine, each morphism σ : O Y σ O Y is completely determined by p (σ) : p O Y p σ O Y. In fact, σ is determined already by its action on B = p O Y (X). The action is Galois if and only if the structure map B is Gequivariant, and this is equivalent to the Gequivariance of p O Y. Thus the Galois Gaction on Y is equivalent to a morphism G Aut OX (p O Y ) under which p O Y is Gequivariant. We investigate the extent to which global sections of O Y are defined over k. If U X is affine then trivially O Y (U ) = O X (U) and O Y (U ) G = O X (U), by Theorem 9.2. If U X is not affine then we obtain the same result, but only because p : Y X is flat: Proposition Suppose Y = Spec B is an object of C [G], X = Y/G, and p : Y X is the quotient map. If U X is any open set, not necessarily affine, then O Y (U ) has the
21 GALOIS DESCENT 21 kstructure O X (U), hence admits a Galois Gaction uniquely determined by the action on B, and O Y (U ) G = O X (U). Thus we have a natural sheaf isomorphism O X (p O Y ) G where (p O Y ) G denotes the subsheaf of p O Y kalgebra O Y (U ) G. that associates to each open set U of X the Proof. Since Y is in C [G] we have a Galois Gaction on B. Let A = BG, so X = Spec A. By definition if {D(f i ) : f i A} is a (finite) cover of U by basic kopen sets then O Y (U ) is the equalizer of the pair of maps i B[1/f i] i<j B[1/f if j ]. This map has the kstructure i A[1/f i] i<j A[1/f if j ], whose equalizer is O X (U), since O X is a sheaf. Since p is flat we conclude that O Y (U ) equals O X (U). Therefore by Theorem 7.2 O Y (U ) admits a left Galois action, which is evidently uniquely determined by the action on B, and we have O Y (U ) G = O X (U). It follows immediately that the canonical morphism O X p O Y actually maps isomorphically onto the subsheaf (p O Y ) G. Remark The familiar Galois action on algebras carries over to global sections on the structure sheaf: If U Y is any open set, a, and x O Y (U), then σ(ax) = σ(a)σ(x). For if {D(g i )} is a finite cover of U then the restriction from U to each basic open set induces an injective algebra homomorphism O Y (U) i B[1/g i]. Thus x = (x i ) O Y (U), and σ(ax) = (σ(ax i )) = σ(a)(σ(x i )) σ O Y (U) i B[1/σ(g i )] The next result gives a criterion for a localization T 1 B to admit a Galois Gaction extending a Galois action on B. We won t actually use this result in the sequel. Proposition Suppose B is an object of C [G] and T B a multiplicative subset. Then T 1 B admits a Galois Gaction extending the action on B if and only if the saturation ˆT of T is stable under the Gaction on B, in which case (T 1 B) G = ( ˆT G ) 1 B G, and (( ˆT G ) 1 B G ) = T 1 B. Proof. We may assume T is saturated, then its complement is the union of prime ideals of B that do not intersect T. Suppose T 1 B admits a Galois Gaction extending the action on B. If P is a prime ideal of B that does not intersect T then T 1 P is a prime of T 1 B, and since the Galois action on algebras preserves prime ideals, G must take P to another prime ideal not intersecting T. Thus G stabilizes T. Conversely if G stabilizes T then T 1 B admits the obvious Galois Gaction extending that on B. For the second statement, suppose that T 1 B admits a Galois Gaction extending the action on B, A = B G, and S = T G = T A. Then clearly (T 1 B) G contains S 1 A, and (S 1 A) is contained in T 1 B. But a prime P of B intersects T if and only if P intersects S: For if f P T then the product of the conjugates of f under G is in P,
22 22 ERIC BRUSSEL CAL POLY MATHEMATICS since P is an ideal, and in T, since T is stable under G. Since the product of conjugates is fixed by G, it is in T A = S, hence P S is nonempty. On the other hand it is obvious that a prime P of B intersects S if and only if P A intersects S, and we conclude P T is empty if and only if P S is empty. This shows that T is in the saturation of S, hence that S 1 B = T 1 B. Now since the tensor product commutes with localization, and A = B, we have (S 1 A) = T 1 B. Now since F E = id Ck by Theorem 7.2, we conclude (T 1 B) G = S 1 A. 11. Galois Descent of QuasiCoherent Sheaves Over Affine Schemes Definition Let k be a field, let X be a kscheme of finite type, and let QCoh : Fields Cat of quasi be the functor that assigns to each field containing k the category QCoh X coherent O X modules, and to each morphism k the scalar extension E : QCoh X QCoh X given on objects by E(M ) = p (M ) and on morphisms E(φ) = p (φ), where p : X X is the projection. We say these objects are defined over k. We sometimes substitute the notations M and φ Sheafification. If B is a fixed kalgebra of finite type and Y = Spec B, the process of module sheafification defines a category equivalence Mod B QCoh Y associating to each Bmodule N the uniquely determined O Y module N whose restriction to each basic open set D(g) is the B[1/g]module N[1/g] = N B B[1/g], and to each Bmorphism φ : N N the uniquely determined O Y morphism φ : N N whose restriction to each D(g) is the B[1/g]module morphism N[1/g] N [1/g] induced by φ. Since these categories are equivalent we have a theory of Galois descent and twisted forms for quasicoherent sheaves over affine schemes, by Theorem 8.2. We recall how the module properties used in Theorem 8.2 sheafify, so that we can correctly translate the theorem. Suppose /k is a finite Galois extension with group G, and Y = Spec B is an affine scheme of finite type that admits a left Galois G action, with quotient the affine kscheme Y/G = X = Spec A, where A = B G. Let p : Y X be the quotient map. Then O Y = p O X is the sheafification of the algebra B = A k, and if M is the O X module corresponding to the Amodule M then p M is the O Y module corresponding to M A B = M k. If N is the quasicoherent O Y module corresponding to the Bmodule N then p N is the O X module corresponding to N viewed as an Amodule via the map A B.
