GALOIS DESCENT AND SEVERI-BRAUER VARIETIES. 1. Introduction

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1 GALOIS DESCENT AND SEVERI-BRAUER 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 base-extended 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. Meta-Theorem 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 G-action on V, and whose morphisms are G-equivariant morphisms in C. Then E maps into C [G], and the G-action 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 pointed-set 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 G-action 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 graduate-level background in algebra, and for some topics a course in algebraic geometry, including basics on schemes and projective varieties. We prove the meta-theorem in categories of vector spaces with tensor (e.g., quadratic spaces, algebras, commutative algebras, and central simple algebras), quasi-projective varieties, quasi-coherent sheaves, and locally free sheaves of fixed rank. We use this background to work out the basic facts about Severi-Brauer varieties, as presented in Artin s classic paper [2]. All of this material is classical. We rely on the sources [4] and [5] (k-rationality), [8] (tensors), [10] (Galois descent), [15] (Galois descent and torsors), and [2] (Severi-Brauer varieties). Serre gives a polished account of Galois descent in [15] and an introduction to Severi-Brauer varieties in [14], and Artin gives the canonical in-depth 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. Gille-Szamuely 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 G-Action 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 G-Objects. Definition 2.1. Let G be a group, and let C be a concrete category. We say a left G-action 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 G-action on V in C, and whose morphisms are G-equivariant 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 G-isomorphism 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 G-isomorphic. For V, V in C [G] the set Isom C (V, V ) admits a left G-action 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 G-sets Cohomology Sets. Let G be a finite group and let A be a left G-group, 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 pointed-set structure Torsors. Let G be a finite group, and let A be a left G-group. A principal homogeneous G-set over A, or A-torsor, is a left G-set P that admits a principal transitive right A-action, such that σ (x a) = σ x σa for all x P, σ G, and a A. We say two A-torsors are isomorphic if they admit a G- and A-equivariant set bijection. Let A-Tors G denote the pointed set of isomorphism classes of A-torsors, with distinguished element the left G-set A, with the obvious right A-action. Main Example. For each c Z 1 (G, A), let c A denote the set A with c-twisted G-action σ c x df = c σ σ x and right A-action x a = xa. Then c A is an A-torsor, an affine set for A. Proposition 3.1. There is a pointed-set isomorphism g : A-Tors G H 1 (G, A)

4 4 ERIC BRUSSEL CAL POLY MATHEMATICS taking an A-torsor 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 A-torsor 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 well-defined, 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 G-invariant, g(a) = [1], so g is a map of pointed sets. We show g : H 1 (G, A) A-Tors 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 A-torsor 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 c-twisted 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 A-Tors 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 A-torsor isomorphism. It is G-equivariant: φ( σ (x a)) = φ(x c σ σ a) = c σ σ a = σ c x It is A-equivariant: 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 A-torsor A, g is an isomorphism of pointed sets, and g = g 1, as desired.

