# A result of Gabber. by A.J. de Jong

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 A result of Gabber by A.J. de Jong 1 The result Let X be a scheme endowed with an ample invertible sheaf L. See EGA II, Definition In particular, X is supposed quasi-compact and separated. 1.1 Theorem. The cohomological Brauer group of X is equal to the Brauer group of X. The purpose of this note is to publish a proof of this result, which was prove by O. Gabber (private communication). The cohomological Brauer group Br (X) of X is the torsion in the étale cohomology group H 2 (X, G m ). The Brauer group of X is the group of Azumaya algebras over X up to Morita equivalence. See [Hoobler] for more precise definitions. Our proof, which is different from Gabber s proof, uses twisted sheaves. Indeed, a secondary goal of this paper is to show how using them some of the questions regarding the Brauer group are simplified. We do not claim any originality in defining α-twisted sheaves; these appear in work of Giraud, Caldararu, and Lieblich. Experts may skip Section 2 where we briefly explain what little we need about this notion. 2 Preliminaries 2.1 Let (X, L) be a pair as in the Section 1. There exists a directed system of pairs (X i, L i ) as in Section 1 such that (X, L) is the inverse limit of the (X i, L i ) and such that each X i is of finite type over Spec Z. Furthermore, all the transition mappings in the system are affine. See [Thomason]. 2.2 Let X, X i be as in the previous paragraph. Then Br (X) is the direct limit of Br (X i ). This because étale cohomology commutes with limits. See??.Similarly, Br(X) is the limit of the groups Br(X i ). 2.3 Let X be a scheme and let α H 2 (X, G m ). Suppose we can represent α by a Čech cocycle α ijk Γ(U i X U j X U k, G m ) on some étale covering U : {U i X} of X. An α-twisted sheaf is given by a system (M i, ϕ ij ), where each M i is a quasi-coherent O Ui -module on U i, and where ϕ ij : M i O Uij M j O Uij are isomorphisms such that ϕ jk ϕ ij = α ijk ϕ ik over U ijk. Since we will be working with quasi-projective schemes all our cohomology classes will be represented by Čech cocycles, see [Artin]. In the general case one can define the category of α-twisted sheaves using cocycles with respect to hypercoverings; in 2.9 below we will suggest another definition and show that it is equivalent to the above in the case that there is a Čech cocycle. 1

2 2.4 We say that the α-twisted sheaf is coherent if the modules M i are coherent. We say an α-twisted sheaf locally free if the modules M i are locally free. Similarly, we can talk about finite or flat α-twisted sheaves. 2.5 Let α H 2 (X, G m ). Let X = X α be the G m -gerb over X defined by the cohomology class α. This is an algebraic stack X endowed with a structure morphism X X. Here is a quick way to define this gerb. Take an injective resolution G m I 0 I 1 I 2... of the sheaf G m on the big fppf site of X. The cohomology class α corresponds to a section τ Γ(X, I 2 ) with τ = 0. The stack X is a category whose objects are pairs (T X, σ), where T is a scheme over X and σ Γ(T, I 1 ) is a section with boundary σ = τ T. A morphism in X is defined to be a pair (f, ρ) : (T, σ) (T, σ ), where f : T T is a morphism of schemes over X and ρ Γ(T, I 0 ) has boundary ρ = σ f (σ ). Given morphisms (f, ρ) : (T, σ) (T, σ ), and (f, ρ ) : (T, σ ) (T, σ ), the composition is defined to be (f f, ρ + f (ρ )). We leave it to the reader to check that the forgetful functor p : X Sch makes X into a category fibred in groupoids over the category of schemes. In fact, for T T there is a natural pullback functor X T X T coming from the restriction maps on the sheaves I j. To show that X is a stack for the fppf topology, you use that the I j are sheaves for the fppf topology. To show that X is an Artin algebraic stack we find a presentation U X. Namely, let U X be an étale surjective morphism such that α restricts to zero on U. Thus we can find a σ Γ(U, I 1 ) whose boundary is τ. The result is a smooth surjective morphism U X. Details left to the reader (the fibre product U X U is a G m -torsor over U X U). 2.6 By construction, for any object (T, σ) the autmorphism sheaf Aut T (σ) of σ on Sch/T is identified with G m,t. Namely, it is identified with the sheaf of pairs (id, u), where u is a section of G m. 2.7 There is a general notion of a quasi-coherent sheaf on an algebraic Artin stack. In our case a quasi-coherent sheaf F on X is given by a quasi coherent sheaf F σ on T for every object (T, σ) of X, and an isomorphism i(ρ) : f F σ F for every morphism (f, ρ) : (T, σ) (T, σ ) of X. These data are subject to the condition i(ρ + f (ρ)) = f (i(ρ )) i(ρ) in case of a composition of morphisms as above. In particular, any quasi-coherent sheaf F on X comes equipped with an action of G m,x. Namely, the sheaves F σ are endowed with the endomorphisms i(u). (More precisely we should write i((id, u)).) 2.8 By the above a quasi-coherent O X -module F can be written canonically as a direct sum F = m Z F (m). Namely, the summand F (m) is the piece of F where the action of G m is via the character λ λ m. This follows from the representation theory of the group scheme G m. See??. 2.9 The alternative definition of an α-twisted sheaf we mentioned above is a quasi-coherent O X -module F such that F = F (1).In case both definitions make sense they lead to equivalent notions. 2

