Introduction to Chiral Algebras


 Bryan Snow
 3 years ago
 Views:
Transcription
1 Introduction to Chiral Algebras Nick Rozenblyum Our goal will be to prove the fact that the algebra End(V ac) is commutative. The proof itself will be very easy  a version of the Eckmann Hilton argument  once the machinery of chiral algebras is set up. 1 Chiral Algebras Let X be a smooth curve over C. A nonunital chiral algebra on X is a Dmodule A along with a chiral bracket map µ : j j (A A)! (A) where j : U X 2 Z : are the inclusion of the complement of the diagonal and the diagonal respectively. We require that µ be antisymmetric and satisfy a version of the Jacobi identity: Antisymmetry: µ = σ 1,2 µ σ 1,2, where σ 1,2 is induced action on A by permuting the variables of X 2. Jacobi Identity: we have three maps µ 1(23), µ (12)3 and µ 2(13) : j j (A A A)! (A) where (somewhat abusing notation) j is the inclusion of the open in X 3 which is the complement of all the diagonals and is the inclusion of X as the diagonal. We have that µ 1(23) is defined as the composition j j (A A A) (x2 =x 3 )!j j (A A)! A and the others are defined similarly. We then demand that µ 1(23) = µ (12)3 + µ 2(13) 1
2 Example 1. Let A = ω X. We then have the canonical exact sequence 0 ω X ω X j j (ω X ω X )! (ω X ) 0 which gives ω X a chiral bracket. It is clearly antisymmetric. We need to check that it satisfies the Jacobi identity. To do that, consider the Cousin complex for ωx 3 on X3 for the stratification given by the diagonals. It gives the exact sequence 0 ω 3 X j j (ω 3 X ) The three maps in the complex x1 =x 2!j j (ωx 2) x1 =x 3!j j (ωx 2) x2 =x 3!j j (ωx 2)! (ω X ) 0 j j (ω 3 X )! (ω X ) are exactly the maps in the Jacobi identity and the fact that the above Cousin complex is a complex at the term between those is exactly the condition that the Jacobi identity is satisfied. We can now finish defining a chiral algebra. Definition 2. A (unital) chiral algebra A is a nonunital chiral algebra together with a map of chiral algebras ω X A such that the restriction of the chiral bracket µ : j j (ω X A)! (A) is the canonical map coming from the complex 0 ω X A j j (ω X A)! (A). As with any kind of algebra, given a chiral algebra A, we can consider modules over it. 2
3 Definition 3. Let A be a chiral algebra. A chiral A module is a Dmodule M on X together with an action map satisfying the unit and Lie identity: ρ : j j (A M)! (M) Unit: we require that the restriction of ρ to ω X be the canonical map. Lie action: ρ : j j (ω X M)! (M) ρ(µ id) = ρ(id ρ) σ 12 ρ((id ρ) σ 12 ) as maps j j (A A M)! (M). Example 4. Let A be a chiral algebra. Then A is canonically a chiral A module. Let M be a Dmodule on X. Then the canonical map j j (ω M)! (M) makes M into a chiral ω module. In fact, because of the unit axiom, this is the unique structure of a chiral ω module on M. Thus we have an equivalence of categories {Dmodules} {chiral ω modules}. For our purposes, we will assume that A is flat as an O X module. However, we will not make a similar assumption on M. For instance, we will often be interested in modules supported at a point x X. A typical example of this is the vacuum module M = i x! i! x(a)[1]. Recall that for Dmodules we have the derham functor h : Dmods {sheaves} 3
4 given by modding out by the action of vector fields: h(m) = M/M Θ. Now, for a chiral algebra A, consider the composition Applying the derham functor, we get which by adjunction gives the map A A j j (A A)! (A). h(a) h(a) (h(a)) h(a) h(a) h(a) which makes h(a) into a sheaf of Lie algebras. Now suppose that M is a chiral A module supported at a point x X. Let M = i! x(m) be the underlying vector space. Pushing forward the action map j j (A M)! (M) along the first projection we get DR(X x, A) M M which is an action of the Lie algebra DR(X x, A) on M. In fact, we can shrink the curve X to get an action of the topological Lie algebra DR(D x, A) on M where DR(D x, A) = lim i! x(j x j x(a)/a ξ ) where the inverse limit is taken over submodules A ξ j x j x(a) such that the quotient j x j x(a)/a ξ is supported at x and j x is the inclusion the open set X x. 2 Factorization There is another equivalent description of chiral algebras in terms of factorization which given what we ve been doing in the seminar might be more 4
5 familiar. For the moment, let X be a topological space. We can then consider the Ran space of X defined as Ran(X) = {nonempty finite subsets of X} It is topologized so that the maps X n Ran(X) are continuous. There is a very important fact which will not be relevant for now, but is very important when dealing with homology of chiral algebas. Theorem 5. If X is connected, the topological space Ran(X) is weakly contractible. We will be interested in doing algebraic geometry on Ran(X) for X an algebraic curve. Unfortunately, it is not possible to define Ran(X) as any kind of algebraic space but we will be able to make sense of quasicoherent sheaves on Ran(X). So let s return to X being an algebraic curve over C. Definition 6. A quasicoherent sheaf F on Ran(X) is a collection of quasi coherent sheaves F I for each finite set I together with isomorphisms ν (π) : (J/I) F J F I for every surjection π : J I, where J/I : X I X J is the corresponding diagonal. We require that the ν (π) be compatible with composition of surjections. Moreover, we demand that the F (I) have no sections supported on the diagonals. Remark 7. Because of the condition requiring no sections supported on the diagonals, quasicoherent sheaves on Ran(X) do not form an abelian category. Definition 8. A nonunital factorization algebra B is a quasicoherent sheaf on Ran(X) along with isomorphisms c α : j α( B (I i) ) j α(b (I) ) for a partition α : I = I 1... I n a partition of I and j α is the inclusion of the open set U = {x i x j if i and j are in different I j }. We require that the c α be compatible with subpartitions and with the ν (π). 5
