The Structure of AS-regular Algebras
|
|
- Beatrix Crawford
- 5 years ago
- Views:
Transcription
1 Department of Mathematics, Shizuoka University Shanghai Workshop 2011, 9/12
2 Noncommutative algebraic geometry Classify noncommutative projective schemes Classify finitely generated graded algebras Classify quantum projective spaces Classify AS-regular algebras
3 For simplicity, we assume that k = k, and A is a graded right coherent algebra over k. gr A = the abelian category of finitely presented graded right A-modules. tors A = the full subcategory of finite dimensional modules. Definition (Artin-Zhang) The noncommutative projective scheme associated to A is defined by tails A := gr A/ tors A.
4 AS-regular algebras Definition (Artin-Schelter) An N-graded algebra A is AS-regular of dimension d and of Gorenstein parameter l if A 0 = k (connected graded), gldim A = d, and Ext i A (k, A) 0 if i d = k(l) if i = d. A quantum projective space is a noncommutative projective scheme associated to an AS-regular algebra.
5 Theorem (Zhang) Every AS-regular algebra of dimension 2 and of Gorenstein parameter l is isomorphic to where n 2, k x 1,..., x n /( deg x 1 deg x n, n x i σ(x n+1 i )) i=1 deg x i + deg x n+1 i = l for all i, and σ Aut k k x 1,..., x n.
6 Theorem (Artin-Tate-Van den Bergh) Quadratic AS-regular algebras of dimension 3 and of finite GKdimension were classified by geometric triples (E, σ, L) where E P 2, σ Aut k E, and L Pic E.
7 Representation theory Classify finite dimensional algebras Classify finite dimensional algebras of finite global dimensions Classify Fano algebras
8 Theorem (Gabriel) Every finite dimensional algebra of global dimension 1 is Morita equivalent to a path algebra of a finite acyclic quiver. Example ( ) Q = 1 α 2 kq ke 1 kα + kβ = β 0 ke 2 ke 1 kα k(αβ) Q = 1 α 2 β 3 kq = 0 ke 2 kβ 0 0 ke 3
9 The double Q of a quiver Q is defined by Q 0 = Q 0 Q 1 = {α : i j, α : j i α Q 1 }. The preprojective algebra of Q is defined by ΠQ := kq/( α Q 1 αα α α). Example Q = 1 α 2 Q = 1 β α β α ΠQ = kq/(αα + ββ, α α + β β). β 2
10 Fano algebras Let R be a finite dimensional algebra. D := D b (mod R) has a standard t-structure D 0 := {M D h i (M) = 0 for all i < 0} D 0 := {M D h i (M) = 0 for all i > 0}. For s Aut k D, we define D s, 0 := {M D s i (M) D 0 for all i 0} D s, 0 := {M D s i (M) D 0 for all i 0}.
11 Definition (Minamoto) s Aut k D is ample if s i (R) D 0 D 0 = mod R for all i 0, and (D s, 0, D s, 0 ) is a t-structure for D. Theorem (Minamoto) If s Aut k D is ample, then (R, s) is ample for H := D s, 0 D s, 0 in the sense of Artin-Zhang.
12 Definition (Minamoto) An algebra R is Fano of dimension d if gldim R = d, and L R ω 1 R Aut k D is ample where DR := Hom k (R, k) and ω R := DR[ d]. The preprojective algebra of a Fano algebra R is defined by ΠR := T R (ω 1 R ).
13 Example R is a Fano algebras of dimension 0 R is a semi-simple algebra In this case, ΠR = R[x] Example R is a basic Fano algebras of dimension 1 R = kq where Q is a finite acyclic non-dynkin quiver. In this case, ΠR = ΠQ.
14 Definition For a graded algebra A = i Z A i and r N +, we define the r-th quasi-veronese algebra of A by A ri A ri+1 A ri+r 1 A [r] := A ri 1 A ri A ri+r 2. i Z A ri r+1 A ri r+2 A ri
15 Definition The Beilinson algebra of an AS-regular algebra A of Gorenstein parameter l is defined by Lemma A := (A [l] ) 0 For any graded algebra A and r N +, gr A [r] = gr A. Lemma For any algebra R, R-R bimodule M and σ Aut k R, gr T R (M σ ) = gr T R (M).