3 2.10 Lemma. Suppose α is given by a Čhech cocylce. There is an equivalence of the category of α-twisted sheaves with the category of O X -modules F such that F = F (1). Proof. Namely, suppose that α is given by the cocycle α ijk as in Subsection 2.3 and by the section τ of I 2 as in Subsection 2.5. This means that on each U i we can find a section σ i with σ i = τ, on each U ij we can find ρ ij such that and that in Γ(U ijk, I 0 ). ρ ij = σ i Uij σ j Uij α ijk = ρ ij Uijk + ρ jk Uijk ρ ik Uijk Let us show that a quasi-coherent sheaf F on X such that F = F (1) gives rise to an α-twisted sheaf. The reverse construction will be left to the reader. For each i the pair (U i X, σ i ) is an object of X. Hence we get a quasi-coherent module M i = F σi on U i. For each pair (i, j) the element ρ ij defines a morphism (id, ρ ij ) : (U ij, σ i Uij ) (U ij, σ j Uij ). Hence, i(id, ρ ij ) is an isomorphism we write as ϕ ij : M i O Uij M j O Uij. Finally, we have to check the α-twisted cocycle condition of 2.3. The point here is that the equality α ijk + ρ ik Uijk = ρ ij Uijk + ρ jk Uijk implies that ϕ jk ϕ ij differs from ϕ ik by the action of α ijk on M k. Since F = F (1) this will act on (U ijk U i ) M i as multiplication by α ijk as desired Generalization. Suppose we have α and β in H 2 (X, G m ). We can obviously define a G m G m -gerb X α,β with class (α, β). Every quasi-coherent sheaf F will have a Z 2 grading. The (1, 0) graded part will be a α-twisted sheaf and the (0, 1) graded part will be a β-twisted sheaf. More generally, the (a, b) graded part is an aα + bβ-twisted sheaf. On X α,β we can tensor quasi-coherent sheaves. Thus we see deduce that there is a tensor functor (F, G) F G which takes as input an α-twisted sheaf and a β-twisted sheaf and produces an α + β-twisted sheaf. (This is also easily seen using cocyles.) Similarly (F, G) Hom(F, G) produces an β α twisted sheaf. In particular a 0-twisted sheaf is just a quasi-coherent O X -module. Therefore, if F, G are α-twisted sheaves, then the sheaf Hom(F, G) is an O X -module On a Noetherian algebraic Artin stack any quasi-coherent sheaf is a direct limit of coherent sheaves. See??. In particular we see the same holds for α-twisted sheaves Azumaya algebras and α-twisted sheaves. Suppose that F is a finite locally free α-twisted sheaf. Then A = Hom(F, F) is a sheaf of O X -algebras. Since it is étale locally isomorphic to the endormophisms of a finite locally free module, we see that A is an Azumaya algebra. It is easy to verify that the Brauer class of A is α. Conversely, suppose that A is an Azumaya algebra over X. Consider the category X (A) whose objects are pairs (T, M, j), where T X is a scheme over X, M is a finite locally free O T -module, and j is an isomorphism j : Hom(M, M) A T. A morphism is defined to be a pair (f, i) : (T, M, j) (T, M, j ) where f : T T is a morphism of schemes over X, and i : f M M is an isomorphism compatible with j and j. Composition 3

4 of morphisms are defined in the obvious manner. As before we leave it to the reader to see that X (A) Sch is an algebraic Artin stack. Note that each object (T, M, j) has naturally G m (acting via the standard character on M) as its automorphism sheaf over Sch/T. Not only is it an algebraic stack, but also the morphism X (A) X presents X (A) as a G m -gerb over X. Since gerbes are classified by H 2 (X, G m ) we deduce that there is a unique cohomology class α such that X (A) is equivalent to the gerb X constructed in Subsection 2.5. Clearly, the gerb X carries a finite locally free sheaf F such that A = Hom(F, F), namely on X (A) it is the quasi-coherent module F whose value on the object (T, M, j) is the sheaf M (compare with the description of quasi-coherent sheaves in 2.7). Working backwards, we conclude that α is the Brauer class of A. The following lemma is a consequence of the discussion above Lemma. The element α H 2 (X, G m ) is in Br(X) if and only if there exists a finite locally free α-twisted sheaf of positive rank Let us use this lemma to reprove the following result (see Hoobler, Proposition 3): If α H 2 (X, G m ) and if there exists a finite locally free morphism ϕ : Y X such that ϕ (α) ends up in Br(Y ), then α in Br(X). Namely, this means there exists a finite locally free α-twisted sheaf F of positive rank over Y. Let Y be the G m -gerb associated to α Y and let us think of F as a sheaf on Y. Let ϕ : Y X be the obvious morphism of G m -gerbs lifting ϕ. The pushforward ϕ F is the desired flat and finitely presented α-twisted sheaf over X Suppose that x is a geometric point of X. Then we can lift the morphism x X to a morphism l : x X and all such lifts are isomorphic (not canonically). The fibre of a coherent sheaf F on X at x is simply defined to be l F. (Not the stalk!) Notation: F κ( x). This is a functor from coherent α-twisted modules to the category of finite dimensional κ( x)-vector spaces Suppose that s X is a point whose residue field is finite. Then, similarly to the above, we can find a lift l : s X, and we define the fibre F κ(s) := l F in the same way. 3 The proof 3.1 The only issue is to show that the map Br(X) Br (X) is surjective. Thus let us assume that α H 2 (X, G m ) is torsion, say nα = 0 for some n N. 3.2 As a first step we reduce to the case where X is a quasi-projective scheme of finite type over Spec Z. This is standard. See 2.1 and 2.2. In particular X is Noetherian, has finite dimension and is a Jacobson scheme. 3.3 Our method does not immediately produce an Azumaya algebra over X. Instead we look at the schemes X R = X R = X Spec Z Spec R, where Z R is a finite flat ring extension. We will always assume that R is actually a normal domain. An α-twisted sheaf on X R means an α XR -twisted sheaf. 4

