Connecting Coinvariants

Size: px
Start display at page:

Download "Connecting Coinvariants"

Transcription

1 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, Giorgia showed us how to use this vector space to define the fusion tensor product on category O κ. In this paper, we will show how to paste the spaces (0.1) together as S varies, and introduce a connection on the corresponding space. Throughout this paper, we will work with smooth schemes X. We will always let : X X X denote the diagonal map, and j : X X \ X X denote the inclusion of the complement. In section 1.2, we will generalize the notion of diagonal and then use the notations j + (resp. j ) and + (resp. ) will mean something more general. Unless we explicitly state the contrary, a D module will refer to a right D module on X. For such a D module M, we let h denote the right exact functor: {D modules} h {sheaves}, M h M/M T X. The de Rham functor is defined as DR(X, M) = RΓ(X, Lh(M)). In certain sections we will set X = P 1, though we will explicitly remind the reader when we will do this. Whenever this will be the case, we will encounter the objects: O = C[[t]], K = C((t)) and if x is a point on X, then we let O x and K x be the ring of functions on the formal (resp. punctured formal) disk around x. They are non-canonically isomorphic to O and K.

2 2 Connecting Coinvariants 1. Chiral Algebras 1.1. All the constructions of this section will be on an arbitrary smooth scheme X. Definition 1.1. On a smooth scheme X, a chiral algebra A is a D module together with a map of D modules: j j (A A) {, }! A, (1.2) called the chiral bracket, which satisfies antisymmetry and the Jacobi identity 1. We call our chiral algebra unital if it comes with a map of D modules: Ω 1 A (1.3) that is compatible with the chiral bracket 2. The vector space DR(X, A) inherits a structure of Lie algebra. Definition 1.4. A chiral D module 3 M is a D module with a map of D modules: j j (A M)! M, (1.5) which must be compatible with (1.2) and with (1.3). For any open subset U X, the vector space DR(U, M) becomes a module of the Lie algebra DR(U, A). Definition 1.6. A chiral O module M is a O module with a map of O modules: j j (A M) O! M, (1.7) that is compatible with (1.2) and with (1.3) in the same sense as before. The objects in the above relation are D X O X modules, and the push-forward in the RHS is taken in this category. Explicitly, it is given by: O! M = M where is the usual push-forward for O modules. O X O X D X O X, 1 These are spelled out explicitly in [2], section The compatibility relation is states in [2], section We usually call this a chiral module, for short

3 Connecting Coinvariants In [3], Nick shows us how to generalize the construction of chiral modules. We will now give a hint of that construction, which is enough for our purposes. As before, let A be a chiral algebra over a smooth scheme X and let n 1 be a natural number. Definition 1.8. A chiral module on X n is a D module M on X n with a map of D modules: j + j +(A M) {, } +!! +p! 2(M), Here the maps are as in the diagram: Z X X n p 1 X where is the diagonal consisting of points {x i, x 1,..., x i,..., x n }, i {1,..., n} and j denotes the complement. Naturally, we ask the chiral bracket (1.10) to be antisymmetric and to satisfy the Jacobi identity just like before. p 2 X n 1.3. We will not need chiral modules on X n, but a closely related concept. Let X n denote the complement to all diagonals, and we will repeat the above with the diagonals of X n removed. Definition 1.9. A chiral module on X n is a D module M on X n with a map of D modules: j j (A M) {, }! M (1.10) Here the maps are as in the diagram: X n... X n X X n, X where denotes the only remaining diagonal and j denotes the complement to it. We ask the chiral bracket (1.10) to be antisymmetric and to satisfy the Jacobi identity just like before. p 1 p 2 X n A particular example of chiral module on X n is M 1... M n X n, where M 1,..., M n are chiral modules on X. In this case it is clear how to construct the chiral bracket (1.10) out of the chiral brackets of each M i, because the locus splits up nicely into n connected components corresponding to each i {1,..., n}. If we had not restricted to X n, then it would have been impossible to define the chiral bracket.

