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Modules and Vector Spaces
Modules and Vector Spaces R. C. Daileda October 16, 2017 1 Modules Definition 1. A (left) R-module is a triple (R; M; ·) consisting of a ring R, an (additive) abelian group M and a binary operation · : R × M ! M (simply written as r · m = rm) that for all r; s 2 R and m; n 2 M satisfies • r(m + n) = rm + rn ; • (r + s)m = rm + sm ; • r(sm) = (rs)m. If R has unity we also require that 1m = m for all m 2 M. If R = F , a field, we call M a vector space (over F ). N Remark 1. One can show that as a consequence of this definition, the zeros of R and M both \act like zero" relative to the binary operation between R and M, i.e. 0Rm = 0M and r0M = 0M for all r 2 R and m 2 M. H Example 1. Let R be a ring. • R is an R-module using multiplication in R as the binary operation. • Every (additive) abelian group G is a Z-module via n · g = ng for n 2 Z and g 2 G. In fact, this is the only way to make G into a Z-module. Since we must have 1 · g = g for all g 2 G, one can show that n · g = ng for all n 2 Z. Thus there is only one possible Z-module structure on any abelian group. • Rn = R ⊕ R ⊕ · · · ⊕ R is an R-module via | {z } n times r(a1; a2; : : : ; an) = (ra1; ra2; : : : ; ran): • Mn(R) is an R-module via r(aij) = (raij): • R[x] is an R-module via X i X i r aix = raix : i i • Every ideal in R is an R-module. -
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 . -
On Fusion Categories
Annals of Mathematics, 162 (2005), 581–642 On fusion categories By Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik Abstract Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any (not nec- essarily hermitian) modular category is unitary. We also show that the cate- gory of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion cat- egory. At the end of the paper we generalize some of these results to positive characteristic. 1. Introduction Throughout this paper (except for §9), k denotes an algebraically closed field of characteristic zero. By a fusion category C over k we mean a k-linear semisimple rigid tensor (=monoidal) category with finitely many simple ob- jects and finite dimensional spaces of morphisms, such that the endomorphism algebra of the neutral object is k (see [BaKi]). Fusion categories arise in several areas of mathematics and physics – conformal field theory, operator algebras, representation theory of quantum groups, and others. This paper is devoted to the study of general properties of fusion cate- gories. This has been an area of intensive research for a number of years, and many remarkable results have been obtained. -
On the Cohen-Macaulay Modules of Graded Subrings
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 2, Pages 735{756 S 0002-9947(04)03562-7 Article electronically published on April 27, 2004 ON THE COHEN-MACAULAY MODULES OF GRADED SUBRINGS DOUGLAS HANES Abstract. We give several characterizations for the linearity property for a maximal Cohen-Macaulay module over a local or graded ring, as well as proofs of existence in some new cases. In particular, we prove that the existence of such modules is preserved when taking Segre products, as well as when passing to Veronese subrings in low dimensions. The former result even yields new results on the existence of finitely generated maximal Cohen-Macaulay modules over non-Cohen-Macaulay rings. 1. Introduction and definitions The notion of a linear maximal Cohen-Macaulay module over a local ring (R; m) was introduced by Ulrich [17], who gave a simple characterization of the Gorenstein property for a Cohen-Macaulay local ring possessing a maximal Cohen-Macaulay module which is sufficiently close to being linear, as defined below. A maximal Cohen-Macaulay module (abbreviated MCM module)overR is one whose depth is equal to the Krull dimension of the ring R. All modules to be considered in this paper are assumed to be finitely generated. The minimal number of generators of an R-module M, which is equal to the (R=m)-vector space dimension of M=mM, will be denoted throughout by ν(M). In general, the length of a finitely generated Artinian module M is denoted by `(M)(orby`R(M), if it is necessary to specify the ring acting on M). -
A NOTE on SHEAVES WITHOUT SELF-EXTENSIONS on the PROJECTIVE N-SPACE
A NOTE ON SHEAVES WITHOUT SELF-EXTENSIONS ON THE PROJECTIVE n-SPACE. DIETER HAPPEL AND DAN ZACHARIA Abstract. Let Pn be the projective n−space over the complex numbers. In this note we show that an indecomposable rigid coherent sheaf on Pn has a trivial endomorphism algebra. This generalizes a result of Drezet for n = 2: Let Pn be the projective n-space over the complex numbers. Recall that an exceptional sheaf is a coherent sheaf E such that Exti(E; E) = 0 for all i > 0 and End E =∼ C. Dr´ezetproved in [4] that if E is an indecomposable sheaf over P2 such that Ext1(E; E) = 0 then its endomorphism ring is trivial, and also that Ext2(E; E) = 0. Moreover, the sheaf is locally free. Partly motivated by this result, we prove in this short note that if E is an indecomposable coherent sheaf over the projective n-space such that Ext1(E; E) = 0, then we automatically get that End E =∼ C. The proof involves reducing the problem to indecomposable linear modules without self extensions over the polynomial algebra. In the second part of this article, we look at the Auslander-Reiten quiver of the derived category of coherent sheaves over the projective n-space. It is known ([10]) that if n > 1, then all the components are of type ZA1. Then, using the Bernstein-Gelfand-Gelfand correspondence ([3]) we prove that each connected component contains at most one sheaf. We also show that in this case the sheaf lies on the boundary of the component. -