5 3.4 Lemma. For any finite set of closed points T X R there exists a positive integer n and a coherent α-twisted sheaf F on X R such that F is finite locally free of rank n in a neighbourhood of each point t T. Proof. We can find a section s Γ(X, L N ) such that the open subscheme X s is affine and contains the image of T in X. See EGA II, Corollary By Gabber s result [Hoobler, Theorem 7] we get an Azumaya algebra A over X s representing α over X s. Thus we get a finite locally free α-twisted sheaf F s of positive rank over X s, see Subsection If F s does not have constant rank then we may modify it to have constant rank. (Namely, F s will have constant rank on the connected components of X s. Take suitable direct summands.) Let j : X s X be the open immersion which is the pullback of the open immersion X s X. Then j F s is a quasi-coherent α-twisted sheaf. It is the direct limit of its coherent subsheaves, see Thus there is a suitable F j F s such that j F = F s. The pullback of F to an α-twisted sheaf over X R is the desired sheaf. 3.5 For a coherent α-twisted sheaf F let Sing(F) denote the set of points of X R at which F is not flat. This is a closed subset of X R. We will show that by varying R we can increase the codimension of the singularity locus of F. 3.6 Let c 1 be an integer. Induction Hypothesis H c : For any finite subset of closed points T X R there exist (a) a finite flat extension R R, and (b) a coherent α-twisted sheaf F on X R such that (i) The codimension of Sing(F) in X R is c, (ii) the rank of F over X R Sing(F) constant and positive, and (iii) the inverse image T R of T in X R is disjoint from Sing(F). 3.7 Note that the case c = dimx + 1 and T = implies the theorem in the introduction. Namely, by Subsection 2.14 this implies that α is representable by an Azumaya algebra over X R. By [Hoober, Proposition 3] (see also our 2.15) it follows that α is representable by an Azumaya algebra over X. 3.8 The start of the induction, namely the case c = 1, follows easily from Lemma 3.4. Now we assume the hypothesis holds for some c 1 and we prove it for c Therefore, let T X be a finite subset of closed points. Pick a pair (R R 1, F 1) satisfying H c with regards to the subset T X. Set T 1 = T R1 S 1, where S 1 Sing(F 1) is a choice of a finite subset of closed points with the property that S 1 contains at least one point from each irreducible component of Sing(F 1) that has codimension c in X R 1. Next, let (R 1 R 2, F 2) be a pair satisfying H c with regards to the subset T 1 X R1. Set T 2 = (T 1 ) R2 S 2, where S 2 Sing(F 2) Sing(F 1) R 2 is a finite subset of closed points which contains at least one point of each irreducible component of Sing(F 2) that has codimension c in X R 2. Such a set S 2 exists because by construction the irreducible components of codimension c of Sing(F 2) are not contained in Sing(F 1 ) R 2. Choose a pair (R 2 R 3, F 3) adapted to T 2. Continue like this until you get a pair (R n+1, F n+1) adapted to T n X Rn. (Recall that n is a fixed integer such that nα = 0.) For clarity, we 5

