Differential Graded Algebras and Applications
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3. Some Commutative Algebra Definition 3.1. Let R Be a Ring. We
3. Some commutative algebra Definition 3.1. Let R be a ring. We say that R is graded, if there is a direct sum decomposition, M R = Rn; n2N where each Rn is an additive subgroup of R, such that RdRe ⊂ Rd+e: The elements of Rd are called the homogeneous elements of order d. Let R be a graded ring. We say that an R-module M is graded if there is a direct sum decomposition M M = Mn; n2N compatible with the grading on R in the obvious way, RdMn ⊂ Md+n: A morphism of graded modules is an R-module map φ: M −! N of graded modules, which respects the grading, φ(Mn) ⊂ Nn: A graded submodule is a submodule for which the inclusion map is a graded morphism. A graded ideal I of R is an ideal, which when considered as a submodule is a graded submodule. Note that the kernel and cokernel of a morphism of graded modules is a graded module. Note also that an ideal is a graded ideal iff it is generated by homogeneous elements. Here is the key example. Example 3.2. Let R be the polynomial ring over a ring S. Define a direct sum decomposition of R by taking Rn to be the set of homogeneous polynomials of degree n. Given a graded ideal I in R, that is an ideal generated by homogeneous elements of R, the quotient is a graded ring. We will also need the notion of localisation, which is a straightfor- ward generalisation of the notion of the field of fractions. -
The Geometry of Syzygies
The Geometry of Syzygies A second course in Commutative Algebra and Algebraic Geometry David Eisenbud University of California, Berkeley with the collaboration of Freddy Bonnin, Clement´ Caubel and Hel´ ene` Maugendre For a current version of this manuscript-in-progress, see www.msri.org/people/staff/de/ready.pdf Copyright David Eisenbud, 2002 ii Contents 0 Preface: Algebra and Geometry xi 0A What are syzygies? . xii 0B The Geometric Content of Syzygies . xiii 0C What does it mean to solve linear equations? . xiv 0D Experiment and Computation . xvi 0E What’s In This Book? . xvii 0F Prerequisites . xix 0G How did this book come about? . xix 0H Other Books . 1 0I Thanks . 1 0J Notation . 1 1 Free resolutions and Hilbert functions 3 1A Hilbert’s contributions . 3 1A.1 The generation of invariants . 3 1A.2 The study of syzygies . 5 1A.3 The Hilbert function becomes polynomial . 7 iii iv CONTENTS 1B Minimal free resolutions . 8 1B.1 Describing resolutions: Betti diagrams . 11 1B.2 Properties of the graded Betti numbers . 12 1B.3 The information in the Hilbert function . 13 1C Exercises . 14 2 First Examples of Free Resolutions 19 2A Monomial ideals and simplicial complexes . 19 2A.1 Syzygies of monomial ideals . 23 2A.2 Examples . 25 2A.3 Bounds on Betti numbers and proof of Hilbert’s Syzygy Theorem . 26 2B Geometry from syzygies: seven points in P3 .......... 29 2B.1 The Hilbert polynomial and function. 29 2B.2 . and other information in the resolution . 31 2C Exercises . 34 3 Points in P2 39 3A The ideal of a finite set of points . -
Lecture 3. Resolutions and Derived Functors (GL)
Lecture 3. Resolutions and derived functors (GL) This lecture is intended to be a whirlwind introduction to, or review of, reso- lutions and derived functors { with tunnel vision. That is, we'll give unabashed preference to topics relevant to local cohomology, and do our best to draw a straight line between the topics we cover and our ¯nal goals. At a few points along the way, we'll be able to point generally in the direction of other topics of interest, but other than that we will do our best to be single-minded. Appendix A contains some preparatory material on injective modules and Matlis theory. In this lecture, we will cover roughly the same ground on the projective/flat side of the fence, followed by basics on projective and injective resolutions, and de¯nitions and basic properties of derived functors. Throughout this lecture, let us work over an unspeci¯ed commutative ring R with identity. Nearly everything said will apply equally well to noncommutative rings (and some statements need even less!). In terms of module theory, ¯elds are the simple objects in commutative algebra, for all their modules are free. The point of resolving a module is to measure its complexity against this standard. De¯nition 3.1. A module F over a ring R is free if it has a basis, that is, a subset B ⊆ F such that B generates F as an R-module and is linearly independent over R. It is easy to prove that a module is free if and only if it is isomorphic to a direct sum of copies of the ring. -
Depth, Dimension and Resolutions in Commutative Algebra