4 4 Connecting Coinvariants 2. Lie Algebras Definition 2.1. A Lie algebra L is a D module with a map: L L {, }! L, (2.2) called the Lie bracket, which satisfies antisymmetry and the Jacobi identity just like in the case of chiral algebras. Note that any chiral algebra is automatically a Lie algebra. Definition 2.3. A Lie (respectively chiral) module for L is a D module M with a map of D modules: L M! M (respectively j j (L M)! M) that is compatible with the Lie bracket (2.2) In the study of finite dimensional Lie algebras, a major role is played by universal enveloping algebras, which are defined by a universal property. The exact same construction works in the chiral setting, where to any Lie algebra L we associate a unital chiral algebra U(L) defined by the same universal property: Hom Lie (L, A) = Hom chiral (U(L), A), (2.4) functorially in the chiral algebra A. Our goal now is to present an explicit construction of U(L), but first we need to see what it should be. One expects its! fiber over x X to be 4 the universal enveloping algebra of L! x. One can write this as a cokernel of Lie algebras: DR(X, L) DR(X\x, L) L! x[1] 0 and therefore we expect that the! fiber of U(L) at x should equal: U(DR(X\x, L)) C (2.5) U(DR(X,L)) 2.2. Let p 1 : X X X be the first projection, and then the D modules: L small = p 1 (Ω L) p 1 j j (Ω L) = L big. have fibers over x equal to DR(X, L) and DR(X\x, L), respectively. They are both Lie algebras in the category of D modules, so we can construct their classical universal enveloping algebras in the category of D modules. Then we define: U(L) = U(L big ) Ω X, U(L small ) 4 in some vague, motivational sense

5 Connecting Coinvariants 5 as an algebra in the category of D modules. It has precisely the fibers we want from (2.5), and this is the explicit description of the universal enveloping chiral algebra. It has a PBW filtration coming from U(L big ) such that: U(L) 0 = Ω X, U(L) 1 = L Ω X. 3. Examples 3.1. Specialize now X to be a curve. Given a finite-dimensional Lie algebra g and a non-degenerate symmetric bilinear form κ : g g C, we construct the Kac-Moody Lie algebra: where the Lie bracket kills Ω X and is otherwise given by: L g,κ = g D X ΩX, (3.1) {x 1, x 1} = [x, x ] 1 + κ(x, x ) 1, where 1 D X! (D X ) and 1 is the section of! (Ω X ) given by: 1 = dt dt (t t ) 2 (Ω X Ω X )(2 ) (Ω X Ω X )( ) =! (Ω X ) Ω X Ω X Ω X Ω X Note that this section does not depend on the choice of the coordinate t on X; we will encounter it again when we will discuss the Sugawara construction. For any point x X, we have: DR 0 (Spec(O x ), L g,κ ) = g O x, DR 0 (Spec(K x ), L g,κ ) = g K x C, (3.2) where the latter has the Lie algebra structure of our good old Kac-Moody Lie algebra ĝ k. Moreover, for any finite subset S X we have: DR 0 (X\S, L g,κ ) = ĝ S κ, (3.3) as introduced by Sasha and Giorgia. Finally, our main player will be the chiral Kac-Moody algebra 5 : A g,κ = U(L g,κ )/(1 1 ), where we are setting equal two units 1, 1 : Ω X A g,κ. The first comes from the definition of a chiral algebra, and the second one comes from the central factor Ω X of L g,κ. 5 The two units 1, 1 : Ω A g,κ we are setting equal to each other are (1.3) and the one coming from (3.1)

6 6 Connecting Coinvariants 3.2. Still taking X to be a curve, consider T = T X OX D X and look at the map: T T = (T X OX D X ) (T X OX D X ) T X OX X D X X =! (T), where the middle arrow is minus the usual Lie bracket on vector fields. This endows T with a structure of a Lie algebra, such that: DR 0 (Spec(K x ), T) = Vir, DR 0 (Spec(O x ), T) = Vir +, (3.4) where Vir + and Vir are the Lie algebras of infinitesimal automorphisms of the formal, respectively punctured formal, disk. 4. Conformal blocks 4.1. Let us now go back to a chiral module M for a chiral algebra A on a smooth projective scheme X. If we take (1.2) and push it forward under the second projection p 2, we get a map of D modules on X: p 2 j j (A M) ρ M. (4.1) The cokernel of the above map is denoted by H (X, M) and is naturally a D module on X called conformal blocks 6. The effect of p 2 above is to kill the tangent vectors in the direction of the first factor of X X, so the fiber of (4.1) at some point x X gives us an exact sequence: DR 0 (X\x, A) M x ρ x Mx H (X, ρ) x 0. (4.2) The reason why we have X\x is j. The Jacobi identity for the chiral bracket implies that ρ x is a Lie algebra action on the vector space M x, and so we can write conformal blocks as a space of coinvariants: H (X, ρ) x = M x /(DR 0 (X\x, A) M x ). A particularly important case is the vacuum module M = A, in which case H (X, A) is the trivial D module on X. In other words, it is not only globally free as a O module, but the connection is also trivial. This is actually true for any algebra in the category of D modules, as we proved in Proposition 3 of my talk in the Fall of If M were only a chiral O module instead of a chiral D module, then the above discussion would go through with the only exception that H (X, M) would only be an O module on X

