Representations of Quivers
|
|
- Vanessa Lambert
- 5 years ago
- Views:
Transcription
1 MINGLE 2012 Simon Peacock 4th October, 2012
2 Outline 1 Quivers Representations 2 Path Algebra Modules 3 Modules Representations
3 Quiver A quiver, Q, is a directed graph.
4 Quiver A quiver, Q, is a directed graph. Example α ζ ε β γ δ 4 5
5 Representation A representation of a quiver over a field k is an assignment of a k-vector space, α V i, to each vertex i and a k-linear map, f α : V i V j to each edge i j.
6 Representation A representation of a quiver over a field k is an assignment of a k-vector space, α V i, to each vertex i and a k-linear map, f α : V i V j to each edge i j. Example f γ V 4 f α V 1 V 2 V 3 f ζ f ε f β f δ V 5
7 Morphism of representations If V = (V i, f α ) and W = (W i, g α ) are two representations of a quiver Q, then a morphism φ : V W is a set of k-linear maps {φ i : V i W i i a vertex} α such that for each edge i j the square V i f α V j φ i φ j W i g α W j commutes; that is g α φ i = φ j f α.
8 Morphism of representations If V = (V i, f α ) and W = (W i, g α ) are two representations of a quiver Q, then a morphism φ : V W is a set of k-linear maps {φ i : V i W i i a vertex} α such that for each edge i j the square V i f α V j φ i φ j W i g α W j commutes; that is g α φ i = φ j f α. If every φ i is invertible then φ is an isomorphism.
9 Examples Example
10 Examples Example Basic linear algebra
11 Examples Example Basic linear algebra Example
12 Examples Example Basic linear algebra Example Jordan-Normal form
13 Algebra is x sighting A ring, A, is an algebra over a field k, if A also has the structure of a k-vector space. Abelian group under addition, with a (not necessarily commutative) multiplication that distributes over addition. I always assume rings have a unit.
14 Algebra is x sighting A ring, A, is an algebra over a field k, if A also has the structure of a k-vector space. Example C is a 2-dimensional algebra over R. Abelian group under addition, with a (not necessarily commutative) multiplication that distributes over addition. I always assume rings have a unit.
15 Paths A path in a quiver is a sequence of zero or more edges, starting from a given vertex. The product of two path is defined to be the concatenation if this makes sense or zero otherwise.
16 Paths A path in a quiver is a sequence of zero or more edges, starting from a given vertex. The product of two path is defined to be the concatenation if this makes sense or zero otherwise. Example α β The paths are e 1, e 2, e 3, α, β and αβ.
17 Paths A path in a quiver is a sequence of zero or more edges, starting from a given vertex. The product of two path is defined to be the concatenation if this makes sense or zero otherwise. Example α β e 1 α = α The paths are e 1, e 2, e 3, α, β and αβ.
18 Paths A path in a quiver is a sequence of zero or more edges, starting from a given vertex. The product of two path is defined to be the concatenation if this makes sense or zero otherwise. Example α β The paths are e 1, e 2, e 3, α, β and αβ. e 1 α = α α β = αβ
19 Paths A path in a quiver is a sequence of zero or more edges, starting from a given vertex. The product of two path is defined to be the concatenation if this makes sense or zero otherwise. Example α β The paths are e 1, e 2, e 3, α, β and αβ. e 1 α = α α β = αβ α e 3 = 0
20 Path algebra For a quiver Q, the path algebra over a field k, denoted kq, is the vector space with basis all paths. The ring multiplication is then extended linearly from path multiplication.
21 Path algebra For a quiver Q, the path algebra over a field k, denoted kq, is the vector space with basis all paths. The ring multiplication is then extended linearly from path multiplication. Example Q = α
22 Path algebra For a quiver Q, the path algebra over a field k, denoted kq, is the vector space with basis all paths. The ring multiplication is then extended linearly from path multiplication. Example Q = α kq = ke 1 kα kα 2
23 Path algebra For a quiver Q, the path algebra over a field k, denoted kq, is the vector space with basis all paths. The ring multiplication is then extended linearly from path multiplication. Example Q = α kq = ke 1 kα kα 2 = k[x]
24 Path algebra For a quiver Q, the path algebra over a field k, denoted kq, is the vector space with basis all paths. The ring multiplication is then extended linearly from path multiplication. Example Q = α kq = ke 1 kα kα 2 = k[x] e i is the identity element of kq.