6 stipulate at each stage that S j Sing(F j) i<j Sing(F i) R j, contains at least one point from each irreducible component of Sing(F j ) that has codimension c in X R j. We also choose a set S n+1 like this. Let us write F i = F i R i R n+1, and S i = S i R i R n+1. These are coherent α-twisted sheaves on X R n+1 and finite subsets of closed points S i X R n+1. They have the following properties: (a) The subset T R n+1 is disjoint from Sing(F i ). (b) Each F i has a constant positive rank over X Rn+1 Sing(F i ). (c) Each component of Sing(F i ) of codimension c in X R n meets the subset S i. (d) For each s S i the sheaves F j are finite locally free at s for j i. From now on we will only use the coherent α-twisted sheaves F i, i = 1,..., n+1, the subsets S i, i = 1,..., n + 1 and the properties (a), (b), (c) and (d) above. We will use them to construct an α-twisted sheaf F over X R for some finite flat extension R n+1 R whose singularity locus Sing(F) is contained in Sing(F i ) R such that S i R Sing(F) =. It is clear that this will be a sheaf as required in H c+1, and it will prove the induction step For ease of notation we write R in stead of R n+1 from now on, so that the coherent α-twisted sheaves F i are defined over X R, and so that the S i X R. Note that we may replace F i by direct sums F m i i for suitable integers m i such that each F i has the same rank r over the open X R Sing(F i ). We may also assume that r is a large integer Consider the α-twisted sheaf and since (n + 1)α = α the α-twisted sheaf Consider also the sheaf of homomorphisms G 1 := F1 rn... Fn+1 rn G 2 := F 1 F 2... F n+1. H := Hom(G 1, G 2 ) which is a coherent O X R -module (see 2.11). Recall that L is our ample invertible sheaf. Since X R X is finite, L R is ample on X R as well. For a very large integer N we are going to take a section ψ of the space Γ N := Γ(X R, H L N ) = Γ ( X R, Hom(G 1, G 2 L N ) ). The idea of the proof is that for a general ψ as above the kernel of the map of α-twisted sheaves G 1 G 2 L N is a solution to the problem we are trying to solve. However, it is not so easy to show there are any general sections. This is why we may have to extend our ground ring R a little. 6

7 3.12 Claim. The kernel of a ψ Γ N is a solution to the problem if it satisfies the following properties: (a) For every geometric point x X, x Sing(F i ) the map ψ x : G 1 κ( x) G 2 L N κ( x) is a surjection. (See 2.16 for the fibre functor.) (b) Let s S i for some i. Then the composition F rn i is an isomorphism. (See 2.17.) κ(s) G 1 κ(s) G 2 L N κ(s) We will study the sheaf H and its sections. Along the way we will show that the claim holds and we will show that there is a finite extension R R such that a section ψ Γ N R can be found satisfying (a), (b) above. This will prove the theorem Local study of the sheaf Hom(G 1, G 2 ). Let Spec(B) X R be a morphism of schemes. Assume that α Spec(B) = 0. Thus there is a 1-morphism l : Spec(B) X lifting Spec(B) X. In other words the sheaves F i give rise to finitely generated B-modules M i. Then G 1 corresponds to M1 rn... Mn+1 rn and G 2 corresponds to M 1 B M 2 B... B M n+1. Finally, the sheaf H = Hom(G 1, G 2 ) pulls back to the quasi-coherent sheaf associated to the B-module Hom B (M rn 1... M rn n+1, M 1 B M 2 B... B M n+1 ) In particular, if x X is a point and O X,x B is an étale local ring extension, then the stalk H x is an O X,x -module H x such that H x B is isomorphic to the module displayed above Suppose that the point x is not in any of the closed subsets Sing(F i ). Then, with the notation above, we can choose isomorphisms M i = B r for all i, and we see that H x B = Hom B (B (n+1)rn+1, B rn+1 ) = Mat((n + 1)r n+1 r n+1, B). We conclude that the condition (a) at a geometric point x over x is defined by a cone C( x) in H κ( x) of codimension (n + 1)r n+1 r n nr n Let H be the vector bundle over U := X Sing(F i ) whose sheaf of sections is H U. (So H is the spectrum of the symmetric algebra on the dual of H.) The description above shows that there is a cone C H whose fibre at each point is the forbidden cone of lower rank maps. (In particular the codimension of the fibres C x H x is large.) Namely, the description above defines this cone étale locally over U. But the rank condition describing C is clearly preserved by the gluing data and hence C descends. 7

8 3.17 Suppose that the point x (as in 3.14) is one of the closed points s S i. In this case we can find an extension O X,x B as in 3.14 with trivial residue field extension. Namely, the restriction of α to the henselization OX,s h will be trivial. Namely, the Brauer group of a henselian local ring is easily seen to be equal to the Brauer group of the residue field which is trivial in this case. In this case the modules M j for j i are finite locally free, but M i is not. We choose isomorphisms M j = B r for j i and we set M i = M. Thus we observe that in this case H s B = Hom B (M rn, M rn ) Hom B (B nrn+1, M rn ). The condition (b) for our point s simply means that if we write ψ B = ψ 1 ψ 2, then ψ 1 κ(s) should be a surjection from M rn κ(s) to itself. By Nakayama s Lemma and the fact that a surjective self map of a finitely generated module over a Noetherian local ring is an isomorphism, this is equivalent to the condition that ψ 1 is an isomorphism Let ψ s H κ(s) be an element which corresponds to the image id M r n 0 in the fibre under an isomorphism as above. Fix an isomorphism L κ(s) = κ(s) so that we can identify the fibre of H and H L N. The following observation will be used below: If we have a global section ψ H L N which reduces modulo the maximal ideal m s to a nonzero multiple of ψ s for every s S i and every i then ψ has property (b) of Proof of Claim Suppose that ψ satisfies (a) and (b). Over the open subscheme U we see that ker(ψ) is the kernel of a surjective map of finite locally free α-twisted sheaves. Hence ker(ψ) is finite locally free over U. At each point s S we see that ker(ψ) B is going to be isomorphic to the factor B nrn+1 of G 1,B. Thus it is finite locally free at each s as well As a last step we still have to show that there exists a section ψ satisfying (a), (b) of 3.12 possibly after enlarging R. This will occupy the rest of the paper. It has nothing to do with Brauer groups. In fact, according to Subsection 3.18, we can formulate the problem as follows Situation. Let X be a quasi-projective scheme of finite type over R finte flat over Z as above. Let L be an ample invertible sheaf. Suppose that H is a coherent O X -module which is finite locally free over an open subscheme U X. Let H be the vector bundle U whose sheaf of sections is H U. Suppose that C H is closed cone in H. Let c N be such that the codimension of C u in H u is bigger than c for all u U. Furthermore, suppose S X is a finite set of closed points not in U, and suppose that we are given prescribed values ψ s H κ(s) (compare with 3.18) Claim. In the situation above, suppose that c > dimx + 1. There exists an N and a finite flat extension R R and a section ψ of H L N R R such that: (a) The set of points u of U R R such that ψ u C u L N u is empty. (b) For each closed point s of X R lying over a point of s S, the value of ψ κ(s ) of ψ at s is a nonzero multiple of ψ s. (Where the multiplier is actually an element of L N κ(s ).) Proof. First we remark that it suffices to prove the claim for R = Z. Namely, given a situation 3.21 and a solution ψ relative to X/Z on X Z R then we just set R = R Z R. So now we assume that R = Z. 8