5 GALOIS DESCENT Twisted G-Action. 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 G-group 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 G-action and right A-action 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 A-torsor. Proof. The A-action is G-equivariant: 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 A-action 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 A-action 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 G-isomorphism. 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 G-action 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 G-isomorphic 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, A-Tors G, and H 1 (G, A), respectively. Then we have pointed-set isomorphisms [V ]/G f g A-Tors 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 well-defined. Suppose V, V are in [V ], and (V, α ) is in [(V, α )]. Then any G-isomorphism 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 A-torsor isomorphism. It is clearly a setisomorphism, since ψ is an isomorphism, and it is G-equivariant since ψ is G-equivariant: 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 A-equivariance is trivial, and we conclude Isom C (V, V ) and Isom C (V, V ) are isomorphic A-torsors. 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 A-Tors 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 1-1. Suppose f([(v, α )]) = f([(v, α )]) via an A-torsor 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 G-isomorphism. It is an isomorphism as a composition of isomorphisms. Since θ is A-equivariant and Isom C (V, V ) is an A-torsor, 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 G-equivariant 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 G-equivariant. This shows f is 1-1, hence f is a pointed-set 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 k-structure V if there exists an object V in C k such that V W. A morphism g : W W between objects with k-structures V and V has a k-structure if there is a morphism f : V V such that f = g. If W and g have k-structures 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 G-Galois 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 G-fixed object V in C k and to each G-equivariant morphism a morphism of the G-fixed objects. If C is covariant then every G-invariant map W W should factor through a morphism i : V W, and we think of V as a subobject of W whose elements are G-fixed points. If C is contravariant then every G-invariant 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 G-orbits. 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 pointed-set 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 G-action 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 G-actions 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 k-vector spaces with k-linear 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 k-vector 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 G-action, and whose morphisms are G-equivariant morphisms. Every -vector space admits at least one Galois G-action, 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 G-equivariant 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 G-action, 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 well-defined. If W is an object of C [G] then since the G-action on W is Galois, multiplication by k stabilizes W G, so W G is a k-vector space. Then, since any morphism ψ : W W in C [G] is G-equivariant, 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 well-defined 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 G-action 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 G-action 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 G-action 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 k-structure, 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 G-isomorphic 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) G-action on the right factor. We will show the canonical map φ : W G k W v a av is a G-isomorphism. Let {x 1,..., x n } be a k-basis 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 G-equivariant: φ(σ(v x)) = φ(v σ(x)) = σ(x)v = σ(xv) = σ(φ(v x)). We show φ is 1-1: Since φ is G-equivariant, 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 k-basis 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 G-equivariant -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 G-equivariant -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 k-vector space such that V V then the k-dimension 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 G-action on V is Galois if and only if the G-action on c V is Galois for each c Z 1 (G, GL(V ). Since each W isomorphic to V is G-isomorphic 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 pointed-set isomorphism TF k (V ) [V ]/G given by [V ] [V ] G, whose inverse is the fixed point functor. Composing these isomorphisms yields the induced pointed-set 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 k-structure 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 k-vector 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 G-action on (W, Ψ) in C k is Galois if it is Galois on W, and Ψ is G-equivariant: 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 G-action α, and whose morphisms are G-equivariant 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 G-action 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 pointed-set 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 G-action 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 G-isomorphism (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 G-isomorphism 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) G-action 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 G-action 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 pointed-set 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 k-algebras 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 G-action, then B k is a k-algebra of finite type and B is integral over A = B G, hence A is a k-algebra 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 k-algebras. A finite-dimensional commutative algebra A over a field k is a pair (V, Φ) where V is a finite dimensional k-vector 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 k-algebras, and k-algebra morphisms. If /k is a finite Galois field extension with group G, a left Galois G-action 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 finite-dimensional commutative k-algebra, 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 G-Galois extension, and let n be the -algebra. Then the pointed set of twisted forms TF k (k n ) consists of the étale k-subalgebras of n of degree n, which are precisely the k-algebras 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 k-subalgebra of n. Since k is a field, the ideals of A are finite-dimensional k-vector spaces, hence they satisfy the descending chain condition, hence A is artinian. Since A is artinian it is a direct product of local artinian k-algebras, 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 k-algebras of fixed points of n under Galois actions, hence they are isomorphic to k-subalgebras 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 k-subalgebra 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 G-group, 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 