Meta-Theorem 1. Suppose K/k is a finite G-Galois extension. Let C K C k

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1 Galois Descent and Severi-Brauer Varieties. Eric Brussel Emory University Throughout this paper / will be a finite field extension. The goal of Galois descent is to characterize objects defined over that are isomorphic over, and to give criteria for -objects to be defined over. Ideally, we obtain the following. Meta-Theorem 1. Suppose / is a finite G-Galois extension. Let C C = {-objects admitting a Galois G-action, and G-equivariant -morphisms} = {-objects and -morphisms} Then there is a category equivalence F : C C defined by F (V ) = V G (or F (V ) = V/G) and F (φ) = φ F (V ), whose inverse is base extension to. Once we have Meta-Theorem 1, we fix a distinguished object V in C, and let TF (V ) denote the set of isomorphism classes of objects in C that become -isomorphic to V over. This set is classified as follows. Meta-Theorem 2. In the situation of Meta-Theorem 1, there is a bijection TF (V ) H 1 (G, Aut (V )) [V ] [c] where if φ : V V then c σ = φ 1 σφ, and if c Z 1 (G, Aut(V )) then V = ( c V ) G (or the quotient V = ( c V )/G), where c V is V equipped with the twisted Galois G-action σ c x = c σ σ x. This program sees to generalize Galois theory: In the categories of one-dimensional and -algebras, we have, so is obtained from by scalar extension, and Galois theory asserts that / is G-Galois if and only if is obtained from as the set of G-fixed points. In this case there is only one twisted form. This summary is aimed at graduate students who have had algebra and a course in algebraic geometry, including basics on schemes and projective varieties. We prove both metatheorems 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 ran. We use this bacground to wor out the basic facts about Severi-Brauer varieties, as presented in [A]. We prove nothing new. Our presentation is a mashup of certain sections of [Bor] and [Bour] (-rationality), [GS] (tensors), [Jah] (Galois descent), [S2] (Galois descent and torsors), and [A] (Severi-Brauer varieties). Serre gives a polished account of Galois descent in [S2] and an introduction to Severi-Brauer varieties in [S1], and Artin gives the canonical in-depth account of Severi-Brauer varieties in [A]. Jahnel treats most of our topics in detail, and indly specializes Grothendiec s theory of faithfully flat descent ([EGA IV]) to Galois descent. Gille-Szamuely also give a thorough treatment of all of these subjects (and more)

2 2 in the valuable resource [GS]. We did not consult many of the seminal founding papers, by Weil ([W]), Châtelet ([C]), Amitsur ([Am]), and others; see [A] and [Jah] for historical bacground and bibliography. 1. Rationality. Let / be a finite field extension. Following [Bor] (see also [Bour]), we will define the notion of a -structure on various algebraic and geometric objects defined over, and what it means for objects to be defined over, and rational over. Essentially a -structure of an object V is another object V of the same type as V but defined over, such that V is obtained from V by base extension. Thus V is generated from its -structure. A - morphism f : W V between objects with -structures has a -structure if f(w ) V, i.e., f is defined over. Then f too is obtained by base extension from f df = f W, so that f is generated from its -structure Vector Spaces. Suppose V is a -vector space. A -structure of V is a -vector space V contained in V such that V V. The elements of V are said to be rational over. If f : V W is a morphism and V and W have -structures, we say f is defined over if f(v ) W. We also call f a -morphism, and write Hom (V, W ) for the set of -morphisms. Let 0 U V f W 0 be an exact sequence of -vector spaces, and set U = U V and W = f(v ). We say U is defined over if U is a -structure on U, and W is defined over if W is a -structure on W. It is easy to prove that W is defined over if and only if U is defined over Algebras. A -structure on a commutative -algebra A is a -algebra A that is a -structure on the vector space underlying A. If A and B are -algebras with -structures, a morphism f : A B is defined over if f(a ) B. We also call f a -morphism, and write Hom (A, B) for the set of -morphisms. If J A is an ideal, set J = J A, and say J is defined over if J is a -structure on J. If π : A A/J is the projection we set (A/J) = π(a ), and as before J is defined over if and only if (A/J) is defined over. Since / is flat, J A = J, so J is a -structure on J if and only if J generates J as an ideal. For example J = (f) is defined over if and only if f is associate to some f A. If S A is a multiplicative subset, then A[S 1 ] has -structure A [S 1 ]. In particular, if f A then A[1/f] = A [1/f] Schemes. Let X be a scheme of finite type over. A -structure on X is a scheme X defined over, such that X Spec Spec X. Let π : X X

3 3 denote the projection onto the first factor. A -open set U X is an open set of the form U = π 1 (U ) for U open in X. A -closed set is the complement of a -open set. The -open sets form a topology on X, called the -topology. Since U X is an open immersion, U = U Spec Spec, so U is a -structure for U. A basis of -open sets is given by the distinguished open sets U defined by elements f O X (X ). A morphism φ : Y X of -schemes with -structures is defined over, or a - morphism, if there is a map φ : Y X such that φ = φ id Spec. It follows that φ is continuous with respect to -top(x), and if U X and V Y are -open such that φ(v ) U, then the induced map O X (U) O Y (V ) is defined over. We will need the following theorem, which is proved in [Jah, Lemma 2.12] Theorem (Grothendiec and Dieudonné). Suppose / is a finite extension and X is a -structure on the -scheme X. If X has property P then so does X, where P includes any of the properties reduced, irreducible, quasi-compact, locally of finite type, finite type, locally noetherian, noetherian, proper, quasi-projective, projective, affine, regular. If U = Spec A is affine, then by Grothendiec and Dieudonné s theorem, U is affine, and if U = Spec A then A is a -structure of A. Conversely, if A is a -structure for A, then Spec A is a -structure for Spec A. Example. We examine -structures for affine schemes in more detail. Let X = Spec A be a scheme of finite type over, with -structure X = Spec A. The -topology on X has a basis consisting of the distinguished open sets D(f) with f A, since these form a basis for the topology on X. Clearly O X (D(f)) = A[1/f] has the -structure A [1/f]. If a subset Z X is -closed then since closed subsets of affine schemes are affine, the -structure Z has the form Spec A /J for some ideal J A. Since / is flat, Z = Spec (A/J), where J = J. If U X is -open, then U can be covered by a finite number of D(f i ) s for f i A, and O X (U) inherits a -structure as the ernel of the equalizer map A[1/fi ] A[1/f i f j ], which is a -morphism of vector spaces with -structures. If φ # : A B is a morphism of -algebras with -structures then φ : Y X is a morphism of -schemes with -structures, and φ is defined over if and only if φ # is defined over. Thus the category of affine -schemes with -structures is equivalent to the category of -algebras with -structures Sheaves. Suppose X is a scheme of finite type over with -structure X, and F is a sheaf of O X -modules. A -structure on F is a sheaf F of O X -modules such that π F F. If F has -structure F, then for any -open U we have F (U) = F (U ), so that each F (U) has a -structure. This follows from [Liu, Proposition ] for F = O X, and the proof for general F is the same Twisted Forms. Let / be a finite field extension. We will be given a category C of -objects, where is an initial (or final) object in the category. We will have a method