7 Connecting Coinvariants The setting in which we will be using conformal blocks is for a D module M on X n as in section 1.2. Then we start from the chiral bracket (1.10), push it forward under the projection p 2, and we define conformal blocks as the D module on X n : H (X n, M) = Coker(p 2 j + j +(A M) p 2 +!! +p! 2M) But it we are looking at D modules on X n as in section 1.3, then note that p 2 is simply a trivial n fold cover. Then the above becomes: H (X n, M) = Coker(p 2 j j (A M) M n ). If we take the fiber of this above a point (x 1,..., x n ) X n, we see that: ( ) H (X n, M) (x1,...,x n) = Coker DR(X\{x 1,..., x n }, A) M x1,...,x n ) Ψ M n (x 1,...,x n). (4.3) Lemma 4.4. The above fiber can also be realized as the cokernel: H (X n, M) = Coker ( DR(X\{x 1,..., x n }, A) M x1,...,x n ) M (x1,...,x n)), (4.5) with the map being the sum of the n component maps in (4.3). Proof. Indeed, the addition map M n (x M 1,...,x n) (x 1,...,x n) induces an a priori surjective map between the cokernel of (4.3) and the cokernel (4.5). To prove that it is injective, one needs to show that anything in the kernel of the addition map is in the image of the arrow in (4.3). Since the kernel is spanned by permutations of the following vectors, it s enough to prove that the vectors (m, m, 0,..., 0) M n (x 1,...,x n) lie in the image of the map (4.3). However, it s easy to see that: (m, m, 0,..., 0) = Ψ(ω m), where ω is a differential form on X with the residue 1 at x 1 and 1 at x 2. If M = M 1... M n X n for chiral modules M 1,..., M n on X, then the above lemma says that: H (X n, M) (x1,...,x n) = (M 1 x1... M n xn )/ DR 0 (X\{x 1,...,x n},a) (4.6) with the action of DR 0 being the diagonal one. In the last section, we will show that this description of conformal blocks coincides with the coinvariants introduced by Sasha and Giorgia.

8 8 Connecting Coinvariants 4.3. In the Fall of 2009, we saw that conformal blocks have certain functorial interpretations in the category of D schemes. However, for our purposes now, we are only interested in the fact that (4.6) realizes them as coinvariants, much in the same way as Sasha and Giorgia talked about in their talks. This analogy will be made explicit later, but let us make a technical point. We are in the case A = A g,κ = U(L g,κ ) of section 3.1. Sasha and Giorgia take coinvariants with respect to the Lie algebra, whose chiral analogue is DR(X\x, L g,κ ), whereas in (4.6) we are taking coinvariants with respect to the universal enveloping DR(X\x, A g,κ ). The following proposition shows that the two constructions actually produce the same thing. Proposition 4.7. For a module M over a Lie algebra L, the natural surjection: M x /(DR 0 (X\x, L) M x ) M x /(DR 0 (X\x, A) M x ) is an isomorphism, where A = U(L). Proof. It is enough to look at the PBW filtration of A, under which A 1 = L Ω X, and to prove that the natural surjection: M x /(DR 0 (X\x, A n ) M x ) M x /(DR 0 (X\x, A n+1 ) M x ) is an isomorphism for all n 1. We need to show it is injective, so we need to show that any element α m DR 0 (X\x, A n+1 ) M x also lies in DR 0 (X\x, A n ) M x. The chiral bracket of the universal enveloping algebra respects the PBW filtration, and actually: j j L A n! A n+1. Because X\x is affine, the above is still a surjection when we pass to DR, so there is a section f(t, t )(α 1 α 2 ) which projects onto α under the chiral bracket, where α 1 DR 0 (X\x, L) and α 2 DR 0 (X\x, A n ). The Jacobi identity then implies that α m lies in DR 0 (X\x, A n ) M x. References [1] Beilinson, Alexander; Drinfeld, Vladimir, Chiral Algebras [2] Gaitsgory, Dennis, Notes on 2D Conformal Field Theory and String Theory [3] Rozenblyum, Nick, Modules over a Chiral Algebra

Conformal blocks for a chiral algebra as quasi-coherent sheaf on Bun G.