k-algebra 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 G-action, as in Definition 7.1. Every A -module N may be viewed as an A-module N A via the map A A. We say a left G-action 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 G-action, and whose morphisms are G-equivariant A -module morphisms. If M is an object in Mod A then M admits a Galois G-action 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 well-defined. If N is an object of Mod A [G] then since the G-action 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 G-equivariant, 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 k-algebra of finite type. Let Mod A and Mod A [G] be the categories of A and A -modules-with-galoisaction 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 pointed-set 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 G-action 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 G-equivariant A -module isomorphism for any object N in Mod A, which is immediate, and that for each G-equivariant 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 G-action, 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 k-schemes. 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 k-scheme X, which is then a k-structure for Y, and a morphism ψ : Y Y of objects with k-structures in C is defined over k if the map arises from a morphism of k-structures 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 G-action 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 k-scheme 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 well-defined 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 G-equivariant under (9.1). It now follows that Spec induces a pointed-set 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 G-action is given by a homomorphism α : G Aut Ck (B) such that the structure map B is G-equivariant, 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 G-equivariant, 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 G-equivariant. 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 k-scheme structure. The scheme quotient Y/G of Y by a group G is a k-scheme X together with a G-invariant k- morphism p : Y k X such that X represents the functor Hom C k (Y k, ) G on the category of affine k-schemes with trivial G-action, 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 G-action, X = Spec B G, p : Y X is the map induced by B G B, and f : Y k Z = Spec C is a G-invariant morphism of affine k-schemes. 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 k-schemes and affine -schemes with Galois G-action. (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 pointed-set isomorphism H 1 (G, Aut C (X )) TF k (X) [d] [( d X )/G] where the (left) G-action 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 locally-ringed-space-theoretic 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 k-scheme. 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] k-topology. 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 k-open set. Similarly if Z X is closed, we call Z a k-closed set. We call the collection of k-open sets of X the k-topology on X. It is generated by the basic k-open 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 k-open set, hence a basic k-open set. However, the k-open sets do not in general generate the topology of X, because not every open set of X is a union of k-open sets. For example if a prime ideal p of A factors into prime ideals P 1,..., P r in A then every k-open set either contains all or none of the set {P 1,..., P r }; the k-open 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 G-equivariant. 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 G-equivariant, and this is equivalent to the G-equivariance of p O Y. Thus the Galois G-action on Y is equivalent to a morphism G Aut OX (p O Y ) under which p O Y is G-equivariant. 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 k-structure O X (U), hence admits a Galois G-action 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 k-algebra 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 G-action 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 k-open 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 k-structure 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 G-action 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 G-action extending the action on B if and only if the saturation ˆT of T is stable under the G-action 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 G-action 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 G-action extending that on B. For the second statement, suppose that T 1 B admits a Galois G-action 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 Quasi-Coherent Sheaves Over Affine Schemes Definition Let k be a field, let X be a k-scheme 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 k-algebra of finite type and Y = Spec B, the process of module sheafification defines a category equivalence Mod B QCoh Y associating to each B-module 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 B-morphism 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 quasi-coherent 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 k-scheme 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 A-module M then p M is the O Y -module corresponding to M A B = M k. If N is the quasi-coherent O Y -module corresponding to the B-module N then p N is the O X -module corresponding to N viewed as an A-module via the map A B.

23 GALOIS DESCENT Galois Action. If N has k-structure M, then as in Proposition 10.3, since k is flat, for any k-open U on X we have N (U ) = M (U), so that each N (U ) has a k-structure. 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 B-module 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 A-module N G Let QCoh Y [G] denote the category whose objects are pairs (N, α), where α is a Galois G-action on N, and whose morphisms are G-equivariant 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 G-equivariant morphism its k-structure. Theorem Suppose /k is a finite Galois extension with group G, X is a k-scheme 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 quasi-coherent O X -module. Then there is a pointed-set 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 k-structures. 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 k-topology on X is the collection {U } of open sets defined over k under the canonical morphism p : X X. We say U is k-open, and its complement is k-closed. The morphism p is finite, hence faithfully flat, open, and closed, by Lemma Since p is closed, every open set of X contains a k-open set, hence an affine k-open set. If U = p 1 (U) is k-open then O X (U ) has the k-structure 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 G-stable affine cover of U. If φ : X X is defined over k then φ is continuous with respect to the k-topologies, and if U X is k-open 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 k-scheme formed by composing Y Spec with Spec Spec k. We say the action is Galois if (a) Y admits a G-stable affine cover, and (b) the structure map Y Spec is G-equivariant, i.e., for each σ G we have a commutative diagram Y σ Y Spec σ Spec The G-stable 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 G-equivariant if φ Hom D (Y, X) G. Let D [G]