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. -
Modules and Lie Semialgebras Over Semirings with a Negation Map 3
MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP GUY BLACHAR Abstract. In this article, we present the basic definitions of modules and Lie semialgebras over semirings with a negation map. Our main example of a semiring with a negation map is ELT algebras, and some of the results in this article are formulated and proved only in the ELT theory. When dealing with modules, we focus on linearly independent sets and spanning sets. We define a notion of lifting a module with a negation map, similarly to the tropicalization process, and use it to prove several theorems about semirings with a negation map which possess a lift. In the context of Lie semialgebras over semirings with a negation map, we first give basic definitions, and provide parallel constructions to the classical Lie algebras. We prove an ELT version of Cartan’s criterion for semisimplicity, and provide a counterexample for the naive version of the PBW Theorem. Contents Page 0. Introduction 2 0.1. Semirings with a Negation Map 2 0.2. Modules Over Semirings with a Negation Map 3 0.3. Supertropical Algebras 4 0.4. Exploded Layered Tropical Algebras 4 1. Modules over Semirings with a Negation Map 5 1.1. The Surpassing Relation for Modules 6 1.2. Basic Definitions for Modules 7 1.3. -morphisms 9 1.4. Lifting a Module Over a Semiring with a Negation Map 10 1.5. Linearly Independent Sets 13 1.6. d-bases and s-bases 14 1.7. Free Modules over Semirings with a Negation Map 18 2. -
Irreducible Representations of Finite Monoids
U.U.D.M. Project Report 2019:11 Irreducible representations of finite monoids Christoffer Hindlycke Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Mars 2019 Department of Mathematics Uppsala University Irreducible representations of finite monoids Christoffer Hindlycke Contents Introduction 2 Theory 3 Finite monoids and their structure . .3 Introductory notions . .3 Cyclic semigroups . .6 Green’s relations . .7 von Neumann regularity . 10 The theory of an idempotent . 11 The five functors Inde, Coinde, Rese,Te and Ne ..................... 11 Idempotents and simple modules . 14 Irreducible representations of a finite monoid . 17 Monoid algebras . 17 Clifford-Munn-Ponizovski˘ıtheory . 20 Application 24 The symmetric inverse monoid . 24 Calculating the irreducible representations of I3 ........................ 25 Appendix: Prerequisite theory 37 Basic definitions . 37 Finite dimensional algebras . 41 Semisimple modules and algebras . 41 Indecomposable modules . 42 An introduction to idempotents . 42 1 Irreducible representations of finite monoids Christoffer Hindlycke Introduction This paper is a literature study of the 2016 book Representation Theory of Finite Monoids by Benjamin Steinberg [3]. As this book contains too much interesting material for a simple master thesis, we have narrowed our attention to chapters 1, 4 and 5. This thesis is divided into three main parts: Theory, Application and Appendix. Within the Theory chapter, we (as the name might suggest) develop the necessary theory to assist with finding irreducible representations of finite monoids. Finite monoids and their structure gives elementary definitions as regards to finite monoids, and expands on the basic theory of their structure. This part corresponds to chapter 1 in [3]. The theory of an idempotent develops just enough theory regarding idempotents to enable us to state a key result, from which the principal result later follows almost immediately. -
Math 615: Lecture of April 16, 2007 We Next Note the Following Fact
Math 615: Lecture of April 16, 2007 We next note the following fact: n Proposition. Let R be any ring and F = R a free module. If f1, . , fn ∈ F generate F , then f1, . , fn is a free basis for F . n n Proof. We have a surjection R F that maps ei ∈ R to fi. Call the kernel N. Since F is free, the map splits, and we have Rn =∼ F ⊕ N. Then N is a homomorphic image of Rn, and so is finitely generated. If N 6= 0, we may preserve this while localizing at a suitable maximal ideal m of R. We may therefore assume that (R, m, K) is quasilocal. Now apply n ∼ n K ⊗R . We find that K = K ⊕ N/mN. Thus, N = mN, and so N = 0. The final step in our variant proof of the Hilbert syzygy theorem is the following: Lemma. Let R = K[x1, . , xn] be a polynomial ring over a field K, let F be a free R- module with ordered free basis e1, . , es, and fix any monomial order on F . Let M ⊆ F be such that in(M) is generated by a subset of e1, . , es, i.e., such that M has a Gr¨obner basis whose initial terms are a subset of e1, . , es. Then M and F/M are R-free. Proof. Let S be the subset of e1, . , es generating in(M), and suppose that S has r ∼ s−r elements. Let T = {e1, . , es} − S, which has s − r elements. Let G = R be the free submodule of F spanned by T . -