25 Module A module over a ring generalises the idea of a vector space over a field. M is a module over a ring R if M is an abelian group, there is a map M R M that is compatible with the ring and group operations.
26 Module A module over a ring generalises the idea of a vector space over a field. M is a module over a ring R if M is an abelian group, there is a map M R M that is compatible with the ring and group operations. Example Z n is a module over Z with the obvious action: (a 0,..., a n )x (a 0 x,..., a n x)
27 Modules Representations Modules of kq are representations of Q over k.
28 Modules Representations Modules of kq are representations of Q over k. Given a module M:
29 Modules Representations Modules of kq are representations of Q over k. Given a module M: For a vertex i, define V i = Me i.
30 Modules Representations Modules of kq are representations of Q over k. Given a module M: For a vertex i, define V i = Me i. α For an edge i j, define f α to be multiplication by α me i me i α = mαe j Me j
31 Modules Representations Modules of kq are representations of Q over k. Given a module M: For a vertex i, define V i = Me i. α For an edge i j, define f α to be multiplication by α me i me i α = mαe j Me j Example M α Me 1 Me 2 Me 3 ζ β γ ε δ Me 4 Me 5
32 Modules Representations Representations of Q over k are modules of kq.
33 Modules Representations Representations of Q over k are modules of kq. Given a representation (V i, f α ):
34 Modules Representations Representations of Q over k are modules of kq. Given a representation (V i, f α ): M = V i as an abelian group.
35 Modules Representations Representations of Q over k are modules of kq. Given a representation (V i, f α ): M = V i as an abelian group. M kq M is extended from the action
36 Modules Representations Representations of Q over k are modules of kq. Given a representation (V i, f α ): M = V i as an abelian group. M kq M is extended from the action v i e j { vi if i = j 0 otherwise for v i V i
37 Modules Representations Representations of Q over k are modules of kq. Given a representation (V i, f α ): M = V i as an abelian group. M kq M is extended from the action v i e j v i α { vi if i = j 0 otherwise { α fα (v i ) if i 0 otherwise for v i V i
38 Modules Representations These associations are inverses of one another. M (Me i, α)
39 Modules Representations These associations are inverses of one another. M (Me i, α) Me i
40 Modules Representations These associations are inverses of one another. M (Me i, α) Me i = M e i
41 Modules Representations These associations are inverses of one another. M (Me i, α) Me i = M e i = M
42 Modules Representations These associations are inverses of one another. M (Me i, α) Me i = M e i = M (V i, f α ) V i
43 Modules Representations These associations are inverses of one another. M (Me i, α) Me i = M e i = M (V i, f α ) V i ( ) ( ) jv j ei, f α i,α
44 Modules Representations These associations are inverses of one another. M (Me i, α) Me i = M e i = M (V i, f α ) V i ( ) ( ) jv j ei, f α = (V i, f α ) i,α
45 Yay MINGLE!
Representations of quivers
Representations of quivers Gwyn Bellamy October 13, 215 1 Quivers Let k be a field. Recall that a k-algebra is a k-vector space A with a bilinear map A A A making A into a unital, associative ring. Notice
More informationA NOTE ON GENERALIZED PATH ALGEBRAS
A NOTE ON GENERALIZED PATH ALGEBRAS ROSA M. IBÁÑEZ COBOS, GABRIEL NAVARRO and JAVIER LÓPEZ PEÑA We develop the theory of generalized path algebras as defined by Coelho and Xiu [4]. In particular, we focus
More information22M: 121 Final Exam. Answer any three in this section. Each question is worth 10 points.
