D-Modules and Representation Theory
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The Jouanolou-Thomason Homotopy Lemma
The Jouanolou-Thomason homotopy lemma Aravind Asok February 9, 2009 1 Introduction The goal of this note is to prove what is now known as the Jouanolou-Thomason homotopy lemma or simply \Jouanolou's trick." Our main reason for discussing this here is that i) most statements (that I have seen) assume unncessary quasi-projectivity hypotheses, and ii) most applications of the result that I know (e.g., in homotopy K-theory) appeal to the result as merely a \black box," while the proof indicates that the construction is quite geometric and relatively explicit. For simplicity, throughout the word scheme means separated Noetherian scheme. Theorem 1.1 (Jouanolou-Thomason homotopy lemma). Given a smooth scheme X over a regular Noetherian base ring k, there exists a pair (X;~ π), where X~ is an affine scheme, smooth over k, and π : X~ ! X is a Zariski locally trivial smooth morphism with fibers isomorphic to affine spaces. 1 Remark 1.2. In terms of an A -homotopy category of smooth schemes over k (e.g., H(k) or H´et(k); see [MV99, x3]), the map π is an A1-weak equivalence (use [MV99, x3 Example 2.4]. Thus, up to A1-weak equivalence, any smooth k-scheme is an affine scheme smooth over k. 2 An explicit algebraic form Let An denote affine space over Spec Z. Let An n 0 denote the scheme quasi-affine and smooth over 2m Spec Z obtained by removing the fiber over 0. Let Q2m−1 denote the closed subscheme of A (with coordinates x1; : : : ; x2m) defined by the equation X xixm+i = 1: i Consider the following simple situation. -
Vector Bundles on Projective Space
Vector Bundles on Projective Space Takumi Murayama December 1, 2013 1 Preliminaries on vector bundles Let X be a (quasi-projective) variety over k. We follow [Sha13, Chap. 6, x1.2]. Definition. A family of vector spaces over X is a morphism of varieties π : E ! X −1 such that for each x 2 X, the fiber Ex := π (x) is isomorphic to a vector space r 0 0 Ak(x).A morphism of a family of vector spaces π : E ! X and π : E ! X is a morphism f : E ! E0 such that the following diagram commutes: f E E0 π π0 X 0 and the map fx : Ex ! Ex is linear over k(x). f is an isomorphism if fx is an isomorphism for all x. A vector bundle is a family of vector spaces that is locally trivial, i.e., for each x 2 X, there exists a neighborhood U 3 x such that there is an isomorphism ': π−1(U) !∼ U × Ar that is an isomorphism of families of vector spaces by the following diagram: −1 ∼ r π (U) ' U × A (1.1) π pr1 U −1 where pr1 denotes the first projection. We call π (U) ! U the restriction of the vector bundle π : E ! X onto U, denoted by EjU . r is locally constant, hence is constant on every irreducible component of X. If it is constant everywhere on X, we call r the rank of the vector bundle. 1 The following lemma tells us how local trivializations of a vector bundle glue together on the entire space X. -
Special Sheaves of Algebras
Special Sheaves of Algebras Daniel Murfet October 5, 2006 Contents 1 Introduction 1 2 Sheaves of Tensor Algebras 1 3 Sheaves of Symmetric Algebras 6 4 Sheaves of Exterior Algebras 9 5 Sheaves of Polynomial Algebras 17 6 Sheaves of Ideal Products 21 1 Introduction In this note “ring” means a not necessarily commutative ring. If A is a commutative ring then an A-algebra is a ring morphism A −→ B whose image is contained in the center of B. We allow noncommutative sheaves of rings, but if we say (X, OX ) is a ringed space then we mean OX is a sheaf of commutative rings. Throughout this note (X, OX ) is a ringed space. Associated to this ringed space are the following categories: Mod(X), GrMod(X), Alg(X), nAlg(X), GrAlg(X), GrnAlg(X) We show that the forgetful functors Alg(X) −→ Mod(X) and nAlg(X) −→ Mod(X) have left adjoints. If A is a nonzero commutative ring, the forgetful functors AAlg −→ AMod and AnAlg −→ AMod have left adjoints given by the symmetric algebra and tensor algebra con- structions respectively. 2 Sheaves of Tensor Algebras Let F be a sheaf of OX -modules, and for an open set U let P (U) be the OX (U)-algebra given by the tensor algebra T (F (U)). That is, ⊗2 P (U) = OX (U) ⊕ F (U) ⊕ F (U) ⊕ · · · For an inclusion V ⊆ U let ρ : OX (U) −→ OX (V ) and η : F (U) −→ F (V ) be the morphisms of abelian groups given by restriction. For n ≥ 2 we define a multilinear map F (U) × · · · × F (U) −→ F (V ) ⊗ · · · ⊗ F (V ) (m1, . -