Depth, Dimension and Resolutions in Commutative Algebra Claire Tête PhD student in Poitiers MAP, May 2014 Claire Tête Commutative Algebra This morning: the Koszul complex, regular sequence, depth Tomorrow: the Buchsbaum & Eisenbud criterion and the equality of Aulsander & Buchsbaum through examples. Wednesday: some elementary results about the homology of a bicomplex Claire Tête Commutative Algebra I will begin with a little example. Let us consider the ideal a = hX1, X2, X3i of A = k[X1, X2, X3]. What is "the" resolution of A/a as A-module? (the question is deliberatly not very precise) Claire Tête Commutative Algebra I will begin with a little example. Let us consider the ideal a = hX1, X2, X3i of A = k[X1, X2, X3]. What is "the" resolution of A/a as A-module? (the question is deliberatly not very precise) We would like to find something like this dm dm−1 d1 · · · Fm Fm−1 · · · F1 F0 A/a with A-modules Fi as simple as possible and s.t. Im di = Ker di−1. Claire Tête Commutative Algebra I will begin with a little example. Let us consider the ideal a = hX1, X2, X3i of A = k[X1, X2, X3]. What is "the" resolution of A/a as A-module? (the question is deliberatly not very precise) We would like to find something like this dm dm−1 d1 · · · Fm Fm−1 · · · F1 F0 A/a with A-modules Fi as simple as possible and s.t. Im di = Ker di−1. We say that F· is a resolution of the A-module A/a Claire Tête Commutative Algebra I will begin with a little example. -
DIFFERENTIAL ALGEBRA Lecturer: Alexey Ovchinnikov Thursdays 9
DIFFERENTIAL ALGEBRA Lecturer: Alexey Ovchinnikov Thursdays 9:30am-11:40am Website: http://qcpages.qc.cuny.edu/˜aovchinnikov/ e-mail: [email protected] written by: Maxwell Shapiro 0. INTRODUCTION There are two main topics we will discuss in these lectures: (I) The core differential algebra: (a) Introduction: We will begin with an introduction to differential algebraic structures, important terms and notation, and a general background needed for this lecture. (b) Differential elimination: Given a system of polynomial partial differential equations (PDE’s for short), we will determine if (i) this system is consistent, or if (ii) another polynomial PDE is a consequence of the system. (iii) If time permits, we will also discuss algorithms will perform (i) and (ii). (II) Differential Galois Theory for linear systems of ordinary differential equations (ODE’s for short). This subject deals with questions of this sort: Given a system d (?) y(x) = A(x)y(x); dx where A(x) is an n × n matrix, find all algebraic relations that a solution of (?) can possibly satisfy. Hrushovski developed an algorithm to solve this for any A(x) with entries in Q¯ (x) (here, Q¯ is the algebraic closure of Q). Example 0.1. Consider the ODE dy(x) 1 = y(x): dx 2x p We know that y(x) = x is a solution to this equation. As such, we can determine an algebraic relation to this ODE to be y2(x) − x = 0: In the previous example, we solved the ODE to determine an algebraic relation. Differential Galois Theory uses methods to find relations without having to solve. -
Differential Graded Lie Algebras and Deformation Theory
Differential graded Lie algebras and Deformation theory Marco Manetti 1 Differential graded Lie algebras Let L be a lie algebra over a fixed field K of characteristic 0. Definition 1. A K-linear map d : L ! L is called a derivation if it satisfies the Leibniz rule d[a; b] = [da; b]+[a; db]. n d P dn Lemma. If d is nilpotent (i.e. d = 0 for n sufficiently large) then e = n≥0 n! : L ! L is an isomorphism of Lie algebras. Proof. Exercise. Definition 2. L is called nilpotent if the descending central series L ⊃ [L; L] ⊃ [L; [L; L]] ⊃ · · · stabilizes at 0. (I.e., if we write [L]2 = [L; L], [L]3 = [L; [L; L]], etc., then [L]n = 0 for n sufficiently large.) Note that in the finite-dimensional case, this is equivalent to the other common definition that for every x 2 L, adx is nilpotent. We have the Baker-Campbell-Hausdorff formula (BCH). Theorem. For every nilpotent Lie algebra L there is an associative product • : L × L ! L satisfying 1. functoriality in L; i.e. if f : L ! M is a morphism of nilpotent Lie algebras then f(a • b) = f(a) • f(b). a P an 2. If I ⊂ R is a nilpotent ideal of the associative unitary K-algebra R and for a 2 I we define e = n≥0 n! 2 R, then ea•b = ea • eb. Heuristically, we can write a • b = log(ea • eb). Proof. Exercise (use the computation of www.mat.uniroma1.it/people/manetti/dispense/BCHfjords.pdf and the existence of free Lie algebras). -
About Lie-Rinehart Superalgebras Claude Roger