22M: 121 Final Exam This is 2 hour exam. Begin each question on a new sheet of paper. All notations are standard and the ones used in class. Please write clearly and provide all details of your work. Good
More information4.1. Paths. For definitions see section 2.1 (In particular: path; head, tail, length of a path; concatenation;
4 The path algebra of a quiver 41 Paths For definitions see section 21 (In particular: path; head, tail, length of a path; concatenation; oriented cycle) Lemma Let Q be a quiver If there is a path of length
More information2.1 Modules and Module Homomorphisms
2.1 Modules and Module Homomorphisms The notion of a module arises out of attempts to do classical linear algebra (vector spaces over fields) using arbitrary rings of coefficients. Let A be a ring. An
More informationAuslander-Reiten-quivers of functorially finite subcategories
Auslander-Reiten-quivers of functorially finite subcategories Matthias Krebs University of East Anglia August 13, 2012 Preliminaries Preliminaries Let K be an algebraically closed field, Preliminaries
More informationFrobenius-Perron Theory of Endofunctors
March 17th, 2018 Recent Developments in Noncommutative Algebra and Related Areas, Seattle, WA Throughout let k be an algebraically closed field. Throughout let k be an algebraically closed field. The Frobenius-Perron
More informationMutation classes of quivers with constant number of arrows and derived equivalences
Mutation classes of quivers with constant number of arrows and derived equivalences Sefi Ladkani University of Bonn http://www.math.uni-bonn.de/people/sefil/ 1 Motivation The BGP reflection is an operation
More informationExamples of Semi-Invariants of Quivers
Examples of Semi-Invariants of Quivers June, 00 K is an algebraically closed field. Types of Quivers Quivers with finitely many isomorphism classes of indecomposable representations are of finite representation
More informationA visual introduction to Tilting
A visual introduction to Tilting Jorge Vitória University of Verona http://profs.sci.univr.it/ jvitoria/ Padova, May 21, 2014 Jorge Vitória (University of Verona) A visual introduction to Tilting Padova,
More informationNoetherian property of infinite EI categories
Noetherian property of infinite EI categories Wee Liang Gan and Liping Li Abstract. It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result
More informationFundamental theorem of modules over a PID and applications
Fundamental theorem of modules over a PID and applications Travis Schedler, WOMP 2007 September 11, 2007 01 The fundamental theorem of modules over PIDs A PID (Principal Ideal Domain) is an integral domain
More informationThe preprojective algebra revisited
The preprojective algebra revisited Helmut Lenzing Universität Paderborn Auslander Conference Woodshole 2015 H. Lenzing Preprojective algebra 1 / 1 Aim of the talk Aim of the talk My talk is going to review
More informationGeneralized Matrix Artin Algebras. International Conference, Woods Hole, Ma. April 2011
Generalized Matrix Artin Algebras Edward L. Green Department of Mathematics Virginia Tech Blacksburg, VA, USA Chrysostomos Psaroudakis Department of Mathematics University of Ioannina Ioannina, Greece
More informationThe origin of pictures: near rings and K 3 Z
Brandeis University March 21, 2018 Disclaimer I did not invent pictures. They existed before I was born. were sometimes called Peiffer diagrams. They were used in combinatorial group theory. This talk
More informationMulti-parameter persistent homology: applications and algorithms
Multi-parameter persistent homology: applications and algorithms Nina Otter Mathematical Institute, University of Oxford Gudhi/Top Data Workshop Porquerolles, 18 October 2016 Multi-parameter persistent
More informationConnected Sums of Simplicial Complexes
Connected Sums of Simplicial Complexes Tomoo Matsumura 1 W. Frank Moore 2 1 KAIST 2 Department of Mathematics Wake Forest University October 15, 2011 Let R,S,T be commutative rings, and let V be a T -module.
More informationTHE CLASSIFICATION PROBLEM REPRESENTATION-FINITE, TAME AND WILD QUIVERS
October 28, 2009 THE CLASSIFICATION PROBLEM REPRESENTATION-FINITE, TAME AND WILD QUIVERS Sabrina Gross, Robert Schaefer In the first part of this week s session of the seminar on Cluster Algebras by Prof.
More informationLECTURE 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 information1. Quivers and their representations: Basic definitions and examples.