6 Example 9. Let O be the nonunital factorization algebra given by O (I) = O X I. This is a factorizable algebra in the obvious way. It is the unit factorization algebra. Definition 10. A (unital) factorization algebra B is a nonunital factorization algebra equipped with a map of nonunital factorization algebras O B such that locally for every section b B (1), 1 b j j (B (1) B (1) ) lies in B (2) j j (B (1) B (1) ) and (1 b) = b. Remark 11. In the definition of a factorization algebra, we required that the unit give a map B (1) O X B (2) compatible with restriction to the diagonal. In fact, we leave it to the reader to check that this implies that we have canonical maps B (I 1) O X I 2 B (I 1 I 2 ) compatible with factorization and restrictions to the diagonals. This follows from the condition requiring no sections supported on the diagonals for quasicoherent sheaves on Ran(X). If we consider dgfactorization algebras, then we need to specify all these maps as part of the data of a unital dgfactorization algebra. Theorem 12. There is an equivalence of categories {factorization algebras} {chiral algebras} given by B B (1) ω X Proof. Let B be a factorization algebra. Let s show that each B (I) has a canonical structure of a left Dmodule. Giving B (I) such a structure is equivalent to giving an isomorphism between B (I) O X I and O X I B (I) on the formal completion of the diagonal in X I X I. The unit gives maps B (I) O X I B (I I) O X I B (I) 6
7 which are isomorphisms on the formal neighborhood of the diagonal X I giving the canonical connection. Now, let A (I) = B (I) ω X I be the corresponding right Dmodules and let A = A (1). We then have the Cousin complex for A (2) : 0 A (2) j j (A A)! (A) 0 which gives the chiral bracket. It is clearly antisymmetric and the unit axiom is satisfied. The Cousin complex for A (3) with stratification given by the diagonals in X 3 gives the Jacobi identity. Thus, we have a functor from factorization algebras to chiral algebras. Let us now construct the inverse functor. Let A be a chiral algebra. On X I consider the ChevalleyCousin complex: C I = j j (A} {{ A} ) α! (j j (A A))...! (A) Itimes α P art I 1 (I) where P art n (I) is the set of partitions of I into n subsets and α for α P art n (I) is the corresponding n dimensional diagonal. The terms with n copies of A are in degree n and the differentials are given by the various chiral brackets. This complex is called the ChevalleyCousin complex because it is the Chevalley complex from the point of view of the chiral algebra, and it is the Cousin complex (for the stratification given by the diagonals) from the point of view of the factorization algebra. Evidently, these complexes are factorize and are compatible with restriction to diagonals, i.e. we have isomorphisms for surjections π : J I and for partitions I = I 1... I n. ν (π) : (J/I)! C J C I c α : j α( C I i ) j α(c I ) We will show that H n (C I ) = 0 unless n = I by induction on I. For 7
8 I = 1, the complex is just given by A and there is nothing to prove. Now, for a general I, consider a codimension one diagonal i : X I X I U : v and a given decomposition I = I [1], with U the complement of X I. The open set U is affine and we have a short exact sequence of complexes 0 i! (C I ) C I v v (C I ) 0. By induction, H n (i! (C I )) = 0 unless n = I = I + 1 and by induction and factorization H n (v v (C I )) = 0 unless n = I since U is a union of complements of diagonals. Thus we need to show that the map vanishes. Let H I +1 (i! (C I )) H I +1 (C I ) Z := H I +1 (C I ) = Ker(C I I C I +1 I ) We have a canonical exact sequence with respect to the decomposition X I = X I X 0 Z ω X v v (Z ω X ) i! (Z) 0. Furthermore, we have a commutative diagram v v (Z ω) i! (Z) C I I d C I +1 I where the top map is given by the unit. It follows that the map is indeed zero. H I +1 (i! (C I )) H I +1 (C I ) Now, let B (I) := H I (C I ) ω 1 X I. By above, it is a factorization algebra. Furthermore, it gives the inverse functor {chiral algebras} {factorization algebras}. 8
9 The factorization perspective on chiral algebras is a very useful one. Suppose that B 1 and B 2 are factorization algebras. Then B (I) = B (I) 1 B (I) 2 is also a factorization algebra in the obvious way. In this way, we can consider tensor products of chiral algebras. Another important aspect of factorization algebra is that they admit a nonlinear analogue of factorization spaces. This will allow us to construct chiral algebras from geometry. Definition 13. A factorization space G is a collection of (ind) schemes G I over X I for each finite set I along with isomorphisms (G J ) X I G I for every diagonal embedding X I X J and for every partition I = I 1... I n factorization ismorphisms G I U ( G Ii ) U where U is the open set corresponding to the partition. We require that these isomorphisms be compatible in the usual way. A factorization space is unital if in addition we have maps X I 1 G I2 G I2 I 2 compatible with factorization and restrictions to diagonals. We have already seen an example of a unital factorization space, namely the BeilinsonDrinfeld Grassmannian. Let G be an algebraic group. Recall that the BeilinsonDrinfeld Grassmannian Gr I (G) X I is defined as the moduli space of the following triples (in what follows we omit reference to G in the notation) Gr I = {P Bun G (X), (x 1,..., x I ) X I, φ : P X {(xi )} P triv X {(xi )}} where P triv Bun G (X) is the trivial Gbundle. Now, suppose we are given a unital factorization space G and a linearization functor, i.e. a rule for obtaining a sheaf on X I from each G I which preserves factorization then we can obtain a factorization algebra. Examples 9
10 of such linearization functors are global sections and pushforward of some canonically defined sheaf. In the case of the BeilinsonDrinfeld Grassmannian, consider for each I, S I the sheaf of Dmodule δ functions along the unit section X I Gr I. Now let V I be the pushforward of S I to X I as a quasicoherent sheaf. These form a factorization algebra with fibers given by the vacuum module for every x X. We could also consider δ functions as a twisted Dmodule and in the same way obtain a factorization algebra with fibers given by the vacuum module at the corresponding level. In addition to describing chiral algebras in factorization terms, we can also describe modules. Definition 14. Let B be a factorization algebra. A factorization B module M is a collection of Dmodules M Ĩ on XĨ for each finite set I, where Ĩ = I { }. For every surjection π : J Ĩ such that π( ) =, we have the corresponding diagonal XĨ X J and we are given isomorphisms M ( J) X Ĩ M (Ĩ) and for every partition I = I 1... I n, we are given factorization isomorphisms MĨ U ( B I i I M n ) U 0<i<n where U XĨ is the open set corresponding to the partition Ĩ = I 1... I n 1 Ĩn. We require that the isomorphisms be mutually compatible. Theorem 15. Let A be a chiral algebra and B be the corresponding factorization algebra. There is an equivalence of categories {factorization Bmodules} {chiral Amodules} As a consequence, we can consider tensor products of chiral modules: let B 1 and B 2 be factorization algebras and M 1 and M 2 respective factorization modules. Then M = M 1 M 2 is a factorization module for the factorization algebra B = B 1 B 2. 10