Conformal blocks for a chiral algebra as quasi-coherent sheaf on Bun G. Conformal blocks for a chiral algebra as quasi-coherent 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 information

LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT

LOCAL 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 D-module, unless explicitly stated

More information

Derivations and differentials

Derivations 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 information

Introduction to Chiral Algebras

Introduction to Chiral Algebras 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

More information

FROM VERTEX ALGEBRAS TO CHIRAL ALGEBRAS

FROM VERTEX ALGEBRAS TO CHIRAL ALGEBRAS FROM VERTEX ALGEBRAS TO CHIRAL ALGEBRAS GIORGIA FORTUNA Our goal for today will be to introduce the concept of Chiral Algebras and relate it to the one of quasi-conformal vertex algebras. Vaguely speaking,

More information

QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS

QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS SAM RASKIN 1. Differential operators on stacks 1.1. We will define a D-module of differential operators on a smooth stack and construct a symbol map when

More information

BRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,

BRIAN 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 p-curvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and p-curvature, in terms of maps of sheaves

More information

PERVERSE SHEAVES. Contents

PERVERSE SHEAVES. Contents PERVERSE SHEAVES SIDDHARTH VENKATESH Abstract. These are notes for a talk given in the MIT Graduate Seminar on D-modules and Perverse Sheaves in Fall 2015. In this talk, I define perverse sheaves on a

More information

Determinant Bundle in a Family of Curves, after A. Beilinson and V. Schechtman

Determinant Bundle in a Family of Curves, after A. Beilinson and V. Schechtman Commun. Math. Phys. 211, 359 363 2000) Communications in Mathematical Physics Springer-Verlag 2000 Determinant Bundle in a Family of Curves, after A. Beilinson and V. Schechtman Hélène snault 1, I-Hsun

More information

The Hitchin map, local to global

The 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 information

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

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 information

BIRTHING OPERS SAM RASKIN

BIRTHING 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 information

Vertex algebras generated by primary fields of low conformal weight

Vertex algebras generated by primary fields of low conformal weight Short talk Napoli, Italy June 27, 2003 Vertex algebras generated by primary fields of low conformal weight Alberto De Sole Slides available from http://www-math.mit.edu/ desole/ 1 There are several equivalent

More information

3. The Sheaf of Regular Functions

3. 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 information

Algebraic Geometry Spring 2009

Algebraic 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 information

THE THEOREM OF THE HIGHEST WEIGHT

THE THEOREM OF THE HIGHEST WEIGHT THE THEOREM OF THE HIGHEST WEIGHT ANKE D. POHL Abstract. Incomplete notes of the talk in the IRTG Student Seminar 07.06.06. This is a draft version and thought for internal use only. The Theorem of the

More information

Notes on p-divisible Groups

Notes on p-divisible Groups Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete

More information

Chern classes à la Grothendieck

Chern 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 information

D-MODULES: AN INTRODUCTION

D-MODULES: AN INTRODUCTION D-MODULES: AN INTRODUCTION ANNA ROMANOVA 1. overview D-modules are a useful tool in both representation theory and algebraic geometry. In this talk, I will motivate the study of D-modules by describing

More information

Algebraic Geometry Spring 2009

Algebraic 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 information

Kac-Moody Algebras. Ana Ros Camacho June 28, 2010

Kac-Moody Algebras. Ana Ros Camacho June 28, 2010 Kac-Moody Algebras Ana Ros Camacho June 28, 2010 Abstract Talk for the seminar on Cohomology of Lie algebras, under the supervision of J-Prof. Christoph Wockel Contents 1 Motivation 1 2 Prerequisites 1

More information

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25 FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves 1 2. Quasicoherent sheaves form an abelian category 5 We began by recalling the distinguished affine base. Definition.

More information

Lecture 4 Super Lie groups

Lecture 4 Super Lie groups Lecture 4 Super Lie groups In this lecture we want to take a closer look to supermanifolds with a group structure: Lie supergroups or super Lie groups. As in the ordinary setting, a super Lie group is

More information

PBW for an inclusion of Lie algebras

PBW for an inclusion of Lie algebras PBW for an inclusion of Lie algebras Damien Calaque, Andrei Căldăraru, Junwu Tu Abstract Let h g be an inclusion of Lie algebras with quotient h-module n. There is a natural degree filtration on the h-module

More information

An Atlas For Bun r (X)

An 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 information

Vertex algebras, chiral algebras, and factorisation algebras

Vertex 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 road-plan Definition A

More information

Crystalline Cohomology and Frobenius

Crystalline Cohomology and Frobenius Crystalline Cohomology and Frobenius Drew Moore References: Berthelot s Notes on Crystalline Cohomology, discussions with Matt Motivation Let X 0 be a proper, smooth variety over F p. Grothendieck s etale

More information

Factorization of birational maps for qe schemes in characteristic 0

Factorization 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 information

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about

More information

Synopsis 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. 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 quasi-coherent O Y -module.