Recent Developments in Representation Theory
RECENT DEVELOPMENTS IN REPRESENTATION THEORY Wilfried 5chmid* Department of Mathematics Harvard University Cambridge, MA 02138 For the purposes of this lecture, "representation theory" means representation theory of Lie groups, and more specifically, of semisimple Lie groups. I am interpreting my assignment to give a survey rather loosely: while I shall touch upon various major advances in the subject, I am concentrating on a single development. Both order and emphasis of my presentation are motivated by expository considerations, and do not reflect my view of the relative importance of various topics. Initially G shall denote a locally compact topological group which is unimodular -- i.e., left and right Haar measure coincide -- and H C G a closed unimodular subgroup. The quotient space G/H then carries a G- invariant measure, so G acts unitarily on the Hilbert space L2(G/H). In essence, the fundamental problem of harmonic analysis is to decompose L2(G/H) into a direct "sum" of irreducibles. The quotation marks allude to the fact that the decomposition typically involves the continuous ana- logue of a sum, namely a direct integral, as happens already for non- compact Abelian groups. If G is of type I -- loosely speaking, if the unitary representations of G behave reasonably -- the abstract Plan- cherel theorem [12] asserts the existence of such a decomposition. This existence theorem raises as many questions as it answers: to make the decomposition useful, one wants to know it explicitly and, most impor- tantly, one wants to understand the structure of the irreducible sum- mands. In principle, any irreducible unitary representation of G can occur as a constituent of L2(G/H), for some H C G. -
SHEAVES of MODULES 01AC Contents 1. Introduction 1 2
SHEAVES OF MODULES 01AC Contents 1. Introduction 1 2. Pathology 2 3. The abelian category of sheaves of modules 2 4. Sections of sheaves of modules 4 5. Supports of modules and sections 6 6. Closed immersions and abelian sheaves 6 7. A canonical exact sequence 7 8. Modules locally generated by sections 8 9. Modules of finite type 9 10. Quasi-coherent modules 10 11. Modules of finite presentation 13 12. Coherent modules 15 13. Closed immersions of ringed spaces 18 14. Locally free sheaves 20 15. Bilinear maps 21 16. Tensor product 22 17. Flat modules 24 18. Duals 26 19. Constructible sheaves of sets 27 20. Flat morphisms of ringed spaces 29 21. Symmetric and exterior powers 29 22. Internal Hom 31 23. Koszul complexes 33 24. Invertible modules 33 25. Rank and determinant 36 26. Localizing sheaves of rings 38 27. Modules of differentials 39 28. Finite order differential operators 43 29. The de Rham complex 46 30. The naive cotangent complex 47 31. Other chapters 50 References 52 1. Introduction 01AD This is a chapter of the Stacks Project, version 77243390, compiled on Sep 28, 2021. 1 SHEAVES OF MODULES 2 In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Z- modules. Basic references are [Ser55], [DG67] and [AGV71]. We work out what happens for sheaves of modules on ringed topoi in another chap- ter (see Modules on Sites, Section 1), although there we will mostly just duplicate the discussion from this chapter. -
4. Coherent Sheaves Definition 4.1. If (X,O X) Is a Locally Ringed Space
4. Coherent Sheaves Definition 4.1. If (X; OX ) is a locally ringed space, then we say that an OX -module F is locally free if there is an open affine cover fUig of X such that FjUi is isomorphic to a direct sum of copies of OUi . If the number of copies r is finite and constant, then F is called locally free of rank r (aka a vector bundle). If F is locally free of rank one then we way say that F is invertible (aka a line bundle). The group of all invertible sheaves under tensor product, denoted Pic(X), is called the Picard group of X. A sheaf of ideals I is any OX -submodule of OX . Definition 4.2. Let X = Spec A be an affine scheme