11 3 Lie* algebras A Lie* algebra L is a Dmodule on X along with a Lie* bracket µ : L L! (L) satisfying the Jacobi identity (in the same way as in the definition of a chiral algebra). Remark 16. Suppose that a Lie* algebra L is holonomic as a Dmodule. In this case, we have the functor which is left adjoint to! =. In this case, we can consider the *tensor product L L := (L L) and being a Lie* algebra is equivalent to being a Lie algebra µ : L L L in the tensor category of holonomic Dmodules with the *tensor product. An advantage of considering Lie* algebras is that it is relatively easy to construct examples. Suppose g is a Lie algebra. Then g D X is a Lie* algebra. In fact, we could implement this construction for any Lie algebra in the category of quasicoherent sheaves on X by instead tensoring over O X. Let A be a chiral algebra. Then the composition A A j j (A A)! (A) makes A into a Lie* algebra. In fact this functor has a left adjoint. Theorem 17. The above functor {chiral algebras} {Lie* algebras} has a left adjoint L A(L) called the chiral envelope. Proof (Sketch). The statement is local on the curve so we can assume without loss of generality that X is affine. Given a Lie* algebra, we will construct a factorization algebra using auxiliary Lie algebras. For a finite set I, consider the space X I X and let 11
12 p i for i = 1, 2 be the two projection maps to X I and X respectively. Let j : U X I X be the open subset given by Now, consider U = {((x i ), x) X I X x i x for i I}. L (I) 0 = p 1 p 2(L) ω 1 X I and L(I) = p 1 j j p 2(L) ω 1 X I. These are Lie algebras in the category of left Dmodules on X I. We have (I) that the fibers of L 0 and L (I) are given by L (I) 0 (x 1,...,x I ) = H dr(x, L) and L(I) (x 1,...,x I ) = H dr(x {x 1,..., x I }, L). By construction, we have that B(L) (I) = U( L (I) )/U( L (I) ) = Ind L (I) O L (I) X I 0 is a factorization algebra (here U(L) denotes the universal enveloping algebra of the Lie algebra L). Let A(L) be the corresponding chiral algebra. Note that we have an exact sequence of Dmodules on X 0 L (1) 0 L (1) L ω 1 X 0 which gives a map L A(L). The fact that it s a map of Lie* algebras follows from a similar exact sequence on X 2. Now suppose A is a chiral algebra and we have a map of Lie* algebras L A. Taking derham cohomology of the action map j j (L A)! (A) along the first component makes B (1) := A ω 1 X into a Lie L (1) module. The unit section of B (1) gives a map of left Dmodules U( L (1) ) B (1). Furthermore, the following diagram commutes L (1) Ã (1) L ω 1 X A ω 1 X 12
13 where Ã(1) = p 1 j j p 2(A) ω 1 (1). It follows that L X I 0 kills the unit section in B (1). This gives us the desired map A(L) A. A similar argument on X 2 shows that it s a map of chiral algebras. Consider the case where L = g D X. In this case, we have that the fibers A(L) x at x X are given by Since we have a Cartesian square A(L) x = Ind H0 dr (X x,g D X) H 0 dr (X,g D X) C. H 0 dr (X, g D X) H 0 dr (X x, g D X) H 0 dr (D x, g D X ) = g(o x ) H 0 dr (D x, g D X ) = g(k x ) it follows that A(L) x = Ind g(kx) g(o C = V ac x) x. Thus in this case the fibers are given by the vacuum module for g. In fact, this chiral algebra agrees with the one we constructed before using the Beilinson Drinfeld Grassmannian. For Lie* algebras, we can consider two types of modules: Definition 18. Let L be a Lie* algebra. A Lie* L module is a Dmodule M on X along with an action map satisfying the Lie action identity. ρ : L M! (M) A chiral L module is a Dmodule M on X along with a chiral action map ρ : j j (L M)! (M) satisfying the following condition. For j : U X 2 X the complement of the diagonals {x 1 = x} and {x 2 = x}, we have (similarly to the case of chiral modules over a chiral algebra) the maps ρ 1(23) : j j (L L M) id ρ 23! j j (L M)! (M) 13
14 with ρ (12)3 and ρ 2(13) defined similarly. We demand that ρ (12)3 = ρ 1(23) ρ 2(13). We have the following important fact about chiral modules over a Lie* algebra. Theorem 19. Let L be a Lie* algebra. Then there is an equivalence of categories {chiral Lmodules} {chiral A(L)modules}. Now, suppose M is a chiral Lmodule. We have a forgetful functor to Lie* modules given by the composition This functor has a left adjoint L M j j (L M)! (M). Ind : {Lie* Lmodules} {chiral Lmodules} given as follows. Let M be a Lie* Lmodule. Taking derham cohomology of the action map L M! (M) along the first component, we see that M := M ω 1 X Now, let Ind (M) (Ĩ) = Ind LĨ LĨ0p (M ) is a Lie L (1) 0 module. for a finite set I where p : XĨ X is the projection to the last component. We have that Ind (M) := {Ind (M)Ĩ} is a factorization module for the factorization algebra corresponding to A(L). We then define Ind(M) to be the corresponding chiral Lmodule. 4 Commutative Chiral Algebras Let A be a chiral algebra. We say that A is commutative if the composition vanishes. A A j j (A A)! (A) We have the following characterization of commutative chiral algebras. 14
15 Theorem 20. There is an equivalence of categories given by {commutative chiral algebras} {commutative left D X algebras} A A ω 1 X. Proof. Recall that we have a canonical exact sequence 0 A A j j (A A)! (A! A) 0. It follows that for a commutative chiral algebra, we have a map m : A! A A. In fact, as we ll see m makes A l := A ω 1 X a commutative algebra in the category of left Dmodules. Commutativity of m (on A l ) clearly follows from anticommutativity of the chiral bracket. Furthermore, the unit gives the unit section η : O X A l. Note that we can factor the chiral bracket as j j (A A) = (A l A l ) j j (ω X ω X ) (A l A l )! (ω X ) = Thus for a section we have =! (A l A l )! (ω X )! (A l )! (ω X ) =! (A). ((a b) s 2 ) A l A l j j (ω X ω X ) µ((a b) s 2 ) = (a b) µ ω (s 2 ) where (a b) := m(a, b) and µ ω is the chiral bracket for ω. In these terms, the Jacobi identity for A becomes (a (b c)) µ ω 1(23)(s 3 ) = ((a b) c) µ ω (12)3(s 3 ) + (b (a c)) µ ω 2(13)(s 3 ) for a section (a b c) s 3 (A l A l A l ) j j (ω X ω X ω X ). 15