More information

Algebraic Geometry Spring 2009

Algebraic 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 information

SECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS

SECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS SECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS In this section we will prove the Künneth theorem which in principle allows us to calculate the (co)homology of product spaces as soon

More information

LECTURE 3 MATH 261A. Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.

LECTURE 3 MATH 261A. Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment. LECTURE 3 MATH 261A LECTURES BY: PROFESSOR DAVID NADLER PROFESSOR NOTES BY: JACKSON VAN DYKE Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.

More information

TCC Homological Algebra: Assignment #3 (Solutions)

TCC Homological Algebra: Assignment #3 (Solutions) TCC Homological Algebra: Assignment #3 (Solutions) David Loeffler, d.a.loeffler@warwick.ac.uk 30th November 2016 This is the third of 4 problem sheets. Solutions should be submitted to me (via any appropriate

More information

Algebraic Geometry Spring 2009

Algebraic 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 information

Exercises 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 Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

More information

10. Smooth Varieties. 82 Andreas Gathmann

10. Smooth Varieties. 82 Andreas Gathmann 82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It

More information

A TALE OF TWO FUNCTORS. Marc Culler. 1. Hom and Tensor

A TALE OF TWO FUNCTORS. Marc Culler. 1. Hom and Tensor A TALE OF TWO FUNCTORS Marc Culler 1. Hom and Tensor It was the best of times, it was the worst of times, it was the age of covariance, it was the age of contravariance, it was the epoch of homology, it

More information

DERIVED CATEGORIES OF COHERENT SHEAVES

DERIVED 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 information

LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C)

LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) IVAN LOSEV Introduction We proceed to studying the representation theory of algebraic groups and Lie algebras. Algebraic groups are the groups

More information

Formal power series rings, inverse limits, and I-adic completions of rings

Formal power series rings, inverse limits, and I-adic completions of rings Formal power series rings, inverse limits, and I-adic 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 information

A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1

A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 A p-adic 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 information

Non characteristic finiteness theorems in crystalline cohomology

Non characteristic finiteness theorems in crystalline cohomology Non characteristic finiteness theorems in crystalline cohomology 1 Non characteristic finiteness theorems in crystalline cohomology Pierre Berthelot Université de Rennes 1 I.H.É.S., September 23, 2015

More information

PERVERSE SHEAVES ON A TRIANGULATED SPACE

PERVERSE SHEAVES ON A TRIANGULATED SPACE PERVERSE SHEAVES ON A TRIANGULATED SPACE A. POLISHCHUK The goal of this note is to prove that the category of perverse sheaves constructible with respect to a triangulation is Koszul (i.e. equivalent to

More information

Cohomology jump loci of local systems

Cohomology jump loci of local systems Cohomology jump loci of local systems Botong Wang Joint work with Nero Budur University of Notre Dame June 28 2013 Introduction Given a topological space X, we can associate some homotopy invariants to

More information

THE SHIMURA-TANIYAMA FORMULA AND p-divisible GROUPS

THE SHIMURA-TANIYAMA FORMULA AND p-divisible GROUPS THE SHIMURA-TANIYAMA FORMULA AND p-divisible GROUPS DANIEL LITT Let us fix the following notation: 1. Notation and Introduction K is a number field; L is a CM field with totally real subfield L + ; (A,

More information

De Rham Cohomology. Smooth singular cochains. (Hatcher, 2.1)

De Rham Cohomology. Smooth singular cochains. (Hatcher, 2.1) II. De Rham Cohomology There is an obvious similarity between the condition d o q 1 d q = 0 for the differentials in a singular chain complex and the condition d[q] o d[q 1] = 0 which is satisfied by the

More information

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for non-stable-rationality based on the decomposition of the diagonal in the Chow group. This criterion

More information

A bundle-theoretic perspective to Kac Moody groups

A bundle-theoretic perspective to Kac Moody groups A bundle-theoretic perspective to Kac Moody groups Christoph Wockel May 31, 2007 Outline Kac Moody algebras and groups Central extensions and cocycles Cocycles for twisted loop algebras and -groups Computation

More information

A Note on Dormant Opers of Rank p 1 in Characteristic p

A Note on Dormant Opers of Rank p 1 in Characteristic p A Note on Dormant Opers of Rank p 1 in Characteristic p Yuichiro Hoshi May 2017 Abstract. In the present paper, we prove that the set of equivalence classes of dormant opers of rank p 1 over a projective

More information

MATH 233B, FLATNESS AND SMOOTHNESS.