1 Quivers and their representations: Basic definitions and examples 11 Quivers A quiver Q (sometimes also called a directed graph) consists of vertices and oriented edges (arrows): loops and multiple arrows
More information(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d
The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers
More informationQuiver Representations
Quiver Representations Molly Logue August 28, 2012 Abstract After giving a general introduction and overview to the subject of Quivers and Quiver Representations, we will explore the counting and classification
More informationThe ring of global sections of a differential scheme
The ring of global sections of a differential scheme Dmitry Trushin The Einstein Institute of Mathematics The Hebrew University of Jerusalem January 2012 Dmitry Trushin () The ring of global sections January,
More informationIndecomposable Quiver Representations
Indecomposable Quiver Representations Summer Project 2015 Laura Vetter September 2, 2016 Introduction The aim of my summer project was to gain some familiarity with the representation theory of finite-dimensional
More informationThe Structure of AS-regular Algebras
Department of Mathematics, Shizuoka University Shanghai Workshop 2011, 9/12 Noncommutative algebraic geometry Classify noncommutative projective schemes Classify finitely generated graded algebras Classify
More informationQUALIFYING EXAM IN ALGEBRA August 2011
QUALIFYING EXAM IN ALGEBRA August 2011 1. There are 18 problems on the exam. Work and turn in 10 problems, in the following categories. I. Linear Algebra 1 problem II. Group Theory 3 problems III. Ring
More informationREPRESENTATION THEORY. WEEKS 10 11
REPRESENTATION THEORY. WEEKS 10 11 1. Representations of quivers I follow here Crawley-Boevey lectures trying to give more details concerning extensions and exact sequences. A quiver is an oriented graph.
More informationQUIVERS AND LATTICES.
QUIVERS AND LATTICES. KEVIN MCGERTY We will discuss two classification results in quite different areas which turn out to have the same answer. This note is an slightly expanded version of the talk given
More informationProfinite Groups. Hendrik Lenstra. 1. Introduction
Profinite Groups Hendrik Lenstra 1. Introduction We begin informally with a motivation, relating profinite groups to the p-adic numbers. Let p be a prime number, and let Z p denote the ring of p-adic integers,
More informationHilbert function, Betti numbers. Daniel Gromada
Hilbert function, Betti numbers 1 Daniel Gromada References 2 David Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry 19, 110 David Eisenbud: The Geometry of Syzygies 1A, 1B My own notes
More informationRepresentations of quivers
Representations of quivers Michel Brion Lectures given at the summer school Geometric methods in representation theory (Grenoble, June 16 July 4, 2008) Introduction Quivers are very simple mathematical
More informationRelative Affine Schemes
Relative Affine Schemes Daniel Murfet October 5, 2006 The fundamental Spec( ) construction associates an affine scheme to any ring. In this note we study the relative version of this construction, which
More informationAlgebra Qualifying Exam August 2001 Do all 5 problems. 1. Let G be afinite group of order 504 = 23 32 7. a. Show that G cannot be isomorphic to a subgroup of the alternating group Alt 7. (5 points) b.