and let M be an A-module. M~ is the sheaf which assigns to every open subset U ⊂ X, the set of functions a s: U −! Mp; p2U which can be locally represented at p as a=g, a 2 M, g 2 R, p 2= Ug ⊂ U. Lemma 4.3. Let A be a ring and let M be an A-module. Let X = Spec A. ~ (1) M is a OX -module. ~ (2) If p 2 X then Mp is isomorphic to Mp. ~ (3) If f 2 A then M(Uf ) is isomorphic to Mf . Proof. (1) is clear and the rest is proved mutatis mutandis as for the structure sheaf. Definition 4.4. An OX -module F on a scheme X is called quasi- coherent if there is an open cover fUi = Spec Aig by affines and ~ isomorphisms FjUi ' Mi, where Mi is an Ai-module. -
256B Algebraic Geometry
256B Algebraic Geometry David Nadler Notes by Qiaochu Yuan Spring 2013 1 Vector bundles on the projective line This semester we will be focusing on coherent sheaves on smooth projective complex varieties. The organizing framework for this class will be a 2-dimensional topological field theory called the B-model. Topics will include 1. Vector bundles and coherent sheaves 2. Cohomology, derived categories, and derived functors (in the differential graded setting) 3. Grothendieck-Serre duality 4. Reconstruction theorems (Bondal-Orlov, Tannaka, Gabriel) 5. Hochschild homology, Chern classes, Grothendieck-Riemann-Roch For now we'll introduce enough background to talk about vector bundles on P1. We'll regard varieties as subsets of PN for some N. Projective will mean that we look at closed subsets (with respect to the Zariski topology). The reason is that if p : X ! pt is the unique map from such a subset X to a point, then we can (derived) push forward a bounded complex of coherent sheaves M on X to a bounded complex of coherent sheaves on a point Rp∗(M). Smooth will mean the following. If x 2 X is a point, then locally x is cut out by 2 a maximal ideal mx of functions vanishing on x. Smooth means that dim mx=mx = dim X. (In general it may be bigger.) Intuitively it means that locally at x the variety X looks like a manifold, and one way to make this precise is that the completion of the local ring at x is isomorphic to a power series ring C[[x1; :::xn]]; this is the ring where Taylor series expansions live. -
Notes on Principal Bundles and Classifying Spaces
Notes on principal bundles and classifying spaces Stephen A. Mitchell August 2001 1 Introduction Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X, BG]. -
D-Modules on Flag Varieties and Localization of G-Modules
D-modules on flag varieties and localization of g-modules. Jos´eSimental. October 7, 2013 1 (Twisted) Differential operators and D-modules. 1.1 Differential Operators. Let X be an algebraic variety. In this text, we assume all algebraic varieties are over C. Let R be a sheaf of rings on X. For R-modules F and G, define the sheaf HomR(F; G) as follows. For any open set U ⊂ X, the sections of HomR(F; G) on U are HomRjU (FjU ; GjU ). For any algebraic variety X, let CX be the constant sheaf of complex numbers C. Now assume that X is a smooth complex algebraic variety. Let OX be its sheaf of regular functions and VX be its sheaf of vector fields, that is, VX := DerC(OX ) := fθ 2 EndCX (OX ): θ(fg) = fθ(g) + θ(f)g; f; g 2 OX g. The sheaf DX of differential operators on X, is, by definition, the subsheaf (of associative rings) of EndCX (OX ) generated by OX and VX . Theorem 1.1 In the setup of the previous paragraph, if dim X = n, then for every point p 2 X there exists an affine open neighborhood U of p, x1; : : : ; xn 2 OX (U) and @1;:::;@n 2 VX (U) such that [@i;@j] = 0, @i(xj) = δij and VX (U) is free over OX (U) with basis @1;:::;@n. Proof. Let mp be the maximal ideal of OX;p. Since X is smooth of dimension n, there exist functions x1; : : : ; xn generating mp. Then, dx1; : : : ; dxn is a basis of ΩX;p, where ΩX is the cotangent sheaf of X. -
Differentials