4 4 of base extension, so for each object V C an object V in the corresponding category C of objects defined over. Given an object V of C, we define the set of twisted forms of V relative to : TF (V ) df = {V C : φ : V V } Generally there is a distinguished V C. The notions of base extension and isomorphism are interpreted according to category. For example, TF (M n ()) df = {B/ : B M n ()} TF (P n 1 ) = df {Y/ : Y Spec Spec P n 1 } 2. Galois Cohomology. We follow [S2, Section I.5] and introduce the first cohomology set and principal homogeneous spaces in the category of left G-sets, to be applied later with additional structure in the categories of -algebras and schemes. Thus automorphisms and isomorphisms of G-sets, which are just set maps, will assume the additional structure of -automorphisms and -isomorphisms of -algebras or -varieties. A plain G-action will assume the structure of a morphism in a given category, with arrows possibly reversed. The point is that the proofs of the basic results are essentially unchanged when applied to a given category. Let G be a finite group, and A a left G-group, with action written f σ f. 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 (c σ = e) gives H 1 (G, A) the structure of a pointed set. A principal homogeneous set over A, or A-torsor, is a left G-set that admits a (compatible) principal transitive action on the right by the left G-group A. For x P, σ G, and a A, we write σ x and x a, respectively. The required compatibility is σ (x a) = σ x σa. We say two A-torsors are isomorphic if they admit a G and A-equivariant set bijection. Let A-Tors G denote the set of isomorphism classes of A-torsors, within the category of (isomorphism classes of) left G-sets. Main Example. For each c Z 1 (G, A), let c A denote the set A with 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 space for A.

5 Proposition. There is a 1-1 correspondence A-Tors G H 1 (G, A) P [(a σ )] defined by σ x = x a σ for (any) x P, under which P a A. Furthermore, there is a surjection from elements y P to cocycles (c σ ) (a σ ). Proof. This is [S2, Proposition 33]. If P is a torsor over A and x P, then for each σ G we have σ x = x a σ for some a σ A, and we claim (a σ ) Z 1 (G, A), i.e., a στ = a σ σ a τ. Let s chec: στ x = x a στ = σ ( τ x) = σ (x a τ ) = ( σ x) σa τ = x a σ σ a τ Since P is principal, we conclude a στ = a σ σ a τ, as desired. It is easy to see that varying x varies the cocycle in the class [(a σ )], and we obtain a map from the set of A-torsors to H 1 (G, A). In fact we have (c σ ) (a σ ) if and only if c σ = b 1 a σ σ b for some (possibly more than one) b A, hence each (c σ ) (a σ ) arises from (possibly more than one) y = x b. The inverse is given by a a A. If a c, and a σ = b 1 c σ σ b, then b : a A c A is an isomorphism, since b(σ a x) = ba σ σ x = c σ σ b σ x = σ c bx Therefore the map is well defined. Mapping bac to H 1 (G, A), we fix x a A and obtain the cocycle (c σ ) by the formula σ a x = x c σ = a σ σ x. As c σ = x 1 a σ σ x, we conclude (c σ ) (a σ ). Let V be a left G-set that admits a compatible left A-action by the left G-group A. If V, V are both left G-sets, the induced left G-action and right A-action on Hom(V, V ) is given by σ φ df = σ φ σ 1 and φ a = φ a. If c = (c σ ) Z 1 (G, A), we define the c-twisted action on V by σ c x = c σ σ x. Denote the resulting G-set by c V. Since V is a left G-set, A = Aut(V ) is a left G-group under the action σ b = σ b σ 1, in a way that is compatible with the G-action on V. We say two G-sets V and V are isomorphic if they are in 1-1 correspondence. In general an isomorphism φ Isom(V, V ) is not G-linear, that is, two isomorphic left G-sets V and V may not be G-isomorphic. The failure of the resulting diagram to commute is measured by the cocycle φ V V σ σ V φ V c σ = φ 1 σφ

6 6 Twisting the action on V then yields a G-linear commutative diagram φ cv V σ σ cv φ so that c V and V are G-isomorphic. The G-isomorphisms classes of left G-sets isomorphic to a fixed V are in this way classified by cocycles, and in bijection with Aut(V )-torsors as follows Proposition. We have bijections V {G-classes isomorphic to V } A-Tors G H 1 (G, Aut(V )) [ c V ] G Isom(V, c V ) [c] Proof. If V is isomorphic to V, then Isom(V, V ) is a principal homogeneous set for Aut(V ). For if σ G and φ Isom(V, V ) then σ φ = σ φ σ 1 Isom(V, V ), so G acts on the left. If b Aut(V ) then φ b Isom(V, V ), so Aut(V ) acts on Isom(V, V ) on the right, and this action is principal and transitive. Finally, G acts on Aut(V ) as σ b = σ b σ 1, and we compute σ(φ b) = σ φ σb. Proposition 2.1 establishes the second bijection, and states that any Aut(V )-torsor is isomorphic to c Aut(V ) for some c Z 1 (G, Aut(V )). We then identify c Aut(V ) with Isom(V, c V ), under the map φ φ. By Proposition 2.1, Isom(V, V ) and Isom(V, V ) determine the same class [c] if and only if there are elements φ Isom(V, V ) and ψ Isom(V, V ) satisfying c σ = φ 1 σφ = ψ 1 σψ. Equivalently, there are G-isomorphisms φ : c V V and ψ : c V V, hence a G- isomorphism ψ φ 1 : V V. As then Isom(V, V ) Isom(V, V ) Isom(V, c V ), we conclude [V ] G Isom(V, V ) is a well defined injection from G-classes isomorphic to V into A-Tors G, and we have an inverse, taing Isom(V, c V ) to [ c V ] G Right G-Sets. We avoid recasting all of the above into the universe of right G- sets admitting a right action by a right G-group A, by replacing any right action with the corresponding canonical left action. Thus if X is a right G-set, with action written ρ σ : x x σ, then we view X as a left G-set via σ x = df x σ 1. If φ Hom(X, X ) then we replace the induced right action φ φ σ = ρ σ φ ρ 1 σ with the left action σ φ = df φ σ 1, so σ φ = σ φ σ 1 = ρ 1 σ φ ρ σ. If Aut(X) acts on X on the right, we write x b for b Aut(X), and define a left action b x = df x b Galois Descent for Vector Spaces.