About Lie-Rinehart superalgebras Claude Roger To cite this version: Claude Roger. About Lie-Rinehart superalgebras. 2019. hal-02071706 HAL Id: hal-02071706 https://hal.archives-ouvertes.fr/hal-02071706 Preprint submitted on 18 Mar 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. About Lie-Rinehart superalgebras Claude Rogera aInstitut Camille Jordan ,1 , Universit´ede Lyon, Universit´eLyon I, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France Keywords:Supermanifolds, Lie-Rinehart algebras, Fr¨olicher-Nijenhuis bracket, Nijenhuis-Richardson bracket, Schouten-Nijenhuis bracket. Mathematics Subject Classification (2010): .17B35, 17B66, 58A50 Abstract: We extend to the superalgebraic case the theory of Lie-Rinehart algebras and work out some examples concerning the most popular samples of supermanifolds. 1 Introduction The goal of this article is to discuss some properties of Lie-Rinehart structures in a superalgebraic context. Let's first sketch the classical case; a Lie-Rinehart algebra is a couple (A; L) where L is a Lie algebra on a base field k, A an associative and commutative k-algebra, with two operations 1. (A; L) ! L denoted by (a; X) ! aX, which induces a A-module structure on L, and 2. -
Lectures on Supersymmetry and Superstrings 1 General Superalgebra
Lectures on supersymmetry and superstrings Thomas Mohaupt Abstract: These are informal notes on some mathematical aspects of su- persymmetry, based on two ‘Mathematics and Theoretical Physics Meetings’ in the Autumn Term 2009.1 First super vector spaces, Lie superalgebras and su- percommutative associative superalgebras are briefly introduced together with superfunctions and superspaces. After a review of the Poincar´eLie algebra we discuss Poincar´eLie superalgebras and introduce Minkowski superspaces. As a minimal example of a supersymmetric model we discuss the free scalar superfield in two-dimensional N = (1, 1) supersymmetry (this is more or less copied from [1]). We briefly discuss the concept of chiral supersymmetry in two dimensions. None of the material is original. We give references for further reading. Some ‘bonus material’ on (super-)conformal field theories, (super-)string theories and symmetric spaces is included. These are topics which will play a role in future meetings, or which came up briefly during discussions. 1 General superalgebra Definition: A super vector space V is a vector space with a decomposition V = V V 0 ⊕ 1 and a map (parity map) ˜ : (V V ) 0 Z · 0 ∪ 1 −{ } → 2 which assigns even or odd parity to every homogeneous element. Notation: a V a˜ =0 , ‘even’ , b V ˜b = 1 ‘odd’ . ∈ 0 → ∈ 1 → Definition: A Lie superalgebra g (also called super Lie algebra) is a super vector space g = g g 0 ⊕ 1 together with a bracket respecting the grading: [gi , gj] gi j ⊂ + (addition of index mod 2), which is required to be Z2 graded anti-symmetric: ˜ [a,b]= ( 1)a˜b[b,a] − − (multiplication of parity mod 2), and to satisfy a Z2 graded Jacobi identity: ˜ [a, [b,c]] = [[a,b],c] + ( 1)a˜b[b, [a,c]] . -
Linear Resolutions Over Koszul Complexes and Koszul Homology
LINEAR RESOLUTIONS OVER KOSZUL COMPLEXES AND KOSZUL HOMOLOGY ALGEBRAS JOHN MYERS Abstract. Let R be a standard graded commutative algebra over a field k, let K be its Koszul complex viewed as a differential graded k-algebra, and let H be the homology algebra of K. This paper studies the interplay between homological properties of the three algebras R, K, and H. In particular, we introduce two definitions of Koszulness that extend the familiar property originally introduced by Priddy: one which applies to K (and, more generally, to any connected differential graded k-algebra) and the other, called strand- Koszulness, which applies to H. The main theoretical result is a complete description of how these Koszul properties of R, K, and H are related to each other. This result shows that strand-Koszulness of H is stronger than Koszulness of R, and we include examples of classes of algebras which have Koszul homology algebras that are strand-Koszul. Introduction Koszul complexes are classical objects of study in commutative algebra. Their structure reflects many important properties of the rings over which they are de- fined, and these reflections are often encoded in the product structure of their homology. Indeed, Koszul complexes are more than just complexes — they are, in fact, the prototypical examples of differential graded (= DG) algebras in commu- tative ring theory, and thus the homology of a Koszul complex carries an algebra structure. These homology algebras encode (among other things) the Gorenstein condition [5], the Golod condition [15], and whether or not the ring is a complete intersection [1]. -
Dimension Formula for Graded Lie Algebras and Its Applications