MATH 233B, FLATNESS AND SMOOTHNESS. MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)

More information

LIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES

LIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES LIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES BENJAMIN HOFFMAN 1. Outline Lie algebroids are the infinitesimal counterpart of Lie groupoids, which generalize how we can talk about symmetries

More information

Math 248B. Applications of base change for coherent cohomology

Math 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 so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

More information

EILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY

EILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY EILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY CHRIS KOTTKE 1. The Eilenberg-Zilber Theorem 1.1. Tensor products of chain complexes. Let C and D be chain complexes. We define

More information

ALGEBRAIC GROUPS JEROEN SIJSLING

ALGEBRAIC GROUPS JEROEN SIJSLING ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined

More information

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar

More information

which is a group homomorphism, such that if W V U, then

which is a group homomorphism, such that if W V U, then 4. Sheaves Definition 4.1. Let X be a topological space. A presheaf of groups F on X is a a function which assigns to every open set U X a group F(U) and to every inclusion V U a restriction map, ρ UV

More information

Math 121 Homework 5: Notes on Selected Problems

Math 121 Homework 5: Notes on Selected Problems Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements

More information

Hungry, Hungry Homology

Hungry, Hungry Homology September 27, 2017 Motiving Problem: Algebra Problem (Preliminary Version) Given two groups A, C, does there exist a group E so that A E and E /A = C? If such an group exists, we call E an extension of

More information

Lifting the Cartier transform of Ogus-Vologodsky. modulo p n. Daxin Xu. California Institute of Technology

Lifting the Cartier transform of Ogus-Vologodsky. modulo p n. Daxin Xu. California Institute of Technology Lifting the Cartier transform of Ogus-Vologodsky modulo p n Daxin Xu California Institute of Technology Riemann-Hilbert correspondences 2018, Padova A theorem of Deligne-Illusie k a perfect field of characteristic

More information

Tunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society

Tunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society Tunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society Grothendieck Messing deformation theory for varieties of K3 type Andreas Langer and Thomas Zink

More information

INTRO TO TENSOR PRODUCTS MATH 250B

INTRO TO TENSOR PRODUCTS MATH 250B INTRO TO TENSOR PRODUCTS MATH 250B ADAM TOPAZ 1. Definition of the Tensor Product Throughout this note, A will denote a commutative ring. Let M, N be two A-modules. For a third A-module Z, consider the

More information

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Where we were 1 2. Yoneda s lemma 2 3. Limits and colimits 6 4. Adjoints 8 First, some bureaucratic details. We will move to 380-F for Monday

More information

LECTURE 3: RELATIVE SINGULAR HOMOLOGY

LECTURE 3: RELATIVE SINGULAR HOMOLOGY LECTURE 3: RELATIVE SINGULAR HOMOLOGY In this lecture we want to cover some basic concepts from homological algebra. These prove to be very helpful in our discussion of singular homology. The following

More information

ALGEBRA HW 3 CLAY SHONKWILER

ALGEBRA HW 3 CLAY SHONKWILER ALGEBRA HW 3 CLAY SHONKWILER (a): Show that R[x] is a flat R-module. 1 Proof. Consider the set A = {1, x, x 2,...}. Then certainly A generates R[x] as an R-module. Suppose there is some finite linear combination

More information

Duality, Residues, Fundamental class

Duality, 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 information

Introduction and preliminaries Wouter Zomervrucht, Februari 26, 2014

Introduction 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 information

Algebraic Geometry. Andreas Gathmann. Class Notes TU Kaiserslautern 2014

Algebraic Geometry. Andreas Gathmann. Class Notes TU Kaiserslautern 2014 Algebraic Geometry Andreas Gathmann Class Notes TU Kaiserslautern 2014 Contents 0. Introduction......................... 3 1. Affine Varieties........................ 9 2. The Zariski Topology......................