7 Let / be a finite Galois extension with group G. Let V be a -vector space. A left G-action G V V is Galois if σ(aw) = σ(a)σ(w) for each a, w V. A Galois G-action on V and W induce a left Galois action G Hom (V, W ) Hom (V, W ) (σ, φ) σ φ df = σφσ Definition. Let / be a finite Galois extension with group G. Define categories C: -vector spaces and -linear maps; C : -vector spaces with Galois G-action, and G-equivariant -maps Theorem. Suppose / is a finite Galois extension with group G. The functor C C defined by V V and φ φ id is a category equivalence, with inverse the functor defined by W W G and ψ ψ W G. Proof. If V is a -vector space then V = V is a -vector space, and V admits a G-action via the action on the second factor. This action is Galois, since for all a, a, σ(a (v a)) = v σ(a a) = v σ(a )σ(a) = σ(a )(v σ(a)) = σ(a )σ(v a). Thus the forward function is well defined on objects. If φ : V V is a map of -vector spaces then φ id : V V 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 C C is well defined on morphisms. Moreover it clearly taes id V preserves composition of morphisms, so it is a functor. to id V and We divide the rest of the proof into two lemmas Lemma. Suppose / is a finite G-Galois extension, and V is a -vector space with a left Galois G-action. Then V G is a -structure on V. Proof. Since the G-action is Galois, multiplication by stabilizes V G, so V G is a -vector space. Consider the action on V G induced by the G-action on the right factor. We will show the canonical map φ : V G V (v a) av

8 8 is a G-isomorphism. Let {x 1,..., x n } be a -basis for. Write G = {σ 1,..., σ n }, with σ 1 = e. Fix w V, and consider the elements v j = n σ i (x j w) = i=1 n σ i (x j )σ i (w) V G i=1 By linear independence of characters [L, 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. We conclude that φ is onto. We show φ is 1-1. Loo, φ is G-equivariant: φ(σ (v x)) = φ(v σ(x)) = σ(x)v = σ(xv) = σ (φ(v x)). It follows that er(φ) is stable under G: If φ( v i x i ) = 0 then φ(σ ( v i x i )) = σ(φ( v i σ(x i ))) = 0. Let {e i } I be a -basis for V G, then {e i 1} I is a -basis for V G. Let V G 1 denote the -span of the e i 1. Since φ(v 1) = v, er(φ) (V 1) = (0). Suppose er(φ) (0). Choose a nonzero element w er(φ) with expression w = m i=1 a ie i 1 involving the fewest number of nonzero a i. We may assume after reordering if necessary that a 1 is nonzero, and we may then choose w such that a 1 = 1. Since w V G 1, we may assume a 2. Then there exists a σ G such that σ(a 2 ) a 2, hence σ (w) w 0, and σ (w) w er(φ). But σ (w) w has a shorter expression than w, contradiction. This proves er(φ) = (0), and completes the proof Lemma. Suppose / is a finite G-Galois extension, V and V are -vector spaces with left Galois G-actions. Suppose φ : V V is a -morphism. Then the following are equivalent a. φ is G-equivariant. b. φ is defined over, i.e., φ Hom (V, V ). c. φ Hom (V, V ) G. Proof. (a) and (c) are equivalent by definition. (a) (b): Let V = V G and V = (V ) G, the -structures on V and V. Suppose φ is G-equivariant. We must show φ(v ) V. But if v V then φ(v) = (φ σ)(v) = σ(φ(v)), hence φ(v) V, as desired. Conversely if φ is defined over then φ has form φ id, where V = V and V = V. Then φ is G-equivariant since (φ id σ)(v a) = φ id (v σ(a)) = φ(v) σ(a) = (σ φ id )(v a) Clearly the restriction of id V to a -structure V is the identity on V, and the function taing φ Hom(V, V ) to φ V preserves composition. Thus we have a functor. To complete the proof of Theorem 3.2 it remains to show the given functors are inverse.

9 9 If V is a -vector space then we must show V = (V ) G, where the action of G is on the second factor. But if {e i } I is a -basis for V then V = e i, so that every element of v is uniquely expressible as an n-tuple (a 1,..., a n ) with a i (for some n), and the action of G is componentwise. Thus v (V ) G if and only if v V. If φ : V V is a -morphism, then clearly φ = φ id V. If V is a -vector space with Galois G-action, then we have seen V = V G is a - structure, so V = V, and similarly if φ : V V is a G-equivariant -morphism, then φ V id. Therefore both compositions of functors yield the identity, so we have a category equivalence. 4. Galois Descent for Algebras. Since vector spaces are classified by dimension, there is up to isomorphism only one - structure on a given -vector space. With algebras the situation is more complicated; there may be many different -structures, or twisted forms, descended from a given -algebra. To perform Galois descent on algebras we will recast the algebra structure in terms of vector spaces with additional structure represented by a multilinear map on V, copying from [GS]. Example. A -algebra is a -vector space V equipped with a (-bilinear and associative) multiplication map Φ : V V V. As Hom (V V, V ) V V V, we write Φ V V 2 Explicitly, let {e i } be a basis for V dual to a basis {φ j } for V. Then if Φ(e i e j ) = a ije, we write Φ = i,j, a ije φ i φ j V V 2, so that Φ(v 1 v 2 ) = i,j, a ijφ i (v 1 )φ j (v 2 )e. 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 [Bourbai 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 Tensors. More generally, suppose V is a -vector space. 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 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 = a IJ (e j1 e jn ) (φ i1 φ im ), IJ