16 From the Jacobi identity for ω, we deduce that all three terms on the left side of the tensors are equal. Thus, the product m on A l is associative. Now, suppose that B is a commutative left D X algebra. We can define a chiral bracket on A = B ω X by µ : j j (A A)! (A! A)! (A) where the first map is the canonical map and the second is the one given by the multiplication map on B. By a similar argument as above, this makes A into a chiral algebra. Now, suppose that B is a commutative left D X algebra. We will describe B modules in terms of the corresponding chiral algebra. Definition 21. Let A be a chiral algebra. A commutative Amodule is a chiral Amodule M such that the composition vanishes. A M j j (A M)! (M) Suppose B is a commutative left D X algebra and A the corresponding commutative chiral algebra. We then have an equivalence of categories given by {(D X B)modules} {commutative Amodules} M M := M ω X with the chiral action map given by the composition j j (A M)! (A! M)! (M) where the last map comes from the action map B M M. Now, for a commutative D X algebra B, we can describe the corresponding chiral algebra in factorization terms as follows. Let Z = Spec(B l ). Then Z is a D X scheme. In this situation, we can construct a counital factorization space Z I X I by considering multijets. An S point of Z I is given by a map φ : S X I along with a horizontal section ˆX S Z, where ˆX S is the 16
17 completion of X S along the subscheme given by the union of the graphs of φ. When Z is an affine D X scheme, each Z I is an affine D X Ischeme. Let (B) I be the left D X Imodule of global sections of Z I. As quasicoherent sheaves, these form a factorization algebra. Let A be the corresponding chiral algebra. Claim 22. The chiral algebra A is commutative and the chiral bracket is given by the multiplication map on B. To prove the claim, we will make use of the EckmannHilton argument. We will write down a formal algebraic proof, but here s the basic idea of the proof, which is extremely simple. Suppose you wanted to explain to someone that addition was commutative. For instance, say you wanted to show that = Well, is the number of marbles you have if you have two piles of marbles  7 marbles in the left pile and 15 marbles in the right pile. Well, is what you would get by moving one pile of marbles past the other. This is essentially the argument. A slightly more sophisticated version of this argument says that a monoid in the category of monoids is a commutative monoid. In this case, one can denote one multiplication as vertical composition and the other as horizontal and essentially carry out the same argument as with the marbles, moving one multiplication past the other. This is the EckmannHilton argument and it shows for instance that higher homotopy groups of a topological space are commutative. In our context, we have the following theorem. Theorem 23. Let A be a chiral algebra with a compatible unital binary operation m : A! A A Then A is a commutative chiral algebra, m makes A l into a commutative algebra and the chiral bracket on A factors through m. Proof. Clearly, it suffices to show that the chiral bracket factors through m. Compatibility of chiral bracket with the binary operation means that the following diagram is commutative (i.e. m is a map of chiral algebras): j j ((A! A) (A! A)) j j (A A)! (A! A)! (A) 17
18 We have two unit maps α, β : ω X A for the chiral bracket and the binary operation respectively. Let s show that these agree. From the commutativity of the above diagram and the unit axioms, we have the following commutative diagram j j ((ω α! ω β ) (ω β! ω α ))! (ω β! ω β ) =! (ω β )! (A) j j (ω α ω α )! (ω α ) It follows that the two units agree. Now, we can construct a section given by s : j j (A A) j j ((A! A) (A! A)) s : j j (A A) = j j ((A! ω) (ω! A) j j ((A! A) (A! A)) It follows that we the chiral bracket factors as j j (A A)! (A! A)! (A) where the last map is given by! (m). Remark 24. We can actually strengthen the above theorem slightly by not imposing the condition that A is a chiral algebra. All that is necessary is that there is a unital chiral operation and a unital binary operation µ : j j (A A)! (A) m : A! A A which are compatible. In this case, µ makes A into a commutative chiral algebra, m makes A l into a commutative left D X algebra and one determines the other. We leave the proof as an exercise for the reader. Let us now return to considering multijets. In this case, B is a commutative left D X algebra, and A is the chiral algebra corresponding to the factorization space given by multijets of Spec(B). As a right D X module, we clearly have A = B ω X, and the multiplication on B is compatible with the chiral bracket. It follows by EckmannHilton that A is a commutative chiral algebra and the chiral bracket is given by the multiplication map. 18
19 5 Factorization Modules on Higher Powers Let A be a chiral algebra. Thus far, we have considered chiral modules supported at a point and chiral modules on the curve. We can also define modules on higher powers of X. We do so in factorization terms. Let B be the corresponding factorization algebra. Definition 25. Let I 0 be a finite set. A factorization B module M on X I 0 is a collection D X Ĩmodules {MĨ} for finite sets I with Ĩ := I I 0. For every surjection π : J Ĩ such that π I 0 = id, we have the corresponding diagonal XĨ X J are we are given isomorphisms M ( J) X Ĩ M (Ĩ) and for every partition I = I 1... I n, we are given factorization isomorphisms MĨ U ( B (Ii) MĨn ) U 0<i<n where U XĨ is the open set corresponding to the partition Ĩ = I 1... Ĩn. We require that the isomorphisms be mutually compatible. One can also give a definition of factorization modules on X I 0 in terms of chiral operations similarly to the definition of chiral modules on X. In fact as in the case of chiral modules on X, we have that the forgetful functor {factorization Bmodules on X I 0 } {D X I 0 modules} if faithful. Furthermore, the category of factorization modules on X I 0 abelian category. is an Example For any I 0, B (I 0) is a factorization module on X I Suppose M is a module on X I 0, and for a finite set J 0 = J I 0, we have that N = M (J 0) is a factorization module on X J Suppose M is a factorization module on X I 0 then for any surjection π : J 0 I 0, we have the corresponding diagonal i : X I 0 X J 0. We then have i (M) is a factorization module on X J 0. Similarly, if N is a factorization module on X J 0, i (N ) is a factorization module on X I 0. 19