More information

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12 MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

More information

1 Notations and Statement of the Main Results

1 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 information

Level raising. Kevin Buzzard April 26, v1 written 29/3/04; minor tinkering and clarifications written

Level raising. Kevin Buzzard April 26, v1 written 29/3/04; minor tinkering and clarifications written Level raising Kevin Buzzard April 26, 2012 [History: 3/6/08] v1 written 29/3/04; minor tinkering and clarifications written 1 Introduction What s at the heart of level raising, when working with 1-dimensional

More information

Math 210B. Artin Rees and completions

Math 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 A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show

More information

SUMMER SCHOOL ON DERIVED CATEGORIES NANTES, JUNE 23-27, 2014

SUMMER SCHOOL ON DERIVED CATEGORIES NANTES, JUNE 23-27, 2014 SUMMER SCHOOL ON DERIVED CATEGORIES NANTES, JUNE 23-27, 2014 D. GAITSGORY 1.1. Introduction. 1. Lecture I: the basics 1.1.1. Why derived algebraic geometry? a) Fiber products. b) Deformation theory. c)

More information

Factorization spaces and moduli spaces over curves

Factorization spaces and moduli spaces over curves Josai Mathematical Monographs vol. 10 (2017), pp. 97 128 Factorization spaces and moduli spaces over curves Shintarou YANAGIDA Abstract. The notion of factorization space is a non-linear counterpart of

More information

LECTURE 20: KAC-MOODY ALGEBRA ACTIONS ON CATEGORIES, II

LECTURE 20: KAC-MOODY ALGEBRA ACTIONS ON CATEGORIES, II LECTURE 20: KAC-MOODY ALGEBRA ACTIONS ON CATEGORIES, II IVAN LOSEV 1. Introduction 1.1. Recap. In the previous lecture we have considered the category C F := n 0 FS n -mod. We have equipped it with two

More information

LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES

LECTURE 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 information

8 Perverse Sheaves. 8.1 Theory of perverse sheaves

8 Perverse Sheaves. 8.1 Theory of perverse sheaves 8 Perverse Sheaves In this chapter we will give a self-contained account of the theory of perverse sheaves and intersection cohomology groups assuming the basic notions concerning constructible sheaves

More information

MIXED HODGE MODULES PAVEL SAFRONOV

MIXED HODGE MODULES PAVEL SAFRONOV MIED HODGE MODULES PAVEL SAFRONOV 1. Mixed Hodge theory 1.1. Pure Hodge structures. Let be a smooth projective complex variety and Ω the complex of sheaves of holomorphic differential forms with the de

More information

370 INDEX AND NOTATION

370 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 information

Lecture 6: Etale Fundamental Group

Lecture 6: Etale Fundamental Group Lecture 6: Etale Fundamental Group October 5, 2014 1 Review of the topological fundamental group and covering spaces 1.1 Topological fundamental group Suppose X is a path-connected topological space, and

More information

Equivalence Relations

Equivalence Relations Equivalence Relations Definition 1. Let X be a non-empty set. A subset E X X is called an equivalence relation on X if it satisfies the following three properties: 1. Reflexive: For all x X, (x, x) E.

More information

CONFORMAL FIELD THEORIES

CONFORMAL FIELD THEORIES CONFORMAL FIELD THEORIES Definition 0.1 (Segal, see for example [Hen]). A full conformal field theory is a symmetric monoidal functor { } 1 dimensional compact oriented smooth manifolds {Hilbert spaces}.

More information

where 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

where 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 quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring

More information

An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations

An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations Mircea Mustaţă University of Michigan Mainz July 9, 2018 Mircea Mustaţă () An overview of D-modules Mainz July 9, 2018 1 The

More information

Symplectic varieties and Poisson deformations

Symplectic varieties and Poisson deformations Symplectic varieties and Poisson deformations Yoshinori Namikawa A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere non-degenerate d-closed 2-form ω on the

More information

A global version of the quantum duality principle

A global version of the quantum duality principle A global version of the quantum duality principle Fabio Gavarini Università degli Studi di Roma Tor Vergata Dipartimento di Matematica Via della Ricerca Scientifica 1, I-00133 Roma ITALY Received 22 August

More information

3.1. Derivations. Let A be a commutative k-algebra. Let M be a left A-module. A derivation of A in M is a linear map D : A M such that

3.1. Derivations. Let A be a commutative k-algebra. Let M be a left A-module. A derivation of A in M is a linear map D : A M such that ALGEBRAIC GROUPS 33 3. Lie algebras Now we introduce the Lie algebra of an algebraic group. First, we need to do some more algebraic geometry to understand the tangent space to an algebraic variety at

More information

MATH 221 NOTES BRENT HO. Date: January 3, 2009.