10 10 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. We consider the set of 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 ). A map f : (V, Φ) (W, Ψ) is a map 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. Equivalently, if Φ and Ψ are viewed as elements of V n V m, f(φ) = Ψ. We denote by Hom (Φ, Ψ), Isom (Φ, Ψ), Aut (Φ) the set of all morphisms, isomorphisms, and automorphisms, respectively. If (V, Φ) is a -algebra, these are simply -algebra homomorphisms, isomorphisms, or automorphisms. If, a -structure on (V, Φ) is a -structure on V such that Φ is defined over, i.e., ) V n. If (V, Φ) is a -algebra, the requirement is that V be a -algebra. Φ(V m Let G be a group. A left Galois G-action G (V, Φ) (V, Φ) is a left Galois G action on V such that Φ is G-equivariant: if Φ Hom (V m, V n ) then σ Φ = Φ σ for each σ G, where σ acts on V m and V n by acting on each tensor factor. We say the action is compatible with Φ. If (V, Φ) is an algebra, i.e., Φ Hom (V 2, V ), then this means each σ is a ring automorphism. If G acts on (V, Φ) and (W, Ψ), then we have a left action We set G Hom (Φ, Ψ) Hom (Φ, Ψ) (σ, f) σ f df = σfσ 1 (V, Φ) G df = (V G, Φ (V m ) G) Hom (Φ, Ψ) G df = {G-equivariant morphisms} Twisted Forms of Vector Spaces with Tensors. Suppose (V, Φ) is defined over and / is a field extension. Set V Φ df = V df = Φ id

11 We say (V, Φ) and (W, Ψ) become isomorphic over if (V, Φ ) (W, Ψ ), and then we call (W, Ψ) a twisted form of (V, Φ). By fixing (V, Φ) we define a pointed set TF (V, Φ) df = {(W, Ψ) : (V, Φ ) (W, Ψ )}, isomorphism classes of twisted forms of (V, Φ) with respect to / Definition. Let / be a finite Galois extension with group G. Fix numbers n, m 0, and define categories C: objects are isomorphism classes (V, Φ) where V is a -vector space equipped with a tensor Φ of type (n, m), and morphisms in Hom (Φ, Ψ). C : objects are isomorphism classes (V, Φ) where V is a -vector space with Galois G-action, equipped with a G-equivariant tensor Φ of type (n, m), and morphisms in Hom (Φ, Ψ) G Theorem. Let / be a finite Galois extension with group G. i. We have a category equivalence C C defined by (V, Φ) (V, Φ ) and f f, where G acts on (V, Φ ) via the factor. The inverse functor is (V, Φ) (V, Φ) G and f f V G. ii. Suppose (V, Φ) C. There is a 1-1 correspondence of pointed sets, θ : TF (V, Φ) H 1 (G, Aut (Φ )) (W, Ψ) [(c σ )] df where if g Isom (Φ, Ψ ) then c σ = g 1 σg for each σ G. The inverse is [(c σ )] ( c V, c Φ ) G, where (c σ ) is a normalized cocycle, and ( c V, c Φ ) C is (V, Φ ) with Galois G-action σ v = df c σ σ(v). Proof. We first setch (i). Note the given action of G on V is compatible with Φ since the elements of G are ring automorphisms of. Suppose (V, Φ) C. Then it is easy to chec that (V, Φ ) C. Furthermore, V G = V, and (Φ ) V = Φ, so that (V, Φ ) G = (V, Φ). Conversely if (V, Φ) C, then by Theorem 3.2, V G is a -structure on V, and since Φ is G-equivariant, Φ V G is defined over. Thus (V, Φ) G C. Furthermore, (V G ) V via the canonical map v a av. This isomorphism is G-equivariant, and it is easy to see it induces (Φ V G) Φ. Thus (V, Φ) G (V, Φ), compatible with the G-action, so it is an isomorphism in C. If (W, Ψ) TF (V, Φ) then Isom (Φ, Ψ ) is nonempty, and defines an element of Aut (Φ )-Tors G, hence a class [c] H 1 (G, Aut (Φ)) by Proposition 2.1, by the rule c σ = g 1 σg for g Isom(Φ, Ψ ). This gives the map θ. The inverse to θ is given as in Proposition 2.2 by µ : H 1 (G, Aut (Φ )) TF (V, Φ) [c] ( c V, c Φ ) G 11

12 12 The twisted action is given as in Section 2 by G (V, Φ ) (V, Φ ) (σ, v) This is a group action, since c e = 1 and σ v df = c σ (σ(v)) (στ) v = c στ (στ)(v) = (c σ σ c τ )(στ)(v) = (c σ σc τ σ 1 στ)(v) = (c σ σ)(c τ τ)(v) = σ (τ v) It is Galois since c σ is -linear for each σ G, so that σ av = c σ σ(av) = σ(a)c σ σ(v) = σ(a)σ v. We chec that c Φ is G-equivariant: Since c σ Aut (Φ ), c σ Φ = Φ c σ, and since Φ is G-equivariant under the old action, σ Φ = Φ σ. By the cocycle condition, σ c σ 1 =c 1 σ. Now compute σ ( c Φ ) = (c σ σ) c Φ (c σ 1σ 1 ) = c Φ (c σ σ)(c σ 1σ 1 ) = c Φ c σ σ(σ 1 c 1 σ σ)σ 1 = c Φ By (i), (W, Ψ) df = ( c V, c Φ ) G is a twisted form of (V, Φ), and it is easy to chec µ and θ are inverse to each other. Example. If Φ = id, then (V, Φ) is just a vector space, and Aut (Φ ) = GL n (). Then H 1 (G, GL n ()) = {1}, reflecting the fact that a vector space is classified up to isomorphism by its dimension. 5. Group Action on Schemes and O X -Modules Basic Structures. Let G be a finite group, and let X be a scheme over. A right action of G on X is a collection of automorphisms ρ σ : X X one for each σ G, such that ρ στ = ρ τ ρ σ and ρ e = id X. If G also acts by automorphisms on, we say the action on X is Galois if the structure map π : X Spec is G-equivariant, i.e., we have a commutative diagram X π ρ σ Spec ρσ X π Spec We will see that a Galois G-action on X is enough to define a quotient scheme X by descent, provided X admits a G-invariant topology.