9 Choose N 0 so that for all N N 0 there is a finite collection of global sections such that the map {Ψ i Γ(X, I S H L N ), i I} i I O X Ψi I S H L N is surjective. For each s S we may choose a global section Ψ s Γ(X, I S {s} H L N ) which reduces to a nonzero scalar multiple of ψ s at s (again we may need to increase N). Let A = Spec Z[x i, x s ; i I, s S]. There is a universal section Ψ = x i Ψ i + x s Ψ s of the pull back of H L N to A Spec Z X. In particular, in a point (a, u) of A U A X the fibre Ψ (a,u) can be seen as a point of H u L N u. In the open subscheme A U we let Z be the closed subset described by the following formula: Z = {(a, u) Ψ (a,u) C u L N u } The fibres of the morphism Z U have dimension at most #S+#I c, by our assumption that c bounds the codimensions of the cones C u from below, and the property that the sections Ψ i generate the sheaf H L N. Thus the dimension of Z is at most dima 2 (note that the dimension of A is #S + #I + 1). Let Z be the closure of the image of Z in A. For each element s S, let t s Spec R be the image of s in Spec R under the morphism X Spec R. Let Z s A t be the closed subscheme defined by x s = 0. This has codimension 2 in A as t is a closed point of Spec R. We want an Z-rational point of A which avoids Z Z s. This may not be possible. However, according to the main result of [1] we can find a finite extension R of Z and morphism Spec R into A which avoids Z Z s as desired. References [1] Moret-Bailly, Laurent, Groupes de Picard et problmes de Skolem. I, II, Ann. Sci. cole Norm. Sup. (4) 22 (1989), no. 2, pp , pp

### SEMINAR NOTES: G-BUNDLES, STACKS AND BG (SEPT. 15, 2009)

SEMINAR NOTES: G-BUNDLES, STACKS AND (SEPT. 15, 2009) DENNIS GAITSGORY 1. G-bundles 1.1. Let G be an affine algebraic group over a ground field k. We assume that k is algebraically closed. Definition 1.2.

### FIBER PRODUCTS AND ZARISKI SHEAVES

FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also

### If I and J are graded ideals, then since Ĩ, J are ideal sheaves, there is a canonical homomorphism J OX. It is equal to the composite

Synopsis of material from EGA Chapter II, 2.5 2.9 2.5. Sheaf associated to a graded module. (2.5.1). If M is a graded S module, then M (f) is an S (f) module, giving a quasi-coherent sheaf M (f) on Spec(S

### Lifting abelian schemes: theorems of Serre-Tate and Grothendieck

Lifting abelian schemes: theorems of Serre-Tate and Grothendieck Gabriel Dospinescu Remark 0.1. J ai décidé d écrire ces notes en anglais, car il y a trop d accents auxquels il faut faire gaffe... 1 The

### Proof. The map. G n i. where d is the degree of D.

7. Divisors Definition 7.1. We say that a scheme X is regular in codimension one if every local ring of dimension one is regular, that is, the quotient m/m 2 is one dimensional, where m is the unique maximal

Algebraic Geometry Keerthi Madapusi Contents Chapter 1. Schemes 5 1. Spec of a Ring 5 2. Schemes 11 3. The Affine Communication Lemma 13 4. A Criterion for Affineness 15 5. Irreducibility and Connectedness

### On The Existence Of Flips

On The Existence Of Flips Hacon and McKernan s paper, arxiv alg-geom/0507597 Brian Lehmann, February 2007 1 Introduction References: Hacon and McKernan s paper, as above. Kollár and Mori, Birational Geometry

### Applying this to X = A n K it follows that Cl(U) = 0. So we get an exact sequence

3. Divisors on toric varieties We start with computing the class group of a toric variety. Recall that the class group is the group of Weil divisors modulo linear equivalence. We denote the class group

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it

### Row Ideals and Fibers of Morphisms

Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

### ANAGRAMS: Dimension of triangulated categories

ANAGRAMS: Dimension of triangulated categories Julia Ramos González February 7, 2014 Abstract These notes were prepared for the first session of "Dimension functions" of ANAGRAMS seminar. The goal is to

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### 3. Prime and maximal ideals. 3.1. Definitions and Examples.