16 Theorem (Minamoto-Mori) If A is an AS-regular algebra of dimension d 1, then S := A is a Fano algebra of dimension d 1. A [l] = T S ((ω 1 S ) σ) for some σ Aut k S. gr A = gr A [l] = gr T S ((ω 1 S ) σ) = gr ΠS. D b (tails A) = D b (tails ΠS) = D b (mod S). Example (Beilinson) Applying to A = k[x 1,..., x n ], deg x i = 1, D b (coh P n 1 ) = D b (tails A) = D b (mod A).
17 Theorem (Minamoto-Mori) Let A, B be AS-regular algebras. 1 The following are equivalent:. gr A = gr B. A = B. Π( A) = Π( B). gr Π( A) = gr Π( B). 2 The following are equivalent: D b (tails A) = D b (tails B). D b (mod A) = D b (mod B).
18 Example A = k[x, y], deg x = 1, deg y = 3 A is an AS-regular algebra of dimension 2 A = kq is a Fano algebra of dimension 1 Q = (extended Dynkin) Q ( is a reduced ) McKay quiver of ξ 0 SL(2, k) where ξ k is a primitive 4-th 0 ξ 3 root of unity.
19 Example A = k x, y, z /(xz + y 2 + zx) deg x = 1, deg y = 2, deg z = 3 A is an AS-regular algebra of dimension 2 A = kq is a Fano algebra of dimension 1 Q = (not extended Dynkin) Q is a reduced McKay quiver of ξ ξ 2 0 GL(3, k) where ξ k is a primitive 0 0 ξ 3 4-th root of unity.
20 Example A = k x, y /(x 2 y yx 2, xy 2 y 2 x), deg x = deg y = 1 A is an AS-regular algebra of dimension 3 A = kq/i is a Fano algebra of dimension 2 Q = ( ) ξ 0 Q is a reduced McKay quiver of GL(2, k) 0 ξ where ξ k is a primitive 4-th root of unity.
21 AS-regular algebras (of dimension 2) can be classified by (reduced) McKay quivers of a finite cyclic subgroups of GL(n, k) up to graded Morita equivalence.
22 Generalizations Definition (Minamoto-Mori) A graded algebra A is AS-regular over R of dimension d and of Gorenstein parameter l if A 0 = R, gldim R <, gldim A = d, and Ext i A (R, A) 0 if i d = (DR)(l) if i = d. An AS-regular algebra A is symmetric if ω A := D H d m (A) = A( l) as graded A-A bimodules.
23 Theorem (Minamoto-Mori) If A is an AS-regular algebra over R of dimension d 1, then S := A is a Fano algebra of dimension d 1. A [l] = T S ((ω 1 S ) σ) for some σ Aut k S. gr A = gr ΠS. D b (tails A) = D b (mod S). Theorem (Minamoto-Mori) A is a preprojective algebras of Fano algebras of dimension d A is a symmetric AS-regular algebras of dimension d + 1 and of Gorenstein parameter 1.
24 {AS-regular algebras over R of dimension d} Π {Fano algebras of dimension d 1} gr Π( A) = gr A (ΠS) = S Classifying AS-regular algebras over R of dimension d 1 up to graded Morita equivalence Classifying Fano algebras of dimension d 1 up to isomorphism.
25 Graded Frobenius Algebras Definition A finite dimensional graded algebra A is graded Frobenius of Gorenstein parameter l if DA = A(l) as graded A-modules. It is graded symmetric if DA = A(l) as graded A-A bimodules. Example The trivial extension of R is defined by R := R DR = T R (DR)/T R (DR) 2.
26 Theorem (Minamoto-Mori) A is a trivial extensions of finite dimensional algebras A is a graded symmetric algebras of Gorenstein parameter 1. Definition The Beilinson algebra of a graded Frobenius algebra A of Gorenstein parameter l is defined by A := (A [l] ) 0.
27 {graded Frobenius algebras} {finite dimensional algebras} gr ( A) = gr A ( S) = S Classifying graded Frobenius algebras up to graded Morita equivalence Classifying finite dimensional algebras up to isomorphism.