MATH 221 NOTES BRENT HO. Date: January 3, 2009. MATH 22 NOTES BRENT HO Date: January 3, 2009. 0 Table of Contents. Localizations......................................................................... 2 2. Zariski Topology......................................................................

More information

Tree sets. Reinhard Diestel

Tree sets. Reinhard Diestel 1 Tree sets Reinhard Diestel Abstract We study an abstract notion of tree structure which generalizes treedecompositions of graphs and matroids. Unlike tree-decompositions, which are too closely linked

More information

Lecture 7: Etale Fundamental Group - Examples

Lecture 7: Etale Fundamental Group - Examples Lecture 7: Etale Fundamental Group - Examples October 15, 2014 In this lecture our only goal is to give lots of examples of etale fundamental groups so that the reader gets some feel for them. Some of

More information

A generalized Koszul theory and its applications in representation theory

A generalized Koszul theory and its applications in representation theory A generalized Koszul theory and its applications in representation theory A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Liping Li IN PARTIAL FULFILLMENT

More information

Fourier Mukai transforms II Orlov s criterion

Fourier Mukai transforms II Orlov s criterion Fourier Mukai transforms II Orlov s criterion Gregor Bruns 07.01.2015 1 Orlov s criterion In this note we re going to rely heavily on the projection formula, discussed earlier in Rostislav s talk) and

More information

DELIGNE S THEOREMS ON DEGENERATION OF SPECTRAL SEQUENCES

DELIGNE S THEOREMS ON DEGENERATION OF SPECTRAL SEQUENCES DELIGNE S THEOREMS ON DEGENERATION OF SPECTRAL SEQUENCES YIFEI ZHAO Abstract. We present the proofs of Deligne s theorems on degeneration of the Leray spectral sequence, and the algebraic Hodge-de Rham

More information

Discussion Session on p-divisible Groups

Discussion Session on p-divisible Groups Discussion Session on p-divisible Groups Notes by Tony Feng April 7, 2016 These are notes from a discussion session of p-divisible groups. Some questions were posed by Dennis Gaitsgory, and then their

More information

Higher representation theory in algebra and geometry: Lecture II

Higher representation theory in algebra and geometry: Lecture II Higher representation theory in algebra and geometry: Lecture II Ben Webster UVA ebruary 10, 2014 Ben Webster (UVA) HRT : Lecture II ebruary 10, 2014 1 / 35 References or this lecture, useful references

More information

CYCLIC HOMOLOGY AND THE BEILINSON-MANIN-SCHECHTMAN CENTRAL EXTENSION. Ezra Getzler Harvard University, Cambridge MA 02138

CYCLIC HOMOLOGY AND THE BEILINSON-MANIN-SCHECHTMAN CENTRAL EXTENSION. Ezra Getzler Harvard University, Cambridge MA 02138 CYCLIC HOMOLOGY AND THE BEILINSON-MANIN-SCHECHTMAN CENTRAL EXTENSION. Ezra Getzler Harvard University, Cambridge MA 02138 Abstract. We construct central extensions of the Lie algebra of differential operators

More information

EXT, TOR AND THE UCT

EXT, TOR AND THE UCT EXT, TOR AND THE UCT CHRIS KOTTKE Contents 1. Left/right exact functors 1 2. Projective resolutions 2 3. Two useful lemmas 3 4. Ext 6 5. Ext as a covariant derived functor 8 6. Universal Coefficient Theorem

More information

Loop Groups and Lie 2-Algebras

Loop Groups and Lie 2-Algebras Loop Groups and Lie 2-Algebras Alissa S. Crans Joint work with: John Baez Urs Schreiber & Danny Stevenson in honor of Ross Street s 60th birthday July 15, 2005 Lie 2-Algebras A 2-vector space L is a category

More information

INTRODUCTION TO PART IV: FORMAL GEOMTETRY

INTRODUCTION 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 laft-def, and whose morphisms are nil-isomorphisms of

More information

1 Hochschild Cohomology and A : Jeff Hicks

1 Hochschild Cohomology and A : Jeff Hicks 1 Hochschild Cohomology and A : Jeff Hicks Here s the general strategy of what we would like to do. ˆ From the previous two talks, we have some hope of understanding the triangulated envelope of the Fukaya

More information