20 4. Suppose M is a factorization module on X I 0 and N is a factorization module on X J 0, then j j (M N ) is a factorization module on X I 0 J 0, where j : U = {(x i ), (y j ) x i y j } X I 0 X J 0. The last example above allows us to define a pseudotensor structure on the category of chiral Amodules on X by setting Hom({M 1, M 2 }, N) = Hom(j j (M 1 M 2 ), (N)) for M 1, M 2, N factorization A modules on X, where the Hom on the righthand side is in the category of factorization Amodules on X 2. 6 Chiral Algebra of Endomorphisms Let A be a chiral algebra and B the corresponding factorization algebra. We have seen that each B (I) is a factorization Bmodule on X I. Let REnd(B) (I) = RHom(B (I), B (I) ) where RHom is in the derived category of factorization Bmodules on X I. Let REnd(A) (I) = REnd(B) (I) ω X I be the corresponding right Dmodules. We have that REnd(B) is a dgfactorization algebra. Namely, for a diagonal : X I X J, we have L (REnd(B) (J) ) REnd(B) (I) and for a partition I = I 1... I n and j : U X I the corresponding open set j (REnd(B) (I) ) j ( REnd(B) (I i) ). Suppose that we are given a decomposition I = I 0 J. For the factorization B module on X I 0 M = B I 0, we have that M I = B I. It follows that we have maps p (REnd(B) I 0 ) = REnd(B) (I0) O X J REnd(B) (I) which makes REnd(B) into a unital dgfactorization algebra. It is coconnective, i.e. it has no cohomology in negative degrees. 20
21 Lemma 27. Let B be a coconnective unital dgfactorization algebra. Then (B 0 ) (I) = H 0 (B (I) ) is a unital factorization algebra. Proof. Let A (I) = B (I) ω X I and A (I) 0 = B (I) 0 ω X I be the corresponding right Dmodule. The Cousin complex for A (2) gives the triangle A (2) j j (A (2) )! (A (1) ). The long exact sequence in cohomology gives the exact sequence 0 A (2) 0 j j (A (2) 0 )! (A (1) 0 ). The unit gives a commutative diagram 0 A (2) 0 j j (A (2) 0 )! (A (1) 0 ) 0 A (1) 0 ω X j j (A (1) 0 ω X )! (A (1) 0 ) 0 It follows that j j (A (2) 0 ) = j j (A (1) 0 A (1) 0 )! (A (1) 0 ) is surjective. Similar considerations on higher powers of X shows that A (1) 0 is a chiral algebra with B 0 the corresponding factorization algebra. For a chiral algebra A with corresponding factorization algebra B, let End(A) be the chiral algebra corresponding to the factorization algebra given by H 0 (REnd(B)). Composition of morphisms gives us an algebra map End(A)! End(A) End(A) which is compatible with the chiral algebra structure. It follows that End(A) is a commutative chiral algebra. Let A g be the chiral envelope of the Lie* algebra D X g for a Lie algebra g. We want to show that the algebra of endomorphisms of the vacuum module of A g supported at a point is commutative. Even though the chiral algebra End(A) is commutative for any chiral algebra A, the corresponding statement for the vacuum module supported at a point is not necessarily true. However, it is true in the case of vertex operator algebras. 21
22 Definition 28. A vertex operator algebra V is an assignment X V X of a chiral algebra V X on every smooth curve X, along with compatible isomorphisms φ (V Y ) V X for etale maps φ : X Y. Example 29. The chiral algebra A g is defined for any smooth curve X and these form a vertex operator algebra. Now, let s show that in the case of a vertex operator algebra V, the algebra of endomorphisms of the vacuum module is commutative. Since the statement is etale local, we can restrict ourselves without loss of generality to the case that X = A 1. Since V is a vertex operator algebra, we have that A := V A 1 is a translation equivariant Dmodule on A 1. Let i : {0} A 1 be the inclusion map and V ac A := i! i! (A)[1] be the vacuum module supported at {0}. We have that the algebra of endomorphisms of V ac A is given by End(V ac A ) = End(H 0 (Ri! (A)[1])) = H 0 (REnd(Ri! (A)[1])) = H 1 (REnd(Ri! (A))). The Grothendieck spectral sequence gives the exact sequence 0 H 1 (Ri! (REnd(A))) H 1 (REnd(i! (A))) i! (H 1 (REnd(A))). Since A is translation equivariant so is H 1 (REnd(A)). In particular, H 1 (REnd(A)) is flat. It follows that i! (H 1 (REnd(A))) = 0 and therefore H 1 (Ri! (REnd(A))) H 1 (REnd(i! (A))) End(V ac A ). By commutativity of End(A), the algebra H 1 (Ri! (REnd(A))) is commutative and therefore so is End(V ac A ). 22
Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)
Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and
More informationConformal blocks for a chiral algebra as quasicoherent sheaf on Bun G.
Conformal blocks for a chiral algebra as quasicoherent sheaf on Bun G. Giorgia Fortuna May 04, 2010 1 Conformal blocks for a chiral algebra. Recall that in Andrei s talk [4], we studied what it means
More informationQUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS
QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS SAM RASKIN 1. Differential operators on stacks 1.1. We will define a Dmodule of differential operators on a smooth stack and construct a symbol map when
More informationVertex algebras, chiral algebras, and factorisation algebras
Vertex algebras, chiral algebras, and factorisation algebras Emily Cliff University of Illinois at Urbana Champaign 18 September, 2017 Section 1 Vertex algebras, motivation, and roadplan Definition A
More informationLOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT
LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT DENNIS GAITSGORY 1. Statement of the problem Throughout the talk, by a chiral module we shall understand a chiral Dmodule, unless explicitly stated
More informationConnecting Coinvariants
Connecting Coinvariants In his talk, Sasha taught us how to define the spaces of coinvariants: V 1,..., V n = V 1... V n g S out (0.1) for any V 1,..., V n KL κg and any finite set S P 1. In her talk,
More informationIndCoh Seminar: Indcoherent sheaves I
IndCoh Seminar: Indcoherent sheaves I Justin Campbell March 11, 2016 1 Finiteness conditions 1.1 Fix a cocomplete category C (as usual category means category ). This section contains a discussion of
More informationPART II.1. INDCOHERENT SHEAVES ON SCHEMES
PART II.1. INDCOHERENT SHEAVES ON SCHEMES Contents Introduction 1 1. Indcoherent sheaves on a scheme 2 1.1. Definition of the category 2 1.2. tstructure 3 2. The direct image functor 4 2.1. Direct image
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More information1 Notations and Statement of the Main Results
An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main
More informationFORMAL GLUEING OF MODULE CATEGORIES
FORMAL GLUEING OF MODULE CATEGORIES BHARGAV BHATT Fix a noetherian scheme X, and a closed subscheme Z with complement U. Our goal is to explain a result of Artin that describes how coherent sheaves on
More informationAFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES
AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES YEHAO ZHOU Conventions In this lecture note, a variety means a separated algebraic variety over complex numbers, and sheaves are Clinear. 1.