13 The right action on X is a right (continuous) action on the underlying topological space X, together with a left action on the structure sheaf, given by σ df = ρ # σ : O X ρ σ O X Thus στ = ρ τ (σ) τ, e : O X O X is the identity, and the restriction maps are G- equivariant. If the action on X is Galois, then the action on the global sections of O X is Galois, i.e., σ(ax) = σ(a)σ(x) for a and x O X (U), for all U X. As ρ σ is an automorphism of X, the sheaf ρ σ O X is a coherent O X -module, and the action of O X (U) on ρ σ O X (U) = O X (ρ 1 σ U) is given by x y = ρ # σ (x)y = σ(x)y, for x O X (U) and y O X (ρ 1 σ U). In other words, the map O X (U) O X (ρ 1 σ U) itself is the action, sending x O X (U) to σ(x). If Y and X are -schemes admitting right Galois G-actions, the induced right G-action on Hom(Y, X) is defined by φ σ = ρ σ φ ρ 1 σ We say φ is G-equivariant if φ Hom(Y, X) G O X -Modules. Suppose F is an O X -module, and G acts on the right on X. A left Galois action of G on F is a collection of O X -module isomorphisms σ : F ρ σ F one for each σ G, such that στ = ρ τ (σ) τ, e : F F is the identity, the restriction maps are G-equivariant, and the structure map ı : O X F is G-equivariant. Thus we have a commutative diagram of O X -linear maps F ı σ ρ σ F ρσ ı O X σ ρ σ O X The O X (U) structure on (ρ σ )F (U) is given by a x = σ(a)x. On sections the above diagram just states that σ(a x) = σ(a) σ(x), eeping in mind the adjoint action on open sets, so that if a and x are sections over U, then σ(a) and σ(x) are sections over ρσ 1 (U) Locally Free Sheaves. Suppose that E is a locally free sheaf of ran n and structure map ı : O X E. By definition there exists an open cover {U i } of X and O Ui -isomorphisms φ i : E Ui O n U i, such that φ i φ 1 j = g ij O n X (U i U j ). Suppose there is a left G-action on E and a right Galois action on X. We claim the action on E is Galois if and only if the φ i are G-equivariant. For the structure map θ i : O Ui O n U i is G-equivariant, so since φ i ı i = θ i, φ i is G-equivariant if and only if ı i is as well. If E admits a Galois G-action, we may define a new (twisted) Galois action on X via some c Z 1 (G, Aut l (X)), as in Section 2. There is no reason why this action should extend to E. 13

14 Quotient Schemes. Instead of a subobject, a right Galois action by the finite group G on a scheme X should determine a quotient, whose points correspond to G-orbits. By definition, a quotient Y = X/G is a scheme with trivial G-action and G-equivariant morphism π : X Y, such that any G-equivariant morphism τ : X Z, where Z has trivial G-action, factors uniquely through π: There exists a unique τ : Y Z such that τ π = τ. A quotient does not always exist in the category of schemes (though it exists in the larger category of ringed topological spaces ([Liu, Exercise ])). We show it exists for affine schemes, so that if X = Spec A then X/G = Spec (A G ), and for quasi-projective varieties, by virtue of a G-stable affine cover. 6. Galois Descent for Affine Schemes. The theory of descent for affine -schemes is equivalent to the theory for -algebras. However, since the arrows switch directions, we obtain quotient schemes instead of subalgebras. Once we write down the details we will have the basic mechanism we need to define descent on more general schemes, modulo the gluing of the descent data for an affine cover Definition. Let / be a finite Galois extension with group G. Define categories C : affine -schemes and -morphisms. C : affine -schemes with right Galois G-action and G-equivariant morphisms. We will replace the given right G-action with the corresponding left G-action in order to use machinery we ve already developed. For x X, write σ x = ρ σ 1(x) = x σ 1, and for φ Hom(X, Y ) write σ φ = φ σ 1 = ρ 1 σ φ ρ σ. Let Aut l (X) denote the group of left automorphisms of X Theorem. Let / be a finite Galois extension with group G. i. The functor C C defined by X X and φ φ is a category equivalence, with inverse the functor X = Spec A X/G = Spec (A G ) and φ φ/g, dual to φ # A G. ii. Suppose X C. There is a 1-1 correspondence of pointed sets, θ : TF (X) H 1 (G, Aut l (X )) Y [(d σ )] where d σ df = φ 1 σφ for any φ Isom (X, Y ). The inverse is [(d σ )] ( d X )/G, where the (left) G-action on d X is given by σ x = d σ σx. If X = Spec A then d X = Spec c A and ( d X )/G = Spec ( c A) G, where c = (c σ ) Z 1 (G, Aut (A)) is defined by c σ = d 1# σ. Proof. For the functor C C, let X C, and set X = X Spec Spec. Then X admits a right Galois G action via the right factor, hence X C. If φ : Y X is a morphism in C, the map φ : Y X given by φ = φ Spec id Spec is in C.

15 For the other direction, suppose X = Spec A is in C. Then A admits a left Galois G-action. By descent for -algebras, A G is a -structure on A, and we define an element of C by X/G = df Spec (A G ) 15 This is a -structure on X, since X/G Spec Spec = Spec (A G ) = Spec A. Y = Spec B and X = Spec A are in C and φ : Y X is in C, we obtain a map If φ/g : Y/G X/G defined by the restriction of φ # : A B to A G. Clearly (φ/g) = φ, so φ is defined over. We conclude the functor C C defined by X X and φ φ is a category equivalence, with inverse X X/G and φ φ/g. We separate the following as a lemma Lemma. Let X = Spec A be an affine -scheme with left Galois G-action. Then there is a canonical isomorphism δ : H 1 (G, Aut l (X)) H 1 (G, Aut (A)) [(d σ )] [(d 1# σ )] Proof. The map φ φ 1 defines a product-inverting bijection between Aut l (X) and Aut (X), hence we have an isomorphism This isomorphism is G-linear, since δ : Aut l (X) Aut (A) φ φ 1# ( σ φ) 1# = (ρ 1 σ φ ρ σ ) 1# = ρ # σ φ 1# ρ 1# σ = σ φ 1# σ 1 = σ (φ 1# ) Thus the cocycle and coboundary conditions are preserved, and we have the desired canonical isomorphism. We continue the proof of Theorem 6.2. If d = (d σ ) Z 1 (G, Aut l (X)), define the d- twisted action on X by σ x = df d σ σx and write d X for X with the twisted Galois action. It is clear that d X = Spec c A, where c σ = d 1# σ, since the twisted actions correspond: (d σ ρσ 1 ) 1# = d 1# σ ρ # σ = c σ σ. If X, Y are both -schemes with left G-action, then Isom (X, Y ) is a principal homogeneous space for Aut l (X ). An element φ Isom (X, Y ) is G-equivariant if and only if d σ = φ 1 σφ is the identity, but by construction the map φ : d X Y is G-equivariant. As in Proposition 2.2, it follows that there is a 1-1 correspondence