23 GALOIS DESCENT Galois Action. If N has kstructure M, then as in Proposition 10.3, since k is flat, for any kopen U on X we have N (U ) = M (U), so that each N (U ) has a kstructure. A left Galois action of G on N is a homomorphism α : G Aut OX (p N ) extending the Galois action on O Y. If N = Ñ for a Bmodule N and D(f) X is a basic open set, then the action of σ on p N (D(f)) has the form σ : N[1/f] N[1/f]. Let (p N ) G denote the sheaf that assigns to each U X the module of fixed points N (U ) G. It is easy to see that (p N ) G = (N G ), the sheafification of the Amodule N G Let QCoh Y [G] denote the category whose objects are pairs (N, α), where α is a Galois Gaction on N, and whose morphisms are Gequivariant morphisms. Let F : QCoh Y [G] QCoh X denote the functor that assigns to each object (N, α) the O X module (p N ) G, and to each Gequivariant morphism its kstructure. Theorem Suppose /k is a finite Galois extension with group G, X is a kscheme of finite type, and p : X X is the projection, and QCoh X, QCoh X [G] are the categories defined above. Then: (i) The scalar extension and fixed point functors define a category equivalence QCoh E X QCoh X [G] F (ii) Suppose M is a quasicoherent O X module. Then there is a pointedset isomorphism H 1 (G, Aut OX (M ) TF k (M ) [c] [(p ( c M )) G ] whose inverse takes a class [M ] TF k (M ) to the class [c] defined by c σ = φ 1 σφ, for any φ Isom OX (M, M ). Proof. We omit the proof, as it is a straightforward consequence of Theorem 8.2.
24 24 ERIC BRUSSEL CAL POLY MATHEMATICS 12. Galois Descent for Schemes kstructures. Let D : Fields Cat be the functor that assigns to each field the category D of schemes of finite type over, and to each morphism k in Fields the fiber product functor E : D k D taking an object X to X = X k and a morphism φ to φ k id. We say the scheme X is rational over k (or defined over k). The ktopology on X is the collection {U } of open sets defined over k under the canonical morphism p : X X. We say U is kopen, and its complement is kclosed. The morphism p is finite, hence faithfully flat, open, and closed, by Lemma Since p is closed, every open set of X contains a kopen set, hence an affine kopen set. If U = p 1 (U) is kopen then O X (U ) has the kstructure O X (U), since p is flat. The proof of this fact is identical to that of the analogous statement in Proposition 10.3, since O X (U ) is the equalizer with respect to a Gstable affine cover of U. If φ : X X is defined over k then φ is continuous with respect to the ktopologies, and if U X is kopen with preimage U then the induced map O X (U ) O X (U ) is defined over k Galois Action. Suppose /k is a finite Galois extension with group G, and Y is a separated scheme of finite type, which is an object of D. A left action of G on Y is a homomorphism α : G Aut Dk (Y k ) where Y k is the kscheme formed by composing Y Spec with Spec Spec k. We say the action is Galois if (a) Y admits a Gstable affine cover, and (b) the structure map Y Spec is Gequivariant, i.e., for each σ G we have a commutative diagram Y σ Y Spec σ Spec The Gstable affine cover {V i } of Y allows us to define affine schemes {U i = V i /G}, which glue together to form a quotient scheme X = Y/G. If Y and Y are schemes admitting left Galois G actions, the induced right G action on Hom D (Y, Y ) is defined by φ σ = (σ ) 1 φ σ, or σ φ = σ φ σ 1 in composition of functions, as in (9.1), where σ 1 acts as σ. We say φ is Gequivariant if φ Hom D (Y, X) G. Let D [G]
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