Ample group actions on AS-regular algebras. noncommutative graded isolated singularities. Kenta Ueyama
and noncommutative graded isolated singularities Shizuoka Universit, Japan Perspectives of Representation Theor of Algebra Nagoa Universit November 15th 2013 AS-regular, AS-Gorenstein, noncomm graded isolated
More informationStructures of AS-regular Algebras
Structures of AS-regular Algebras Hiroyuki Minamoto and Izuru Mori Abstract In this paper, we define a notion of AS-Gorenstein algebra for N-graded algebras, and show that symmetric AS-regular algebras
More informationON GRADED MORITA EQUIVALENCES FOR AS-REGULAR ALGEBRAS KENTA UEYAMA
ON GRADED MORITA EQUIVALENCES FOR AS-REGULAR ALGEBRAS KENTA UEYAMA Abstract. One of the most active projects in noncommutative algebraic geometry is to classify AS-regular algebras. The motivation of this
More informationGraded maximal Cohen-Macaulay modules over. Noncommutative graded Gorenstein isolated singularities. Kenta Ueyama. ICRA XV, Bielefeld, August 2012
Graded maximal Cohen-Macaulay modules over noncommutative graded Gorenstein isolated singularities Shizuoka University, Japan ICRA XV, Bielefeld, August 2012 Notations Throughout this talk, k : an algebraically
More informationZ-graded noncommutative projective geometry Algebra Seminar
Z-graded noncommutative projective geometry Algebra Seminar Robert Won University of California, San Diego November 9, 2015 1 / 43 Overview 1 Preliminaries Pre-talk catchup Noncommutative things 2 Noncommutative
More informationNONCOMMUTATIVE GRADED GORENSTEIN ISOLATED SINGULARITIES
NONCOMMUTATIVE GRADED GORENSTEIN ISOLATED SINGULARITIES KENTA UEYAMA Abstract. Gorenstein isolated singularities play an essential role in representation theory of Cohen-Macaulay modules. In this article,
More informationNoncommutative invariant theory and Auslander s Theorem
Noncommutative invariant theory and Auslander s Theorem Miami University Algebra Seminar Robert Won Wake Forest University Joint with Jason Gaddis, Ellen Kirkman, and Frank Moore arxiv:1707.02822 November
More informationThe graded module category of a generalized Weyl algebra
The graded module category of a generalized Weyl algebra Final Defense Robert Won Advised by: Daniel Rogalski May 2, 2016 1 / 39 Overview 1 Graded rings and things 2 Noncommutative is not commutative 3
More informationWhat is noncommutative algebraic geometry?
What is noncommutative algebraic geometry? Robert Won University of California, San Diego Graduate Algebraic Geometry Seminar, August 2015 August 14, 2015 1 / 20 Overview In the great tradition of algebra,
More informationGraded Calabi-Yau Algebras actions and PBW deformations
Actions on Graded Calabi-Yau Algebras actions and PBW deformations Q. -S. Wu Joint with L. -Y. Liu and C. Zhu School of Mathematical Sciences, Fudan University International Conference at SJTU, Shanghai
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 informationAuslander s Theorem for permutation actions on noncommutative algebras
Auslander s Theorem for permutation actions on noncommutative algebras (arxiv:1705.00068) Jason Gaddis Miami University Joint with Ellen Kirkman, W. Frank Moore, Robert Won Invariant Theory Throughout,
More informationPreprojective algebras, singularity categories and orthogonal decompositions
Preprojective algebras, singularity categories and orthogonal decompositions Claire Amiot Abstract In this note we use results of Minamoto [7] and Amiot-Iyama-Reiten [1] to construct an embedding of the
More informationSkew Calabi-Yau algebras and homological identities
Skew Calabi-Yau algebras and homological identities Manuel L. Reyes Bowdoin College Joint international AMS-RMS meeting Alba Iulia, Romania June 30, 2013 (joint work with Daniel Rogalski and James J. Zhang)
More informationGraded modules over generalized Weyl algebras
Graded modules over generalized Weyl algebras Advancement to Candidacy Robert Won Advised by: Dan Rogalski December 4, 2014 1 / 41 Overview 1 Preliminaries Graded rings and modules Noncommutative things
More informationGenus zero phenomena in noncommutative algebraic geometry
Genus zero phenomena in noncommutative algebraic geometry Western Washington University April 23, 2017 Part 1 Introduction Goal of talk k=base field Noncommutative algebraic geometry Study k-linear abelian
More informationAuslander s Theorem for permutation actions on noncommutative algebras
Auslander s Theorem for permutation actions on noncommutative algebras (arxiv:1705.00068) Jason Gaddis Miami University Introduction This project is joint work with my collaborators at Wake Forest University.