16 16 between G-isomorphism classes subdividing the -isomorphism class [X ] in C, hence a 1-1 correspondence TF (X) Y H 1 (G, Aut l (X )) [(φ 1 σφ)] Remar. Suppose X = Spec A is an affine -scheme and A admits a Galois G-action, so we have the quotient π : X X/G = Spec A G. Then π(q) = q A G, π 1 (p) = {q A : q A G = p}, and a point q X is -rational if and only if q has a -structure, i.e., (q A G ) generates q over A. 7. Galois Descent for Quasi-Projective Varieties. For non-affine schemes the situation is complicated by the need to construct a -topology for a given X, in particular to construct a cover of G-stable affine open subschemes. By [Liu, Exercise ], if every point of X is contained in a G-stable affine open set, then X/G exists. We show how it wors for quasi-projective varieties Definition. Let / be a finite Galois extension with group G. Define categories D: quasi-projective -varieties and morphisms. D : quasi-projective -varieties with right Galois G-action, and G-equivariant morphisms. We will need to find a G-stable affine cover of an arbitrary X D. We copy the proof in [Jah] Lemma. Suppose P = {p 1,..., p s } is a finite (possibly empty) set of closed points in P n, and Z Pn is a closed subvariety such that Z P =. Then there exists a hypersurface H P n such that Z H and H P =. Proof. We tensor the defining exact sequence 0 I P O P n ı O P 0 with I Z over to obtain the exact sequence I P I Z I Z ı O P I Z 0. Using the hypothesis O P n Z P = and looing at stals, we easily show the sequence is injective on the left, and that the canonical map ı O P ı O P I Z is an isomorphism. Thus for each r we have a short exact sequence 0 I P I Z (r) I Z (r) ı O P (r) 0. Since I P OP n I Z is coherent, for r 0 we have H 1 (P n, I P I Z (r)) = 0 by Serre s Theorem, hence a surjection s Γ(P n, I Z (r)) Γ(P n, ı O P (r)) = (p i ). Thus there exists f Γ(P n, I Z(r)) such that f(p i ) 0 for each i, and we tae H = V (f). Since f vanishes on Z we have Z H, and since f(p i ) 0, H P =. i=1

17 7.3. Lemma. Let G be a finite group. Any quasi-projective -variety with a right Galois G-action has a G-stable affine cover. Proof. Suppose Y P n is quasi-projective. Let Z = Y \Y, a (possibly empty) closed subvariety of P n. We will show each p Y has a G-stable affine neighborhood in Y. Let P = {σ(p) : σ G}. By Lemma 7.2 there exists a hypersurface H P n such that Z H and H P =. The distinguished open set D = P n H is an affine variety, and since Z H, Y D is a closed subvariety of D, hence an affine variety. Now consider U = σ(y D) σ G Then σ(u) = U for all σ, and since σ(p) Y D for all σ, p U. Thus U is a G-stable affine open subset of Y containing p Theorem. Suppose / is a finite Galois extension with group G. i. The functor D D defined by X X and φ φ is a category equivalence. ii. Suppose X D is a quasi-projective -variety. Then there is an bijective map of pointed sets θ : TF (X) H 1 (G, Aut l (X )) Y [(d σ )] where d σ df = φ 1 σφ for φ Isom (X, X ). The inverse is given by [(d σ )] ( d X )/G, where the (left) G action on d X is given by σ d x = d σ σx. If X is any -variety (not necessarily quasi-projective) then θ is injective. Proof. The restriction to quasi-projective varieties in (i) and (ii) ensures the existence of a G-stable open cover for X (Lemma 7.3), so that the quotient X /G = ( d X )/G may be constructed. To prove (i) we construct an inverse functor D D. Suppose X D. By Lemma 7.3 there is a cover X = r i=1 U i by G-stable open affine subschemes. By Theorem 6.2 we obtain -structures U i /G for each i, such that (U i /G) U i. Since X is separated, each U ij = U i U j is affine, with -structure U ij /G = U i /G U j /G. The U i are glued together by the identity map id X : U ij U ji, and the cocycle condition is trivial. Since clearly id X is G-equivariant, and the action of G commutes with composition of functions, the U i /G glue together to give a scheme Y over, such that Y = X. By Grothendiec and Dieudonné s theorem, Y is quasi-projective since Y is quasi-projective. Suppose φ : Y X is a morphism in D. Since the affine distinguished open sets form a basis for the -topology on V i, each y Y is contained in a G-stable affine open subset V i of V that maps via φ into a G-stable affine open subset U i of X. Since φ is G-equivariant, (φ Vi )/G taes V i /G to U i /G by Theorem 6.2, such that ((φ Vi )/G) = φ Vi and the various 17

18 18 (φ Vi )/G clearly agree on intersections. Thus we have a morphism φ/g : Y/G X/G, such that ((φ Y )/G) = φ. We conclude D D. For part (ii) we just cite [S2, Chapter III, Section 1, Proposition 5]. 8. Galois Descent for Quasi-Coherent Sheaves. Let / be a finite Galois extension with group G. Galois descent for quasi-coherent sheaves over -schemes with -structures is again based on Galois descent for -vector spaces. Let X be a scheme over, let X = X Spec Spec, and let π : X X be projection on the first factor. X admits a right Galois G-action ρ σ : X X via the right factor, and X /G = X Definition. Assume the above notation. Define categories Q: quasi-coherent sheaves over X and O X -morphisms; Q : quasi-coherent sheaves over X with left Galois G-action, and G-equivariant O X -morphisms Theorem. Suppose / is a finite Galois extension with group G, and X is a scheme over, with scalar extension π : X X. The functor Q Q defined by F π F and φ π φ is a category equivalence. Proof. Suppose X = Spec A is affine, and F = M is a quasi-coherent sheaf on X. The hypothesis on F implies M is a -vector space with left Galois G-action, and we set F G = df (M G ). By basic properties, π (F G ) is the sheaf associated to the module M G, so π (F G ) F, as desired. If φ : G F is a G-equivariant map of quasi-coherent O X -modules for X = Spec A, then φ corresponds to a map f : N M of A -modules, and since φ is G-equivariant we have f N G : N G M G, and (f N G) id = f. Taing the associated maps of sheaves we see φ G G : G G F G is a morphism of O X -modules such that π (φ G G) = φ. For the general case, let X = i U i be an affine -open cover of X, with U i = Spec A i. this exists since by definition X is a -structure for X. By the affine case the restrictions F i df = F Ui = M i have -structures F G i such that π F G i F i. It remains to chec that these glue together along the intersections U ij /G = U i /G U j /G. If X is not separated the U ij may not be affine, but for each one we have a -open affine cover U ij = U ij. Descent for morphisms is local, and since the G action on F is Galois, the gluing morphisms F i Uij F j Uij are G-equivariant, and their restrictions to the (F i Uij )/G satisfy the cocycle condition. Thus we obtain a quasi-coherent sheaf F G on X, and it is easy to chec that π (F G ) = F. If φ : G F is a G-equivariant map of quasi-coherent O X -modules, then by definition φ G G : G G F G, and π (φ G G) = φ. This is trivial since the question is local.