More informationMatsumura: Commutative Algebra Part 2
Matsumura: Commutative Algebra Part 2 Daniel Murfet October 5, 2006 This note closely follows Matsumura s book [Mat80] on commutative algebra. Proofs are the ones given there, sometimes with slightly more
More informationLECTURE 16: REPRESENTATIONS OF QUIVERS
LECTURE 6: REPRESENTATIONS OF QUIVERS IVAN LOSEV Introduction Now we proceed to study representations of quivers. We start by recalling some basic definitions and constructions such as the path algebra
More informationConstructions of Derived Equivalences of Finite Posets
Constructions of Derived Equivalences of Finite Posets Sefi Ladkani Einstein Institute of Mathematics The Hebrew University of Jerusalem http://www.ma.huji.ac.il/~sefil/ 1 Notions X Poset (finite partially
More informationABSOLUTELY PURE REPRESENTATIONS OF QUIVERS
J. Korean Math. Soc. 51 (2014), No. 6, pp. 1177 1187 http://dx.doi.org/10.4134/jkms.2014.51.6.1177 ABSOLUTELY PURE REPRESENTATIONS OF QUIVERS Mansour Aghasi and Hamidreza Nemati Abstract. In the current
More informationFUNCTORS AND ADJUNCTIONS. 1. Functors
FUNCTORS AND ADJUNCTIONS Abstract. Graphs, quivers, natural transformations, adjunctions, Galois connections, Galois theory. 1.1. Graph maps. 1. Functors 1.1.1. Quivers. Quivers generalize directed graphs,
More informationThus, the integral closure A i of A in F i is a finitely generated (and torsion-free) A-module. It is not a priori clear if the A i s are locally
Math 248A. Discriminants and étale algebras Let A be a noetherian domain with fraction field F. Let B be an A-algebra that is finitely generated and torsion-free as an A-module with B also locally free
More informationarxiv: v1 [math.ra] 16 Dec 2014
arxiv:1412.5219v1 [math.ra] 16 Dec 2014 CATEGORY EQUIVALENCES INVOLVING GRADED MODULES OVER QUOTIENTS OF WEIGHTED PATH ALGEBRAS CODY HOLDAWAY Abstract. Let k be a field, Q a finite directed graph, and
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to
More informationCombinatorial aspects of derived equivalence
Combinatorial aspects of derived equivalence Sefi Ladkani University of Bonn http://guests.mpim-bonn.mpg.de/sefil/ 1 What is the connection between... 2 The finite dimensional algebras arising from these
More informationCOMPARISON MORPHISMS BETWEEN TWO PROJECTIVE RESOLUTIONS OF MONOMIAL ALGEBRAS
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 59, No. 1, 2018, Pages 1 31 Published online: July 7, 2017 COMPARISON MORPHISMS BETWEEN TWO PROJECTIVE RESOLUTIONS OF MONOMIAL ALGEBRAS Abstract. We construct
More informationSection Higher Direct Images of Sheaves
Section 3.8 - Higher Direct Images of Sheaves Daniel Murfet October 5, 2006 In this note we study the higher direct image functors R i f ( ) and the higher coinverse image functors R i f! ( ) which will
More informationPart II Galois Theory
Part II Galois Theory Theorems Based on lectures by C. Birkar Notes taken by Dexter Chua Michaelmas 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after
More informationNoncommutative compact manifolds constructed from quivers
Noncommutative compact manifolds constructed from quivers Lieven Le Bruyn Universitaire Instelling Antwerpen B-2610 Antwerp (Belgium) lebruyn@wins.uia.ac.be Abstract The moduli spaces of θ-semistable representations
More informationCATEGORY THEORY. Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths.
CATEGORY THEORY PROFESSOR PETER JOHNSTONE Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths. Definition 1.1. A category C consists
More informationQUIVERS, REPRESENTATIONS, ROOTS AND LIE ALGEBRAS. Volodymyr Mazorchuk. (Uppsala University)
QUIVERS, REPRESENTATIONS, ROOTS AND LIE ALGEBRAS Volodymyr Mazorchuk (Uppsala University) . QUIVERS Definition: A quiver is a quadruple Q = (V, A, t, h), where V is a non-empty set; A is a set; t and h
More informationCategories of noncrossing partitions
Categories of noncrossing partitions Kiyoshi Igusa, Brandeis University KIAS, Dec 15, 214 Topology of categories The classifying space of a small category C is a union of simplices k : BC = X X 1 X k k
More informationGrothendieck duality for affine M 0 -schemes.