Section 2.8 - Differentials Daniel Murfet October 5, 2006 In this section we will define the sheaf of relative differential forms of one scheme over another. In the case of a nonsingular variety over C, which is like a complex manifold, the sheaf of differential forms is essentially the same as the dual of the tangent bundle in differential geometry. However, in abstract algebraic geometry, we will define the sheaf of differentials first, by a purely algebraic method, and then define the tangent bundle as its dual. Hence we will begin this section with a review of the module of differentials of one ring over another. As applications of the sheaf of differentials, we will give a characterisation of nonsingular varieties among schemes of finite type over a field. We will also use the sheaf of differentials on a nonsingular variety to define its tangent sheaf, its canonical sheaf, and its geometric genus. This latter is an important numerical invariant of a variety. Contents 1 K¨ahler Differentials 1 2 Sheaves of Differentials 2 3 Nonsingular Varieties 13 4 Rational Maps 16 5 Applications 19 6 Some Local Algebra 22 1 K¨ahlerDifferentials See (MAT2,Section 2) for the definition and basic properties of K¨ahler differentials. Definition 1. Let B be a local ring with maximal ideal m.A field of representatives for B is a subfield L of A which is mapped onto A/m by the canonical mapping A −→ A/m. Since L is a field, the restriction gives an isomorphism of fields L =∼ A/m. Proposition 1. -
The Chern Characteristic Classes
The Chern characteristic classes Y. X. Zhao I. FUNDAMENTAL BUNDLE OF QUANTUM MECHANICS Consider a nD Hilbert space H. Qauntum states are normalized vectors in H up to phase factors. Therefore, more exactly a quantum state j i should be described by the density matrix, the projector to the 1D subspace generated by j i, P = j ih j: (I.1) n−1 All quantum states P comprise the projective space P H of H, and pH ≈ CP . If we exclude zero point from H, there is a natural projection from H − f0g to P H, π : H − f0g ! P H: (I.2) On the other hand, there is a tautological line bundle T (P H) over over P H, where over the point P the fiber T is the complex line generated by j i. Therefore, there is the pullback line bundle π∗L(P H) over H − f0g and the line bundle morphism, π~ π∗T (P H) T (P H) π H − f0g P H . Then, a bundle of Hilbert spaces E over a base space B, which may be a parameter space, such as momentum space, of a quantum system, or a phase space. We can repeat the above construction for a single Hilbert space to the vector bundle E with zero section 0B excluded. We obtain the projective bundle PEconsisting of points (P x ; x) (I.3) where j xi 2 Hx; x 2 B: (I.4) 2 Obviously, PE is a bundle over B, p : PE ! B; (I.5) n−1 where each fiber (PE)x ≈ CP . -
Lie Algebroid Cohomology As a Derived Functor
LIE ALGEBROID COHOMOLOGY AS A DERIVED FUNCTOR Ugo Bruzzo Area di Matematica, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, 34136 Trieste, Italy; Department of Mathematics, University of Pennsylvania, 209 S 33rd st., Philadelphia, PA 19104-6315, USA; Istituto Nazionale di Fisica Nucleare, Sezione di Trieste Abstract. We show that the hypercohomology of the Chevalley-Eilenberg- de Rham complex of a Lie algebroid L over a scheme with coefficients in an L -module can be expressed as a derived functor. We use this fact to study a Hochschild-Serre type spectral sequence attached to an extension of Lie alge- broids. Contents 1. Introduction 2 2. Generalities on Lie algebroids 3 3. Cohomology as derived functor 7 4. A Hochshild-Serre spectral sequence 11 References 16 arXiv:1606.02487v4 [math.RA] 14 Apr 2017 Date: Revised 14 April 2017 2000 Mathematics Subject Classification: 14F40, 18G40, 32L10, 55N35, 55T05 Keywords: Lie algebroid cohomology, derived functors, spectral sequences Research partly supported by indam-gnsaga. u.b. is a member of the vbac group. 1 2 1. Introduction As it is well known, the cohomology groups of a Lie algebra g over a ring A with coeffi- cients in a g-module M can be computed directly from the Chevalley-Eilenberg complex, or as the derived functors of the invariant submodule functor, i.e., the functor which with ever g-module M associates the submodule of M M g = {m ∈ M | ρ(g)(m)=0}, where ρ: g → End A(M) is a representation of g (see e.g. [16]). In this paper we show an analogous result for the hypercohomology of the Chevalley-Eilenberg-de Rham complex of a Lie algebroid L over a scheme X, with coefficients in a representation M of L (the notion of hypercohomology is recalled in Section 2).