19 Remar. We obtain the same theorem if we replace Q and Q with the corresponding categories of locally free sheaves of fixed ran, since a descended sheaf is locally free (of ran n) if and only if the original is locally free (of ran n) ([EGA IV, Proposition 2.5.2]) Severi-Brauer Varieties. A Severi-Brauer variety of degree n over is a twisted form P of P n 1 with respect to some finite /. Since P P n 1, we say P is split by. We call the distinguished case P P n 1 trivial. By the theorem of Grothendiec and Dieudonné, P is smooth. Since Aut(P n 1 ) PGL n(), if / is a finite Galois extension with group G, then by Theorem 7.4 we have a bijection of pointed sets between TF (P n 1 ) and H 1 (G, PGL n ()), and every Severi-Brauer variety may be defined by descent. By convention we identify PGL n () with the left automorphism group, so C PGL n () induces C t on the homogeneous coordinate ring, and acts on points as λ C ([a 0,..., a n 1 ] = C t [a 0,..., a n 1 ] t. The standard left Galois G-action on [x 0,..., x n 1 ] induces a right Galois action on P n 1, and the corresponding left G-action on closed points is σ [a 0,..., a n 1 ] = [σ(a 0 ),..., σ(a n 1 )]. Therefore if c Z 1 (G, PGL n ()), This G-action on P n 1 P = P n 1 σ c [a 0,..., a n 1 ] = c t σ [σ(a 0 ),..., σ(a n 1 )] t produces the quotient P = ( c P n 1 )/G. If [c] = 1, we obtain. If [c] 1, P has no -rational points, as we will see below. A central simple algebra of degree n over is a twisted form of M n () with respect to some finite /. Since PGL n () is also the automorphism group of M n (), isomorphism classes of central simple -algebras are also classified by H 1 (G, PGL n ()), and may be defined by descent. In particular, if A and P are central simple algebras and Severi-Brauer varieties over, respectively, corresponding to the same class in H 1 (G, PGL n ()), we say A is associated to P, and write A = CSA(P ), and we say P is associated to A, and write P = SBV(A). By Theorem 4.4, each c Z 1 (G, PGL n ()) defines a central simple -algebra by the formula ( c M n ()) G = {T M n () : c σ σ T c 1 σ = T, σ G} where the G-action on T is via coefficients. This is a nonempty subset of M n (). For m, n N, the tensor product pairing m n mn induce -linear maps M m M n M mn, GL m GL n GL mn, and PGL m PGL n PGL mn. It is easy to see these maps are G-linear, since G acts via the coefficients, and we obtain a map H 1 (G, PGL m ()) H 1 (G, PGL n ()) H 1 (G, PGL mn ()) ([c], [d]) [c d] If [c] and [d] correspond to central simple -algebras A and B, respectively, then [c d] corresponds to A B, a twisted form of M mn (). If P and Q are the corresponding Severi- Brauer varieties, then [c d] corresponds to the product P Q, a twisted form of P mn 1.

20 20 This can be seen easily by considering the case of a G-stable affine open set. In particular, if A = CSA(P ) then A d = CSA(P d ) We note as in [A, p.202] that if Q has a -rational point, then the embedding P P Q realizes P as a linear subvariety. The special case d = I n yields λ m,n : H 1 (G, PGL m ()) H 1 (G, PGL mn ()) [c] [c I n ] If [c] H 1 (G, PGL m ()) determines the Severi-Brauer variety P, then λ m,n ([c]) = [c n ], which determines the variety P P n 1. Since H 1 (G, GL n ()) = 0, the short exact sequence determines an injection 1 G m GL n PGL n 1 δ n : H 1 (G, PGL n ()) H 2 (G, ) [(c σ )] [(a σ,τ )] where a σ,τ df = c σσ c τ c 1 στ, and c σ represents c σ in GL n (). It is not hard to show the following diagram is commutative H 1 (G, PGL n ) λ m,n δ n H 2 (G, ) H 1 (G, PGL mn ) δ mn H 2 (G, ) We say Severi-Brauer varieties P and P of degrees n and n are Brauer equivalent, and write [P ] = [P ], if δ n (P ) = δ n (P ) in H 2 (G, ). The period of [P ] divides n. For it is easy to see the diagram 1 µ n SL n PGL n 1 1 G m GL n PGL n 1 commutes, consequently the map δ n factors through H 2 (G, µ n ) = n H 2 (G, ), as desired. Rational Bundles. Assume / is a finite Galois extension with group G. Recall we call a line bundle on a -variety P with -structure P rational if it descends to a line bundle on P, or equivalently, if it admits a (left) G-action that is Galois with respect to the (right) G-action on P. Since Pic P n 1 an isomorphism Pic P n 1 Z is generated by O P n 1(1), scalar extension produces Pic P n 1. Thus every line bundle on Pn 1 descends to Pn 1, extends to a Galois action on. This turns out not to hold in the twisted case. which shows the untwisted right Galois action of G on P n 1 every line bundle on P n 1

21 9.1. Proposition. Suppose / is a finite G-Galois extension and P TF (P n 1 ). Then O P (d) is rational if and only if the d-tuple embedding is G-equivariant, if and only if the d-tuple embedding descends to an embedding 21 ν d : P P N 1 (N = ( ) n+d 1 d ) Proof. The last two statements are equivalent by Theorem 8.2. Since P P is flat, for any invertible sheaf L on P we have L (P ) = L (P ) ([Liu, ]), hence if O P (d) is rational, its global sections descend, so the d-tuple embedding on P descends. Conversely, if the d-tuple embedding is defined over, we have an invertible sheaf L = df νd O P N 1 (1) on P, which becomes O P (d) on P Corollary. Let P TF (P n 1 ), with φ : P P n 1 φ O P n 1 (1). Then O P (1) is rational if and only if P P n 1., and let O P (1) = 9.3. Remar. The corollary suggests that if P P n 1, i.e., the G-action on P n 1 is twisted, then not every line bundle on P n 1 is rational. Despite this, the action of G on Pic P is trivial: Since each σ is an automorphism of P, it induces an automorphism on Pic P, hence it taes O P (1) to either O P (1) or O P ( 1). Since the latter has no global sections, it must go to the former. Alternatively, the action by PGL n () can be represented by GL n+1 () acting on the homogeneous coordinate ring [x 0,..., x n 1 ], and the action on Div X can then be traced by the action on forms. As PGL n () preserves degree on divisors, divisor classes are PGL n ()-fixed. Since the Galois action on also preserves degree, we conclude any twisted action of G on P preserves degree. Thus each line bundle O P n 1(d) is G-invariant, whether or not it is rational ([A, p.197]). By Proposition 9.1 we can show every Severi-Brauer variety is projective by simply observing that every smooth -variety has an invertible sheaf, hence that a rational bundle exists for every (twisted) G-action on P n 1. We accomplish this with the canonical sheaf Proposition. Let P TF (P n 1 ). Then we have an n-tuple embedding ν n : P P N 1 where N = ( ) 2n 1 n. In particular, P is projective. Proof. Let Ω 1 X denote the sheaf of ahler differentials on a nonsingular n 1-dimensional df -scheme X of finite type. The canonical sheaf is the wedge product ω X = n Ω 1 X Pic X. We have ω P O P ( n) by [Liu, Corollary ]. Since the canonical sheaf is stable under (flat) base change ([Liu, Theorem 6.4.9]), ω P = ω P O P ( n), hence O P (n) is rational. Since ω P (P ) ω P (P ), the n-tuple embedding descends.