Grothendieck duality for affine M 0 -schemes. A. Salch March 2011 Outline Classical Grothendieck duality. M 0 -schemes. Derived categories without an abelian category of modules. Computing Lf and Rf and
More informationQuiver Representations and Gabriel s Theorem
Quiver Representations and Gabriel s Theorem Kristin Webster May 16, 2005 1 Introduction This talk is based on the 1973 article by Berstein, Gelfand and Ponomarev entitled Coxeter Functors and Gabriel
More informationClassification of semisimple Lie algebras
Chapter 6 Classification of semisimple Lie algebras When we studied sl 2 (C), we discovered that it is spanned by elements e, f and h fulfilling the relations: [e, h] = 2e, [ f, h] = 2 f and [e, f ] =
More informationAUSLANDER-REITEN THEORY FOR FINITE DIMENSIONAL ALGEBRAS. Piotr Malicki
AUSLANDER-REITEN THEORY FOR FINITE DIMENSIONAL ALGEBRAS Piotr Malicki CIMPA, Mar del Plata, March 2016 3. Irreducible morphisms and almost split sequences A algebra, L, M, N modules in mod A A homomorphism
More informationA Categorification of Hall Algebras
A Categorification of Christopher D. Walker joint with John Baez Department of Mathematics University of California, Riverside November 7, 2009 A Categorification of Groupoidification There is a systematic
More informationModules of the Highest Homological Dimension over a Gorenstein Ring
Modules of the Highest Homological Dimension over a Gorenstein Ring Yasuo Iwanaga and Jun-ichi Miyachi Dedicated to Professor Kent R. Fuller on his 60th birthday We will study modules of the highest injective,
More informationDimension of the mesh algebra of a finite Auslander-Reiten quiver. Ragnar-Olaf Buchweitz and Shiping Liu
Dimension of the mesh algebra of a finite Auslander-Reiten quiver Ragnar-Olaf Buchweitz and Shiping Liu Abstract. We show that the dimension of the mesh algebra of a finite Auslander-Reiten quiver over
More informationOn Vector Product Algebras
On Vector Product Algebras By Markus Rost. This text contains some remarks on vector product algebras and the graphical techniques. It is partially contained in the diploma thesis of D. Boos and S. Maurer.
More informationThe Proj Construction
The Proj Construction Daniel Murfet May 16, 2006 Contents 1 Basic Properties 1 2 Functorial Properties 2 3 Products 6 4 Linear Morphisms 9 5 Projective Morphisms 9 6 Dimensions of Schemes 11 7 Points of
More informationAlgebraic varieties and schemes over any scheme. Non singular varieties
Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two
More informationAlgebraic Geometry: MIDTERM SOLUTIONS
Algebraic Geometry: MIDTERM SOLUTIONS C.P. Anil Kumar Abstract. Algebraic Geometry: MIDTERM 6 th March 2013. We give terse solutions to this Midterm Exam. 1. Problem 1: Problem 1 (Geometry 1). When is
More informationAlgebra Exam Fall Alexander J. Wertheim Last Updated: October 26, Groups Problem Problem Problem 3...
Algebra Exam Fall 2006 Alexander J. Wertheim Last Updated: October 26, 2017 Contents 1 Groups 2 1.1 Problem 1..................................... 2 1.2 Problem 2..................................... 2
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #7 09/26/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #7 09/26/2013 In Lecture 6 we proved (most of) Ostrowski s theorem for number fields, and we saw the product formula for absolute values on
More informationSynopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].
Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.
More informationPatterns and Invariants in Mathematics
Patterns and in Mathematics Professor Nicole Snashall 24th June 2017 1 / 12 Outline 1 2 3 4 2 / 12 Pinecones How many spirals are there in each direction on the pinecone? 3 / 12 Pinecones How many spirals
More informationThe Segre Embedding. Daniel Murfet May 16, 2006
The Segre Embedding Daniel Murfet May 16, 2006 Throughout this note all rings are commutative, and A is a fixed ring. If S, T are graded A-algebras then the tensor product S A T becomes a graded A-algebra
More informationWIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES
WIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels,
More informationSEPARABILITY II KEITH CONRAD
SEPARABILITY II KEITH CONRAD 1. Introduction Separability of a finite field extension L/K can be described in several different ways. The original definition is that every element of L is separable over
More informationOn the Homology of the Ginzburg Algebra
On the Homology of the Ginzburg Algebra Stephen Hermes Brandeis University, Waltham, MA Maurice Auslander Distinguished Lectures and International Conference Woodshole, MA April 23, 2013 Stephen Hermes
More informationA proof of the Graph Semigroup Group Test in The Graph Menagerie
A proof of the Graph Semigroup Group Test in The Graph Menagerie by Gene Abrams and Jessica K. Sklar We present a proof of the Graph Semigroup Group Test. This Test plays a key role in the article The
More informationA Primer on Homological Algebra
A Primer on Homological Algebra Henry Y Chan July 12, 213 1 Modules For people who have taken the algebra sequence, you can pretty much skip the first section Before telling you what a module is, you probably
More informationFun with Dyer-Lashof operations
Nordic Topology Meeting, Stockholm (27th-28th August 2015) based on arxiv:1309.2323 last updated 27/08/2015 Power operations and coactions Recall the extended power construction for n 1: D n X = EΣ n Σn
More informationAN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES
AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES DOMINIC L. WYNTER Abstract. We introduce the concepts of divisors on nonsingular irreducible projective algebraic curves, the genus of such a curve,
More informationWhat is the Langlands program all about?