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 11, Pages 4281–4336 S 0002-9947(99)02239-4 Article electronically published on June 29, 1999 DIMENSION FORMULA FOR GRADED LIE ALGEBRAS AND ITS APPLICATIONS SEOK-JIN KANG AND MYUNG-HWAN KIM Abstract. In this paper, we investigate the structure of infinite dimensional Lie algebras L = α Γ Lα graded by a countable abelian semigroup Γ sat- isfying a certain finiteness∈ condition. The Euler-Poincar´e principle yields the denominator identitiesL for the Γ-graded Lie algebras, from which we derive a dimension formula for the homogeneous subspaces Lα (α Γ). Our dimen- sion formula enables us to study the structure of the Γ-graded2 Lie algebras in a unified way. We will discuss some interesting applications of our dimension formula to the various classes of graded Lie algebras such as free Lie algebras, Kac-Moody algebras, and generalized Kac-Moody algebras. We will also dis- cuss the relation of graded Lie algebras and the product identities for formal power series. Introduction In [K1] and [Mo1], Kac and Moody independently introduced a new class of Lie algebras, called Kac-Moody algebras, as a generalization of finite dimensional simple Lie algebras over C. These algebras are ususally infinite dimensional, and are defined by the generators and relations associated with generalized Cartan matrices, which is similar to Serre’s presentation of complex semisimple Lie algebras. In [M], by generalizing Weyl’s denominator identity to the affine root system, Macdonald obtained a new family of combinatorial identities, including Jacobi’s triple product identity as the simplest case. -
Z-Graded Lie Superalgebras of Infinite Depth and Finite Growth 547
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. I (2002), pp. 545-568 Z-graded Lie Superalgebras of Infinite Depth and Finite Growth NICOLETTA CANTARINI Abstract. In 1998 Victor Kac classified infinite-dimensional Z-graded Lie su- peralgebras of finite depth. We construct new examples of infinite-dimensional Lie superalgebras with a Z-gradation of infinite depth and finite growth and clas- sify Z-graded Lie superalgebras of infinite depth and finite growth under suitable hypotheses. Mathematics Subject Classification (2000): 17B65 (primary), 17B70 (secondary). Introduction Simple finite-dimensional Lie superalgebras were classified by V. G. Kac in [K2]. In the same paper Kac classified the finite-dimensional, Z-graded Lie superalgebras under the hypotheses of irreducibility and transitivity. The classification of infinite-dimensional, Z-graded Lie superalgebras of finite depth is also due to V. G. Kac [K3] and is deeply related to the classifi- cation of linearly compact Lie superalgebras. We recall that finite depth implies finite growth. This naturally leads to investigate infinite-dimensional, Z-graded Lie super- algebras of infinite depth and finite growth. The hypothesis of finite growth is central to the problem; indeed, it is well known that it is not possible to classify Z-graded Lie algebras (and thus Lie superalgebras) of any growth (see [K1], [M]). The only known examples of infinite-dimensional, Z-graded Lie superalgebras of finite growth and infinite depth are given by contragredient Lie superalgebras which were classified by V. G. Kac in [K2] in the case of finite dimension and by J.W. van de Leur in the general case [vdL]. -
Graded Lie Algebras and Representations of Supersymmetry Algebras Physics 251 Group Theory and Modern Physics
Graded Lie Algebras and Representations of Supersymmetry Algebras Physics 251 Group Theory and Modern Physics Jaryd Franklin Ulbricht June 14, 2017 J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 1 / 50 Overview 1 Introduction and Definitions Linear Vector Spaces Linear Algebras Lie Algebras 2 Graded Algebras Graded Lie Algebras 3 Supersymmetry Lie Superalgebras Superspace Supersymmetry Algebras 4 Representations of Supersymmetry Algebras Boson and Fermion Number Constructing Massive Representations Constructing Massless Representations J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 2 / 50 How do we construct a graded algebra? It's actually much easier than you think Graded Lie Algebras What is a graded algebra? J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 3 / 50 It's actually much easier than you think Graded Lie Algebras What is a graded algebra? How do we construct a graded algebra? J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 3 / 50 Graded Lie Algebras What is a graded algebra? How do we construct a graded algebra? It's actually much easier than you think J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 3 / 50 Graded Lie Algebras su (2) J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 4 / 50 Graded Lie Algebras su (2) J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 4 / 50 Graded Lie Algebras so (3) J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 5 / 50 Graded Lie Algebras so (3) J. F. Ulbricht Graded Lie Algebras and SUSY June 14, 2017 5 / 50 Graded Lie Algebras E6 J.