More information370 INDEX AND NOTATION
Index and Notation action of a Lie algebra on a commutative! algebra 1.4.9 action of a Lie algebra on a chiral algebra 3.3.3 action of a Lie algebroid on a chiral algebra 4.5.4, twisted 4.5.6 action of
More information1 Replete topoi. X = Shv proét (X) X is locally weakly contractible (next lecture) X is replete. D(X ) is left complete. K D(X ) we have R lim
Reference: [BS] Bhatt, Scholze, The proétale topology for schemes In this lecture we consider replete topoi This is a nice class of topoi that include the proétale topos, and whose derived categories
More informationGeometry of Conformal Field Theory
Geometry of Conformal Field Theory Yoshitake HASHIMOTO (Tokyo City University) 2010/07/10 (Sat.) AKB Differential Geometry Seminar Based on a joint work with A. Tsuchiya (IPMU) Contents 0. Introduction
More informationExercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti
Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY Email address: alex.massarenti@sissa.it These notes collect a series of
More informationSection Higher Direct Images of Sheaves
Section 3.8  Higher Direct Images of Sheaves Daniel Murfet October 5, 2006 In this note we study the higher direct image functors R i f ( ) and the higher coinverse image functors R i f! ( ) which will
More informationThe Hitchin map, local to global
The Hitchin map, local to global Andrei Negut Let X be a smooth projective curve of genus g > 1, a semisimple group and Bun = Bun (X) the moduli stack of principal bundles on X. In this talk, we will present
More informationHomology and Cohomology of Stacks (Lecture 7)
Homology and Cohomology of Stacks (Lecture 7) February 19, 2014 In this course, we will need to discuss the ladic homology and cohomology of algebrogeometric objects of a more general nature than algebraic
More informationMODULI TOPOLOGY. 1. Grothendieck Topology
MODULI TOPOLOG Abstract. Notes from a seminar based on the section 3 of the paper: Picard groups of moduli problems (by Mumford). 1. Grothendieck Topology We can define a topology on any set S provided
More informationChern classes à la Grothendieck
Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces
More informationThe Hecke category (part II Satake equivalence)
The Hecke category (part II Satake equivalence) Ryan Reich 23 February 2010 In last week s lecture, we discussed the Hecke category Sph of spherical, or (Ô)equivariant Dmodules on the affine grassmannian
More informationDerived Algebraic Geometry IX: Closed Immersions
Derived Algebraic Geometry I: Closed Immersions November 5, 2011 Contents 1 Unramified Pregeometries and Closed Immersions 4 2 Resolutions of TStructures 7 3 The Proof of Proposition 1.0.10 14 4 Closed
More informationFormal power series rings, inverse limits, and Iadic completions of rings
Formal power series rings, inverse limits, and Iadic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely
More informationMotivic integration on Artin nstacks
Motivic integration on Artin nstacks Chetan Balwe Nov 13,2009 1 / 48 Prestacks (This treatment of stacks is due to B. Toën and G. Vezzosi.) Let S be a fixed base scheme. Let (Aff /S) be the category of
More informationSOME OPERATIONS ON SHEAVES
SOME OPERATIONS ON SHEAVES R. VIRK Contents 1. Pushforward 1 2. Pullback 3 3. The adjunction (f 1, f ) 4 4. Support of a sheaf 5 5. Extension by zero 5 6. The adjunction (j!, j ) 6 7. Sections with support
More information1 Categorical Background
1 Categorical Background 1.1 Categories and Functors Definition 1.1.1 A category C is given by a class of objects, often denoted by ob C, and for any two objects A, B of C a proper set of morphisms C(A,
More informationLIMITS OF CATEGORIES, AND SHEAVES ON INDSCHEMES
LIMITS OF CATEGORIES, AND SHEAVES ON INDSCHEMES JONATHAN BARLEV 1. Inverse limits of categories This notes aim to describe the categorical framework for discussing quasi coherent sheaves and Dmodules
More informationwhere Σ is a finite discrete Gal(K sep /K)set unramified along U and F s is a finite Gal(k(s) sep /k(s))subset
Classification of quasifinite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasifinite étale separated schemes X over a henselian local ring
More informationON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DGMANIFOLDS
ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DGMANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical
More informationConstructible isocrystals (London 2015)
Constructible isocrystals (London 2015) Bernard Le Stum Université de Rennes 1 March 30, 2015 Contents The geometry behind Overconvergent connections Construtibility A correspondance Valuations (additive
More informationModules over a Scheme
Modules over a Scheme Daniel Murfet October 5, 2006 In these notes we collect various facts about quasicoherent sheaves on a scheme. Nearly all of the material is trivial or can be found in [Gro60]. These
More informationPERVERSE SHEAVES. Contents
PERVERSE SHEAVES SIDDHARTH VENKATESH Abstract. These are notes for a talk given in the MIT Graduate Seminar on Dmodules and Perverse Sheaves in Fall 2015. In this talk, I define perverse sheaves on a
More informationLINKED HOM SPACES BRIAN OSSERMAN
LINKED HOM SPACES BRIAN OSSERMAN Abstract. In this note, we describe a theory of linked Hom spaces which complements that of linked Grassmannians. Given two chains of vector bundles linked by maps in both
More informationUNIVERSAL DERIVED EQUIVALENCES OF POSETS
UNIVERSAL DERIVED EQUIVALENCES OF POSETS SEFI LADKANI Abstract. By using only combinatorial data on two posets X and Y, we construct a set of socalled formulas. A formula produces simultaneously, for
More informationDERIVED CATEGORIES OF STACKS. Contents 1. Introduction 1 2. Conventions, notation, and abuse of language The lisseétale and the flatfppf sites
DERIVED CATEGORIES OF STACKS Contents 1. Introduction 1 2. Conventions, notation, and abuse of language 1 3. The lisseétale and the flatfppf sites 1 4. Derived categories of quasicoherent modules 5
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves
More informationBIRTHING OPERS SAM RASKIN
BIRTHING OPERS SAM RASKIN 1. Introduction 1.1. Let G be a simply connected semisimple group with Borel subgroup B, N = [B, B] and let H = B/N. Let g, b, n and h be the respective Lie algebras of these
More informationCategories and functors
Lecture 1 Categories and functors Definition 1.1 A category A consists of a collection ob(a) (whose elements are called the objects of A) for each A, B ob(a), a collection A(A, B) (whose elements are called
More informationLECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES
LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EAMPLES VIVEK SHENDE Historically, sheaves come from topology and analysis; subsequently they have played a fundamental role in algebraic geometry and certain
More informationDeformation theory of representable morphisms of algebraic stacks
Deformation theory of representable morphisms of algebraic stacks Martin C. Olsson School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, molsson@math.ias.edu Received:
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationIntroduction and preliminaries Wouter Zomervrucht, Februari 26, 2014
Introduction and preliminaries Wouter Zomervrucht, Februari 26, 204. Introduction Theorem. Serre duality). Let k be a field, X a smooth projective scheme over k of relative dimension n, and F a locally
More informationA padic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1
A padic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection
More informationINDCOHERENT SHEAVES AND SERRE DUALITY II. 1. Introduction
INDCOHERENT SHEAVES AND SERRE DUALITY II 1. Introduction Let X be a smooth projective variety over a field k of dimension n. Let V be a vector bundle on X. In this case, we have an isomorphism H i (X,
More informationDuality, Residues, Fundamental class
Duality, Residues, Fundamental class Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu May 22, 2011 Joseph Lipman (Purdue University) Duality, Residues, Fundamental class
More informationINTRODUCTION TO PART IV: FORMAL GEOMTETRY
INTRODUCTION TO PART IV: FORMAL GEOMTETRY 1. What is formal geometry? By formal geometry we mean the study of the category, whose objects are PreStk laftdef, and whose morphisms are nilisomorphisms of
More informationEquivariant Algebraic KTheory
Equivariant Algebraic KTheory Ryan Mickler Email: mickler.r@husky.neu.edu Abstract: Notes from lectures given during the MIT/NEU Graduate Seminar on Nakajima Quiver Varieties, Spring 2015 Contents 1
More information1. Algebraic vector bundles. Affine Varieties
0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasiprojective varieties over a field k Affine Varieties 1.