COMMUTATIVE ALGEBRA 5 3.1. Definitions and Examples. 3. Prime and maximal ideals Definition. An ideal P in a ring A is called prime if P A and if for every pair x, y of elements in A\P we have xy P. Equivalently,

### Analytic cohomology groups in top degrees of Zariski open sets in P n

Analytic cohomology groups in top degrees of Zariski open sets in P n Gabriel Chiriacescu, Mihnea Colţoiu, Cezar Joiţa Dedicated to Professor Cabiria Andreian Cazacu on her 80 th birthday 1 Introduction

### Homological Algebra - Problem Set 3

Homological Algebra - Problem Set 3 Problem 1. Let R be a ring. (1) Show that an R-module M is projective if and only if there exists a module M and a free module F such that F = M M (we say M is a summand

### Kenji Matsuki and Martin Olsson

Mathematical Research Letters 12, 207 217 (2005) KAWAMATA VIEHWEG VANISHING AS KODAIRA VANISHING FOR STACKS Kenji Matsuki and Martin Olsson Abstract. We associate to a pair (X, D), consisting of a smooth

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### Math 231b Lecture 35. G. Quick

Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

### Revision of ring theory

CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic

### A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE

A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE DANIEL A. RAMRAS In these notes we present a construction of the universal cover of a path connected, locally path connected, and semi-locally simply

### NOTES ON CATEGORIES AND FUNCTORS

NOTES ON CATEGORIES AND FUNCTORS These notes collect basic definitions and facts about categories and functors that have been mentioned in the Homological Algebra course. For further reading about category

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### arxiv:math/0402080v1 [math.nt] 5 Feb 2004

BIEXTENSIONS OF 1-MOTIVES BY 1-MOTIVES arxiv:math/0402080v1 [math.nt] 5 Feb 2004 CRISTIANA BERTOLIN Abstract. Let S be a scheme. In this paper, we define the notion of biextensions of 1-motives by 1-motives.

### Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

### DEFORMATIONS OF MODULES OF MAXIMAL GRADE AND THE HILBERT SCHEME AT DETERMINANTAL SCHEMES.

DEFORMATIONS OF MODULES OF MAXIMAL GRADE AND THE HILBERT SCHEME AT DETERMINANTAL SCHEMES. JAN O. KLEPPE Abstract. Let R be a polynomial ring and M a finitely generated graded R-module of maximal grade

### INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 16

INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 16 RAVI VAKIL Contents 1. Valuation rings (and non-singular points of curves) 1 1.1. Completions 2 1.2. A big result from commutative algebra 3 Problem sets back.

### Lecture 4 Cohomological operations on theories of rational type.

Lecture 4 Cohomological operations on theories of rational type. 4.1 Main Theorem The Main Result which permits to describe operations from a theory of rational type elsewhere is the following: Theorem

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### GROUP ALGEBRAS. ANDREI YAFAEV

GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite

### SINGULARITIES APPEARING ON GENERIC FIBERS OF MORPHISMS BETWEEN SMOOTH SCHEMES

SINGULARITIES APPEARING ON GENERIC FIBERS OF MORPHISMS BETWEEN SMOOTH SCHEMES STEFAN SCHRÖER 29 June 2007, final version Abstract. I give various criteria for singularities to appear on geometric generic

### Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces

Communications in Mathematical Physics Manuscript-Nr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan Bechtluft-Sachs, Marco Hien Naturwissenschaftliche

### SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS

SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS Ivan Kolář Abstract. Let F be a fiber product preserving bundle functor on the category FM m of the proper base order r. We deduce that the r-th

### The Euler class groups of polynomial rings and unimodular elements in projective modules

The Euler class groups of polynomial rings and unimodular elements in projective modules Mrinal Kanti Das 1 and Raja Sridharan 2 1 Harish-Chandra Research Institute, Allahabad. Chhatnag Road, Jhusi, Allahabad

### RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific

### DEFINITION OF PURE HODGE MODULES

DEFINITION OF PURE HODGE MODULES DONGKWAN KIM 1. Previously in the Seminar... 1 2. Nearby and Vanishing Cycles 2 3. Strict Support 4 4. Compatibility of Nearby and Vanishing Cycles and the Filtration 5

### Yan Zhao. 27 November 2013 / 4 December 2013

Géométrie Algébrique et Géométrie Analytique Yan Zhao 27 November 2013 / 4 December 2013 In these notes, we give a quick introduction to the main classical results in the theory of Géométrie Algébrique

### BABY VERMA MODULES FOR RATIONAL CHEREDNIK ALGEBRAS

BABY VERMA MODULES FOR RATIONAL CHEREDNIK ALGEBRAS SETH SHELLEY-ABRAHAMSON Abstract. These are notes for a talk in the MIT-Northeastern Spring 2015 Geometric Representation Theory Seminar. The main source

### THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

### DIVISORS AND LINE BUNDLES

DIVISORS AND LINE BUNDLES TONY PERKINS 1. Cartier divisors An analytic hypersurface of M is a subset V M such that for each point x V there exists an open set U x M containing x and a holomorphic function

### EXERCISES FOR THE COURSE MATH 570, FALL 2010

EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime

### OSTROWSKI FOR NUMBER FIELDS

OSTROWSKI FOR NUMBER FIELDS KEITH CONRAD Ostrowski classified the nontrivial absolute values on Q: up to equivalence, they are the usual (archimedean) absolute value and the p-adic absolute values for

### COHOMOLOGY OF GROUPS

Actes, Congrès intern. Math., 1970. Tome 2, p. 47 à 51. COHOMOLOGY OF GROUPS by Daniel QUILLEN * This is a report of research done at the Institute for Advanced Study the past year. It includes some general

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### Fiber bundles and non-abelian cohomology

Fiber bundles and non-abelian cohomology Nicolas Addington April 22, 2007 Abstract The transition maps of a fiber bundle are often said to satisfy the cocycle condition. If we take this terminology seriously

### The Euler class group of a polynomial algebra

The Euler class group of a polynomial algebra Mrinal Kanti Das Harish-Chandra Research Institute, Allahabad. Chhatnag Road, Jhusi, Allahabad - 211 019, India. e-mail : mrinal@mri.ernet.in 1 Introduction

### The topology of smooth divisors and the arithmetic of abelian varieties

The topology of smooth divisors and the arithmetic of abelian varieties Burt Totaro We have three main results. First, we show that a smooth complex projective variety which contains three disjoint codimension-one

### Notes on Localisation of g-modules

Notes on Localisation of g-modules Aron Heleodoro January 12, 2014 Let g be a semisimple Lie algebra over a field k. One is normally interested in representation of g, i.e. g-modules. If g were commutative,

### Seminar: Stacks and Construction of M g Winter term 2016/17 Time: Thursday 4-6pm Room: Room 0.008

Seminar: Stacks and Construction of M g Winter term 2016/17 Time: Thursday 4-6pm Room: Room 0.008 Contact: Ulrike Rieß Professor: Prof. Dr. Daniel Huybrechts First session for introduction (& organization):

### Factoring of Prime Ideals in Extensions

Chapter 4 Factoring of Prime Ideals in Extensions 4. Lifting of Prime Ideals Recall the basic AKLB setup: A is a Dedekind domain with fraction field K, L is a finite, separable extension of K of degree

### NOTES ON DEDEKIND RINGS

NOTES ON DEDEKIND RINGS J. P. MAY Contents 1. Fractional ideals 1 2. The definition of Dedekind rings and DVR s 2 3. Characterizations and properties of DVR s 3 4. Completions of discrete valuation rings

### The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

### SMALL SKEW FIELDS CÉDRIC MILLIET

SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### Commutative Algebra seminar talk

Commutative Algebra seminar talk Reeve Garrett May 22, 2015 1 Ultrafilters Definition 1.1 Given an infinite set W, a collection U of subsets of W which does not contain the empty set and is closed under

### ADDITIVE GROUPS OF RINGS WITH IDENTITY

ADDITIVE GROUPS OF RINGS WITH IDENTITY SIMION BREAZ AND GRIGORE CĂLUGĂREANU Abstract. A ring with identity exists on a torsion Abelian group exactly when the group is bounded. The additive groups of torsion-free

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).

### arxiv: v2 [math.ag] 10 Jul 2011

DEGREE FORMULA FOR THE EULER CHARACTERISTIC OLIVIER HAUTION arxiv:1103.4076v2 [math.ag] 10 Jul 2011 Abstract. We give a proof of the degree formula for the Euler characteristic previously obtained by Kirill

### Algebraic Geometry. Andreas Gathmann. Notes for a class. taught at the University of Kaiserslautern 2002/2003

Algebraic Geometry Andreas Gathmann Notes for a class taught at the University of Kaiserslautern 2002/2003 CONTENTS 0. Introduction 1 0.1. What is algebraic geometry? 1 0.2. Exercises 6 1. Affine varieties

### University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

### FACTORING IN QUADRATIC FIELDS. 1. Introduction. This is called a quadratic field and it has degree 2 over Q. Similarly, set

FACTORING IN QUADRATIC FIELDS KEITH CONRAD For a squarefree integer d other than 1, let 1. Introduction K = Q[ d] = {x + y d : x, y Q}. This is called a quadratic field and it has degree 2 over Q. Similarly,

### SEMINAR NOTES: HITCHIN FIBERS AND AFFINE SPRINGER FIBERS, OCTOBER 29 AND NOVEMBER 5, 2009

SEMINAR NOTES: HITCHIN FIBERS AND AFFINE SPRINGER FIBERS, OCTOBER 29 AND NOVEMBER 5, 2009 ROMAN BEZRUKAVNVIKOV 1. Affine Springer fibers The goal of the talk is to introduce affine Springer fibers and

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### 1 Local Brouwer degree

1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.