What is the Langlands program all about? Laurent Lafforgue November 13, 2013 Hua Loo-Keng Distinguished Lecture Academy of Mathematics and Systems Science, Chinese Academy of Sciences This talk is mainly
More informationIntroduction to abstract algebra: definitions, examples, and exercises
Introduction to abstract algebra: definitions, examples, and exercises Travis Schedler January 21, 2015 1 Definitions and some exercises Definition 1. A binary operation on a set X is a map X X X, (x,
More informationNormal forms in combinatorial algebra
Alberto Gioia Normal forms in combinatorial algebra Master s thesis, defended on July 8, 2009 Thesis advisor: Hendrik Lenstra Mathematisch Instituut Universiteit Leiden ii Contents Introduction iv 1 Generators
More informationMobius Inversion on Partially Ordered Sets
Mobius Inversion on Partially Ordered Sets 1 Introduction The theory of Möbius inversion gives us a unified way to look at many different results in combinatorics that involve inverting the relation between
More informationand this makes M into an R-module by (1.2). 2
1. Modules Definition 1.1. Let R be a commutative ring. A module over R is set M together with a binary operation, denoted +, which makes M into an abelian group, with 0 as the identity element, together
More informationGALOIS THEORY. Contents
GALOIS THEORY MARIUS VAN DER PUT & JAAP TOP Contents 1. Basic definitions 1 1.1. Exercises 2 2. Solving polynomial equations 2 2.1. Exercises 4 3. Galois extensions and examples 4 3.1. Exercises. 6 4.
More information1 Categorical Background
1 Categorical Background 1.1 Categories and Functors Definition 1.1.1 A category C is given by a class of objects, often denoted by ob C, and for any two objects A, B of C a proper set of morphisms C(A,
More informationNOTES ON BASIC HOMOLOGICAL ALGEBRA 0 L M N 0
NOTES ON BASIC HOMOLOGICAL ALGEBRA ANDREW BAKER 1. Chain complexes and their homology Let R be a ring and Mod R the category of right R-modules; a very similar discussion can be had for the category of
More informationWe can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a pre-specified embedding in a projective space. However, an ample line bundle
More informationPULLBACK MODULI SPACES
PULLBACK MODULI SPACES FRAUKE M. BLEHER AND TED CHINBURG Abstract. Geometric invariant theory can be used to construct moduli spaces associated to representations of finite dimensional algebras. One difficulty
More informationRAPHAËL ROUQUIER. k( )
GLUING p-permutation MODULES 1. Introduction We give a local construction of the stable category of p-permutation modules : a p- permutation kg-module gives rise, via the Brauer functor, to a family of
More informationJune 2014 Written Certification Exam. Algebra
June 2014 Written Certification Exam Algebra 1. Let R be a commutative ring. An R-module P is projective if for all R-module homomorphisms v : M N and f : P N with v surjective, there exists an R-module
More informationCLASSIFYING TANGENT STRUCTURES USING WEIL ALGEBRAS
Theory and Applications of Categories, Vol. 32, No. 9, 2017, pp. 286 337. CLASSIFYING TANGENT STRUCTURES USING WEIL ALGEBRAS POON LEUNG Abstract. At the heart of differential geometry is the construction
More informationAlgebraic Structures Exam File Fall 2013 Exam #1
Algebraic Structures Exam File Fall 2013 Exam #1 1.) Find all four solutions to the equation x 4 + 16 = 0. Give your answers as complex numbers in standard form, a + bi. 2.) Do the following. a.) Write
More informationHomotopical versions of Hall algebras
University of California, Riverside January 7, 2009 Classical Hall algebras Let A be an abelian category such that, for any objects X and Y of A, Ext 1 (X, Y ) is finite. Then, we can associate to A an