More informationWhat is an indcoherent sheaf?
What is an indcoherent sheaf? Harrison Chen March 8, 2018 0.1 Introduction All algebras in this note will be considered over a field k of characteristic zero (an assumption made in [Ga:IC]), so that we
More informationMath 210B. Profinite group cohomology
Math 210B. Profinite group cohomology 1. Motivation Let {Γ i } be an inverse system of finite groups with surjective transition maps, and define Γ = Γ i equipped with its inverse it topology (i.e., the
More informationSynopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].
Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasicoherent O Y module.
More informationDirect Limits. Mathematics 683, Fall 2013
Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry
More informationThus we get. ρj. Nρj i = δ D(i),j.
1.51. The distinguished invertible object. Let C be a finite tensor category with classes of simple objects labeled by a set I. Since duals to projective objects are projective, we can define a map D :
More informationPART II.2. THE!PULLBACK AND BASE CHANGE
PART II.2. THE!PULLBACK AND BASE CHANGE Contents Introduction 1 1. Factorizations of morphisms of DG schemes 2 1.1. Colimits of closed embeddings 2 1.2. The closure 4 1.3. Transitivity of closure 5 2.
More informationVector Bundles on Algebraic Varieties
Vector Bundles on Algebraic Varieties Aaron Pribadi December 14, 2010 Informally, a vector bundle associates a vector space with each point of another space. Vector bundles may be constructed over general
More informationDERIVED CATEGORIES OF COHERENT SHEAVES
DERIVED CATEGORIES OF COHERENT SHEAVES OLIVER E. ANDERSON Abstract. We give an overview of derived categories of coherent sheaves. [Huy06]. Our main reference is 1. For the participants without bacground
More informationCHAPTER I.2. BASICS OF DERIVED ALGEBRAIC GEOMETRY
CHAPTER I.2. BASICS OF DERIVED ALGEBRAIC GEOMETRY Contents Introduction 2 0.1. Why prestacks? 2 0.2. What do we say about prestacks? 3 0.3. What else is done in this Chapter? 5 1. Prestacks 6 1.1. The
More information6. Lecture cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in SuslinVoevodsky s approach to not
6. Lecture 6 6.1. cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in SuslinVoevodsky s approach to not only motivic cohomology, but also to MorelVoevodsky
More informationMILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES
MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the ChernWeil approach to characteristic classes on manifolds, and in particular, the Chern classes.
More informationPART III.3. INDCOHERENT SHEAVES ON INDINFSCHEMES
PART III.3. INDCOHERENT SHEAVES ON INDINFSCHEMES Contents Introduction 1 1. Indcoherent sheaves on indschemes 2 1.1. Basic properties 2 1.2. tstructure 3 1.3. Recovering IndCoh from indproper maps
More informationConstructible Derived Category
Constructible Derived Category Dongkwan Kim September 29, 2015 1 Category of Sheaves In this talk we mainly deal with sheaves of Cvector spaces. For a topological space X, we denote by Sh(X) the abelian
More informationProof of Langlands for GL(2), II
Proof of Langlands for GL(), II Notes by Tony Feng for a talk by Jochen Heinloth April 8, 016 1 Overview Let X/F q be a smooth, projective, geometrically connected curve. The aim is to show that if E is
More informationh M (T ). The natural isomorphism η : M h M determines an element U = η 1
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7 2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch Set be a moduli
More informationThe Hecke category (part I factorizable structure)
The Hecke category (part I factorizable structure) Ryan Reich 16 February 2010 In this lecture and the next, we will describe the Hecke category, namely, the thing which acts on Dmodules on Bun G and
More informationBRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,
CONNECTIONS, CURVATURE, AND pcurvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and pcurvature, in terms of maps of sheaves
More informationDerived Algebraic Geometry III: Commutative Algebra
Derived Algebraic Geometry III: Commutative Algebra May 1, 2009 Contents 1 Operads 4 1.1 Basic Definitions........................................... 5 1.2 Fibrations of Operads.......................................
More information3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection
3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection is called the objects of C and is denoted Obj(C). Given
More informationEnveloping algebras of HomLie algebras
Journal of Generalized Lie Theory and Applications Vol. 2 (2008), No. 2, 95 108 Enveloping algebras of HomLie algebras Donald YAU Department of Mathematics, The Ohio State University at Newark, 1179 University
More informationTHE SMOOTH BASE CHANGE THEOREM
THE SMOOTH BASE CHANGE THEOREM AARON LANDESMAN CONTENTS 1. Introduction 2 1.1. Statement of the smooth base change theorem 2 1.2. Topological smooth base change 4 1.3. A useful case of smooth base change
More informationLectures on Grothendieck Duality II: Derived Hom Tensor adjointness. Local duality.