### ORIENTATIONS. Contents

ORIENTATIONS Contents 1. Generators for H n R n, R n p 1 1. Generators for H n R n, R n p We ended last time by constructing explicit generators for H n D n, S n 1 by using an explicit n-simplex which

### 3. Recurrence Recursive Definitions. To construct a recursively defined function:

3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a

### The Mathematics of Origami

The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................

### Mathematische Zeitschrift

Math. Z. (212 27:871 887 DOI 1.17/s29-1-83-2 Mathematische Zeitschrift Modular properties of nodal curves on K3 surfaces Mihai Halic Received: 25 April 21 / Accepted: 28 November 21 / Published online:

### CLUSTER ALGEBRAS AND CATEGORIFICATION TALKS: QUIVERS AND AUSLANDER-REITEN THEORY

CLUSTER ALGEBRAS AND CATEGORIFICATION TALKS: QUIVERS AND AUSLANDER-REITEN THEORY ANDREW T. CARROLL Notes for this talk come primarily from two sources: M. Barot, ICTP Notes Representations of Quivers,

### Integral Domains. As always in this course, a ring R is understood to be a commutative ring with unity.

Integral Domains As always in this course, a ring R is understood to be a commutative ring with unity. 1 First definitions and properties Definition 1.1. Let R be a ring. A divisor of zero or zero divisor

### Algebraic K-Theory of Ring Spectra (Lecture 19)

Algebraic K-Theory of ing Spectra (Lecture 19) October 17, 2014 Let be an associative ring spectrum (here by associative we mean A or associative up to coherent homotopy; homotopy associativity is not

### Chapter 13: Basic ring theory

Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

### THE FANO THREEFOLD X 10. This is joint work in progress with A. Iliev (Sofia) and L. Manivel (Grenoble).

THE FANO THREEFOLD X 10 OLIVIER DEBARRE This is joint work in progress with A. Iliev (Sofia) and L. Manivel (Grenoble). 1. Introduction A Fano variety is a (complex) projective manifold X with K X ample.

### FIBERS OF GENERIC PROJECTIONS

FIBERS OF GENERIC PROJECTIONS ROYA BEHESHTI AND DAVID EISENBUD Abstract. Let X be a smooth projective variety of dimension n in P r, and let π : X P n+c be a general linear projection, with c > 0. In this

### INTRODUCTION TO MORI THEORY. Cours de M2 2010/2011. Université Paris Diderot

INTRODUCTION TO MORI THEORY Cours de M2 2010/2011 Université Paris Diderot Olivier Debarre March 11, 2016 2 Contents 1 Aim of the course 7 2 Divisors and line bundles 11 2.1 Weil and Cartier divisors......................................

### Appendix A. Appendix. A.1 Algebra. Fields and Rings

Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.

### 1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]

1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not

### An Advanced Course in Linear Algebra. Jim L. Brown

An Advanced Course in Linear Algebra Jim L. Brown July 20, 2015 Contents 1 Introduction 3 2 Vector spaces 4 2.1 Getting started............................ 4 2.2 Bases and dimension.........................

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 24

INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 24 RAVI VAKIL Contents 1. Degree of a line bundle / invertible sheaf 1 1.1. Last time 1 1.2. New material 2 2. The sheaf of differentials of a nonsingular curve

### Group Theory. Contents

Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

### 8.1 Examples, definitions, and basic properties

8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.

### Parametric Domain-theoretic models of Linear Abadi & Plotkin Logic

Parametric Domain-theoretic models of Linear Abadi & Plotkin Logic Lars Birkedal Rasmus Ejlers Møgelberg Rasmus Lerchedahl Petersen IT University Technical Report Series TR-00-7 ISSN 600 600 February 00

### Basic theory of group schemes.

Chapter III. Basic theory of group schemes. As we have seen in the previous chapter, group schemes come naturally into play in the study of abelian varieties. For example, if we look at kernels of homomorphisms

### OPERS. The geometric Langlands correspondence conjectures a correspondence

OPERS JONATHAN BARLEV The geometric Langlands correspondence conjectures a correspondence Qcoloc-sysL Gx)) = D-modBun G )) on the level of derived categories. As remarked previously in theseminar,toeach

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

### Factorization in Polynomial Rings

Factorization in Polynomial Rings These notes are a summary of some of the important points on divisibility in polynomial rings from 17 and 18 of Gallian s Contemporary Abstract Algebra. Most of the important

### A SURVEY OF ALGEBRAIC COBORDISM

A SURVEY OF ALGEBRAIC COBORDISM MARC LEVINE Abstract. This paper is based on the author s lecture at the ICM Satellite Conference in Algebra at Suzhou University, August 30-September 2, 2002, describing

### Finite dimensional topological vector spaces

Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the

### A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number

Number Fields Introduction A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number field K = Q(α) for some α K. The minimal polynomial Let K be a number field and

### The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

### REPRESENTATION THEORY WEEK 12

REPRESENTATION THEORY WEEK 12 1. Reflection functors Let Q be a quiver. We say a vertex i Q 0 is +-admissible if all arrows containing i have i as a target. If all arrows containing i have i as a source,