More informationExercises on chapter 0
Exercises on chapter 0 1. A partially ordered set (poset) is a set X together with a relation such that (a) x x for all x X; (b) x y and y x implies that x = y for all x, y X; (c) x y and y z implies that
More informationKiddie Talk - The Diamond Lemma and its applications
Kiddie Tal - The Diamond Lemma and its applications April 22, 2013 1 Intro Start with an example: consider the following game, a solitaire. Tae a finite graph G with n vertices and a function e : V (G)
More information0.2 Vector spaces. J.A.Beachy 1
J.A.Beachy 1 0.2 Vector spaces I m going to begin this section at a rather basic level, giving the definitions of a field and of a vector space in much that same detail as you would have met them in a
More information18.312: Algebraic Combinatorics Lionel Levine. Lecture 22. Smith normal form of an integer matrix (linear algebra over Z).
18.312: Algebraic Combinatorics Lionel Levine Lecture date: May 3, 2011 Lecture 22 Notes by: Lou Odette This lecture: Smith normal form of an integer matrix (linear algebra over Z). 1 Review of Abelian
More informationUniversal Properties
A categorical look at undergraduate algebra and topology Julia Goedecke Newnham College 24 February 2017, Archimedeans Julia Goedecke (Newnham) 24/02/2017 1 / 30 1 Maths is Abstraction : more abstraction
More information6 Ideal norms and the Dedekind-Kummer theorem
18.785 Number theory I Fall 2016 Lecture #6 09/27/2016 6 Ideal norms and the Dedekind-Kummer theorem Recall that for a ring extension B/A in which B is a free A-module of finite rank, we defined the (relative)
More information7350: TOPICS IN FINITE-DIMENSIONAL ALGEBRAS. 1. Lecture 1
7350: TOPICS IN FINITE-DIMENSIONAL ALGEBRAS RICHARD VALE Abstract. These are the notes from a course taught at Cornell in Spring 2009. They are a record of what was covered in the lectures. Many results
More informationPublic-key Cryptography: Theory and Practice
Public-key Cryptography Theory and Practice Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Chapter 2: Mathematical Concepts Divisibility Congruence Quadratic Residues
More informationThe real root modules for some quivers.
SS 2006 Selected Topics CMR The real root modules for some quivers Claus Michael Ringel Let Q be a finite quiver with veretx set I and let Λ = kq be its path algebra The quivers we are interested in will
More informationCluster varieties for tree-shaped quivers and their cohomology
Cluster varieties for tree-shaped quivers and their cohomology Frédéric Chapoton CNRS & Université de Strasbourg Octobre 2016 Cluster algebras and the associated varieties Cluster algebras are commutative
More informationGroup Theory. 1. Show that Φ maps a conjugacy class of G into a conjugacy class of G.
Group Theory Jan 2012 #6 Prove that if G is a nonabelian group, then G/Z(G) is not cyclic. Aug 2011 #9 (Jan 2010 #5) Prove that any group of order p 2 is an abelian group. Jan 2012 #7 G is nonabelian nite
More informationOutline of the Seminar Topics on elliptic curves Saarbrücken,
Outline of the Seminar Topics on elliptic curves Saarbrücken, 11.09.2017 Contents A Number theory and algebraic geometry 2 B Elliptic curves 2 1 Rational points on elliptic curves (Mordell s Theorem) 5
More informationR S. with the property that for every s S, φ(s) is a unit in R S, which is universal amongst all such rings. That is given any morphism
8. Nullstellensatz We will need the notion of localisation, which is a straightforward generalisation of the notion of the field of fractions. Definition 8.1. Let R be a ring. We say that a subset S of
More information