22 22 Remar. We will produce a proof that does not rely on the canonical sheaf in Linear Subspaces. We follow [A] as closely as we can. Suppose P TF (P n 1 ). A twisted linear subvariety L P is a subvariety that becomes linear when P is split. If L P is a twisted linear subvariety of dimension d 1, we call d the affine dimension of L. Since L P d 1, L is itself a Severi-Brauer variety. Duality. We write P n 1 = P(V ) for the projective space whose points correspond to linear subspaces of a -vector space V of dimension n. Let {e i } be a standard basis for V, with dual basis {x i } for V, so [x 0,..., x n 1 ] is the homogeneous coordinate ring for P n 1. Let P n 1 = P(V ) denote the dual projective space, with coordinate ring [e 0,..., e n 1 ]. Suppose / is a finite Galois extension with group G, and V admits a left Galois G- action. The induced left Galois action on GL(V ) is σ C = σ C σ 1, and since the G-action is Galois, we have an induced left action on PGL(V ). The left G action on V induces a left Galois action on V, such that σ (v (w)) = σ v ( σ w), and hence left actions on GL(V ) and PGL(V ). Since H 1 (G, GL(V )) = 1, every left Galois G-action on V is given by the standard action on coefficients with respect to some basis, and then the induced action on GL n () is by coefficients, and the induced action on a dual basis of V is again on coefficients Duality Lemma. The isomorphism PGL(V ) PGL(V ) induces a bijection H 1 (G, PGL(V )) H 1 (G, PGL(V )) [c] [c t ] If P and P are corresponding Severi-Brauer varieties, we have an inclusion reversing correspondence between twisted linear subvarieties L P of dimension d 1 and those L P of codimension d. Proof. The map W Ann(W ) sets up an inclusion-reversing 1-1 correspondence between subspaces W V of dimension d and subspaces W V of codimension d, hence between the corresponding linear subvarieties L P of dimension d 1 and L P of codimension d. If C GL(V ) stabilizes W, then C t GL(V ) stabilizes W, hence C t stabilizes W. The map GL(V ) GL(V ) sending C to C t is an isomorphism, and since it preserves scalar matrices it induces an isomorphism PGL(V ) PGL(V ), under which c stabilizes L if and only if c t stabilizes L. Similarly if σ stabilizes W then σ = σ t stabilizes W, so if G stabilizes a linear subvariety L P then it stabilizes L P. Since σ t = σ, the map GL(V ) GL(V ) is G-equivariant, so PGL(V ) PGL(V ) is G-linear, and we have the desired bijection H 1 (G, PGL(V )) H 1 (G, PGL(V )). By the compatibility of the induced G and PGL(V ) actions, σ c stabilizes L P if and only if σ c t stabilizes L P. Therefore twisted linear subvarieties of dimension d 1 in the quotient P correspond to twisted linear subvarieties of codimension d in P.

23 9.7. Proposition. A Severi-Brauer variety P over is trivial if and only if P (). Proof. [A, 3.3]. The result is due to Châtelet. If p P () then by Lemma 9.6, P has a twisted hyperplane, hence an invertible sheaf that scalar extends to O P (1). Therefore P is trivial by Proposition 9.1, hence it has a -rational point, and since (P ) P, P has a twisted hyperplane, hence P is trivial. Conversely if P P n 1 then clearly P (). Remar. Proposition 9.7 says that a Severi-Brauer variety P of degree n is trivial if and only if it shares a twisted linear subvariety with P n 1, i.e., there exists a map φ : P that is G-equivariant on a linear subvariety. We will show that similarly two P n 1 Severi-Brauer varieties P and P of degree n are equal if and only if they share a twisted linear subvariety, i.e., there exists an isomorphism φ : P P that is G-equivariant on a linear subvariety Reduction of Structure Group I. Suppose / is a finite Galois extension with group G, and P TF (P n 1 ) has a twisted linear subvariety L of dimension d 1. Let φ : P n 1 P, and fix φ 1 (L ) = P d 1 Pn 1. Let Pd 1 P n 1 be the image in the quotient variety under the standard Galois action on P n 1. There exists a projec- ) is in standard position in P n 1, tive transformation θ PGL n () such that θ 1 (Pd 1 occupying the first d coordinates. Scalar extending to we obtain a new isomorphism ψ = df φ θ : P n 1 θ PGL n (), the corresponding cocycle c, defined by c σ = ψ 1 σψ, is cohomologous to P under which ψ 1 (L ) P n 1 is in the same standard position. As the one defined by φ 1 σφ, and since θ is defined over, each c σ taes P d 1 we may assume the cocycle c defining P from P taes values in the subgroup Γ df = GL d() 0 GL n d () / PGL n () The extra structure provided by L reduces the structure group. Define U = df 1 0 PGL n () 1 to itself. Thus Γ df = GL d() 0 0 GL n d () / PGL n () We have a G-stable decomposition Γ = U Γ, hence we may view H 1 (G, Γ) as a subset of H 1 (G, Γ). The group U is connected, since topologically it is a -vector space, and it is unipotent, since every element is unipotent. Therefore H 1 (G, U) = 0 by the additive version of Hilbert 90 ([D, Section 6, Lemma 1]), hence the inclusion H 1 (G, Γ) H 1 (G, Γ) is surjective, hence H 1 (G, Γ) = H 1 (G, Γ)

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