A Topology for Operator Modules Over W*-Algebras Bojan Magajna
Journal of Functional AnalysisFU3203 journal of functional analysis 154, 1741 (1998) article no. FU973203 A Topology for Operator Modules over W*-Algebras Bojan Magajna Department of Mathematics, University of Ljubljana, Jadranska 19, Ljubljana 1000, Slovenia E-mail: Bojan.MagajnaÄuni-lj.si Received July 23, 1996; revised February 11, 1997; accepted August 18, 1997 dedicated to professor ivan vidav in honor of his eightieth birthday Given a von Neumann algebra R on a Hilbert space H, the so-called R-topology is introduced into B(H), which is weaker than the norm and stronger than the COREultrastrong operator topology. A right R-submodule X of B(H) is closed in the Metadata, citation and similar papers at core.ac.uk Provided byR Elsevier-topology - Publisher if and Connector only if for each b #B(H) the right ideal, consisting of all a # R such that ba # X, is weak* closed in R. Equivalently, X is closed in the R-topology if and only if for each b #B(H) and each orthogonal family of projections ei in R with the sum 1 the condition bei # X for all i implies that b # X. 1998 Academic Press 1. INTRODUCTION Given a C*-algebra R on a Hilbert space H, a concrete operator right R-module is a subspace X of B(H) (the algebra of all bounded linear operators on H) such that XRX. Such modules can be characterized abstractly as L -matricially normed spaces in the sense of Ruan [21], [11] which are equipped with a completely contractive R-module multi- plication (see [6] and [9]). -
On Finite Semifields with a Weak Nucleus and a Designed
On finite semifields with a weak nucleus and a designed automorphism group A. Pi~nera-Nicol´as∗ I.F. R´uay Abstract Finite nonassociative division algebras S (i.e., finite semifields) with a weak nucleus N ⊆ S as defined by D.E. Knuth in [10] (i.e., satisfying the conditions (ab)c − a(bc) = (ac)b − a(cb) = c(ab) − (ca)b = 0, for all a; b 2 N; c 2 S) and a prefixed automorphism group are considered. In particular, a construction of semifields of order 64 with weak nucleus F4 and a cyclic automorphism group C5 is introduced. This construction is extended to obtain sporadic finite semifields of orders 256 and 512. Keywords: Finite Semifield; Projective planes; Automorphism Group AMS classification: 12K10, 51E35 1 Introduction A finite semifield S is a finite nonassociative division algebra. These rings play an important role in the context of finite geometries since they coordinatize projective semifield planes [7]. They also have applications on coding theory [4, 8, 6], combinatorics and graph theory [13]. During the last few years, computational efforts in order to clasify some of these objects have been made. For instance, the classification of semifields with 64 elements is completely known, [16], as well as those with 243 elements, [17]. However, many questions are open yet: there is not a classification of semifields with 128 or 256 elements, despite of the powerful computers available nowadays. The knowledge of the structure of these concrete semifields can inspire new general constructions, as suggested in [9]. In this sense, the work of Lavrauw and Sheekey [11] gives examples of constructions of semifields which are neither twisted fields nor two-dimensional over a nucleus starting from the classification of semifields with 64 elements. -
Modules Over the Steenrod Algebra
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Tcwhcv’ Vol. 10. PD. 211-282. Pergamon Press, 1971. Printed in Great Britain MODULES OVER THE STEENROD ALGEBRA J. F. ADAMS and H. R. MARGOLIS (Received 4 November 1970 ; revised 26 March 197 1) 90. INTRODUCTION IN THIS paper we will investigate certain aspects of the structure of the mod 2 Steenrod algebra, A, and of modules over it. Because of the central role taken by the cohomology groups of spaces considered as A-modules in much of recent algebraic topology (e.g. [ 11, [2], [7]), it is not unreasonable to suppose that the algebraic structure that we elucidate will be of benefit in the study of topological problems. The main result of this paper, Theorem 3.1, gives a criterion for a module !M to be free over the Steenrod algebra, or over some subHopf algebra B of A. To describe this criterion consider a ring R and a left module over R, M; for e E R we define the map e: M -+ M by e(m) = em. If the element e satisfies the condition that e2 = 0 then im e c ker e, therefore we can define the homology group H(M, e) = ker elime. Then, for example, H(R, e) = 0 implies that H(F, e) = 0 for any free R-module F. We can now describe the criterion referred to above: for any subHopf algebra B of A there are elements e,-with ei2 = 0 and H(B, ei) = O-such that a connected B-module M is free over B if and only if H(M, ei) = 0 for all i.