Lectures on Grothendieck Duality II: Derived Hom Tensor adjointness. Local duality. Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu February 16, 2009 Joseph Lipman (Purdue
More informationConstruction of M B, M Dol, M DR
Construction of M B, M Dol, M DR Hendrik Orem Talbot Workshop, Spring 2011 Contents 1 Some Moduli Space Theory 1 1.1 Moduli of Sheaves: Semistability and Boundedness.............. 1 1.2 Geometric Invariant
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationThe Affine Grassmannian
1 The Affine Grassmannian Chris Elliott March 7, 2013 1 Introduction The affine Grassmannian is an important object that comes up when one studies moduli spaces of the form Bun G (X), where X is an algebraic
More informationLECTURE X: KOSZUL DUALITY
LECTURE X: KOSZUL DUALITY Fix a prime number p and an integer n > 0, and let S vn denote the category of v n periodic spaces. Last semester, we proved the following theorem of Heuts: Theorem 1. The BousfieldKuhn
More informationAn Outline of Homology Theory
An Outline of Homology Theory Stephen A. Mitchell June 1997, revised October 2001 Note: These notes contain few examples and even fewer proofs. They are intended only as an outline, to be supplemented
More information3. The Sheaf of Regular Functions
24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice
More informationGroupoid Representation Theory
Groupoid Representation Theory Jeffrey C. Morton University of Western Ontario Seminar on Stacks and Groupoids February 12, 2010 Jeffrey C. Morton (U.W.O.) Groupoid Representation Theory UWO Feb 10 1 /
More informationPART IV.2. FORMAL MODULI
PART IV.2. FORMAL MODULI Contents Introduction 1 1. Formal moduli problems 2 1.1. Formal moduli problems over a prestack 2 1.2. Situation over an affine scheme 2 1.3. Formal moduli problems under a prestack
More informationINTRODUCTION TO PART V: CATEGORIES OF CORRESPONDENCES
INTRODUCTION TO PART V: CATEGORIES OF CORRESPONDENCES 1. Why correspondences? This part introduces one of the two main innovations in this book the (, 2)category of correspondences as a way to encode
More informationMath 210B. Artin Rees and completions
Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an Amodule. In class we defined the Iadic completion of M to be M = lim M/I n M. We will soon show
More informationp I Hecke I ( 1) i Tr(T Λ n i V ) T i = 0.
Literal notes 0.. Where are we? Recall that for each I fset and W I Rep(ǦI ), we have produced an object F I,W I D ind (X I ). It is the proper pushforward along paws of the moduli of Shtukas: F I,W I
More informationDerivations and differentials
Derivations and differentials Johan Commelin April 24, 2012 In the following text all rings are commutative with 1, unless otherwise specified. 1 Modules of derivations Let A be a ring, α : A B an A algebra,
More informationBEN KNUDSEN. Conf k (f) Conf k (Y )
CONFIGURATION SPACES IN ALGEBRAIC TOPOLOGY: LECTURE 2 BEN KNUDSEN We begin our study of configuration spaces by observing a few of their basic properties. First, we note that, if f : X Y is an injective
More informationRelative Affine Schemes
Relative Affine Schemes Daniel Murfet October 5, 2006 The fundamental Spec( ) construction associates an affine scheme to any ring. In this note we study the relative version of this construction, which
More informationNOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE
NOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE GUFANG ZHAO Contents 1. Introduction 1 2. What is a Procesi bundle 2 3. Derived equivalences from exceptional objects 4 4. Splitting of the
More informationCHAPTER III.4. AN APPLICATION: CRYSTALS
CHAPTER III.4. AN APPLICATION: CRYSTALS Contents Introduction 1 0.1. Let s do Dmodules! 2 0.2. Dmodules via crystals 2 0.3. What else is done in this chapter? 4 1. Crystals on prestacks and infschemes
More informationAn Atlas For Bun r (X)
An Atlas For Bun r (X) As told by Dennis Gaitsgory to Nir Avni October 28, 2009 1 Bun r (X) Is Not Of Finite Type The goal of this lecture is to find a smooth atlas locally of finite type for the stack
More informationSEMINAR NOTES: QUANTIZATION OF HITCHIN S INTEGRABLE SYSTEM AND HECKE EIGENSHEAVES (SEPT. 8, 2009)
SEMINAR NOTES: QUANTIZATION OF HITCHIN S INTEGRABLE SYSTEM AND HECKE EIGENSHEAVES (SEPT. 8, 2009) DENNIS GAITSGORY 1. Hecke eigensheaves The general topic of this seminar can be broadly defined as Geometric
More informationFactorization of birational maps for qe schemes in characteristic 0
Factorization of birational maps for qe schemes in characteristic 0 AMS special session on Algebraic Geometry joint work with M. Temkin (Hebrew University) Dan Abramovich Brown University October 24, 2014
More informationLocal Theory: Chiral Basics
CHAPTER 3 Local Theory: Chiral Basics Algebraisty obyqno opredel t gruppy kak mnoжestva s operaci mi, udovletvor wimi dlinnomu r du trudnozapominaemyh aksiom. Pon tь takoe opredelenie, na mo i vzgl d,
More informationEquivalence of the Combinatorial Definition (Lecture 11)
Equivalence of the Combinatorial Definition (Lecture 11) September 26, 2014 Our goal in this lecture is to complete the proof of our first main theorem by proving the following: Theorem 1. The map of simplicial
More informationDerived Morita theory and Hochschild Homology and Cohomology of DG Categories
Derived Morita theory and Hochschild Homology and Cohomology of DG Categories German Stefanich In this talk we will explore the idea that an algebra A over a field (ring, spectrum) k can be thought of
More informationLECTURE 4.5: SOERGEL S THEOREM AND SOERGEL BIMODULES
LECTURE 4.5: SOERGEL S THEOREM AND SOERGEL BIMODULES DMYTRO MATVIEIEVSKYI Abstract. These are notes for a talk given at the MITNortheastern Graduate Student Seminar on category O and Soergel bimodules,
More informationCohomology and Base Change
Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is halfexact if whenever 0 M M M 0 is an exact sequence of Amodules, the sequence T (M ) T (M)
More informationMULTIPLE DISJUNCTION FOR SPACES OF POINCARÉ EMBEDDINGS
MULTIPLE DISJUNCTION FOR SPACES OF POINCARÉ EMBEDDINGS THOMAS G. GOODWILLIE AND JOHN R. KLEIN Abstract. Still at it. Contents 1. Introduction 1 2. Some Language 6 3. Getting the ambient space to be connected
More informationMath 248B. Applications of base change for coherent cohomology
Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the socalled cohomology and base change theorem: Theorem 1.1 (Grothendieck).
More informationEilenbergSteenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )
II.3 : EilenbergSteenrod properties (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, 8.3 8.5 Definition. Let U be an open subset of R n for some n. The de Rham cohomology groups (U are the cohomology groups
More informationCycle groups for Artin stacks
Cycle groups for Artin stacks arxiv:math/9810166v1 [math.ag] 28 Oct 1998 Contents Andrew Kresch 1 28 October 1998 1 Introduction 2 2 Definition and first properties 3 2.1 The homology functor..........................
More information