More informationInvariant Theory of AS-Regular Algebras: A Survey
Invariant Theory of AS-Regular Algebras: A Survey Ellen Kirkman Maurice Auslander Distinguished Lectures and International Conference April 20, 2013 Collaborators Jacque Alev Kenneth Chan James Kuzmanovich
More informationZ-graded noncommutative algebraic geometry University of Washington Algebra/Algebraic Geometry Seminar
Z-graded noncommutative algebraic geometry University of Washington Algebra/Algebraic Geometry Seminar Robert Won Wake Forest University Joint with Jason Gaddis (Miami University) and Cal Spicer (Imperial
More informationDeformations of a noncommutative surface of dimension 4
Deformations of a noncommutative surface of dimension 4 Sue Sierra University of Edinburgh Homological Methods in Algebra and Geometry, AIMS Ghana 2016 In this talk, I will describe the work of my student
More informationQUIVERS AND CALABI-YAU ALGEBRAS
QUIVERS AND CALABI-YAU ALGEBRAS JASON GADDIS Abstract. These lectures are meant as a gentle introduction to the study of (twisted Calabi-Yau algebras, which are related to Calabi-Yau manifolds, or the
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 informationHIGHER DIMENSIONAL AUSLANDER-REITEN THEORY ON MAXIMAL ORTHOGONAL SUBCATEGORIES 1. Osamu Iyama
HIGHER DIMENSIONAL AUSLANDER-REITEN THEORY ON MAXIMAL ORTHOGONAL SUBCATEGORIES 1 Osamu Iyama Abstract. Auslander-Reiten theory, especially the concept of almost split sequences and their existence theorem,
More informationTHE DERIVED CATEGORY OF A GRADED GORENSTEIN RING
THE DERIVED CATEGORY OF A GRADED GORENSTEIN RING JESSE BURKE AND GREG STEVENSON Abstract. We give an exposition and generalization of Orlov s theorem on graded Gorenstein rings. We show the theorem holds
More informationArtin-Schelter regular algebras and the Steenrod algebra
Artin-Schelter regular algebras and the Steenrod algebra J. H. Palmieri and J. J. Zhang University of Washington Los Angeles, 10 October 2010 Exercise Let A(1) be the sub-hopf algebra of the mod 2 Steenrod
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 informationIdeals of three dimensional Artin-Schelter regular algebras. Koen De Naeghel Thesis Supervisor: Michel Van den Bergh
Ideals of three dimensional Artin-Schelter regular algebras Koen De Naeghel Thesis Supervisor: Michel Van den Bergh February 17, 2006 Polynomial ring Put k = C. Commutative polynomial ring S = k[x, y,
More informationAuslander s Theorem for permutation actions on noncommutative algebras
Auslander s Theorem for permutation actions on noncommutative algebras Robert Won Joint with Jason Gaddis, Ellen Kirkman, and Frank Moore AMS Western Sectional Meeting, Pullman, WA April 23, 2017 1 / 22
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 informationAnnihilation of Cohomology over Curve Singularities
Annihilation of Cohomology over Curve Singularities Maurice Auslander International Conference Özgür Esentepe University of Toronto April 29, 2018 Özgür Esentepe (University of Toronto) Annihilation of
More informationSELF-DUAL HOPF QUIVERS
Communications in Algebra, 33: 4505 4514, 2005 Copyright Taylor & Francis, Inc. ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870500274846 SELF-DUAL HOPF QUIVERS Hua-Lin Huang Department of
More informationHOMOLOGICAL TRANSCENDENCE DEGREE
HOMOLOGICAL TRANSCENDENCE DEGREE AMNON YEKUTIELI AND JAMES J. ZHANG Abstract. Let D be a division algebra over a base field k. The homological transcendence degree of D, denoted by Htr D, is defined to
More informationADE Dynkin diagrams in algebra, geometry and beyond based on work of Ellen Kirkman
ADE Dynkin diagrams in algebra, geometry and beyond based on work of Ellen Kirkman James J. Zhang University of Washington, Seattle, USA at Algebra Extravaganza! Temple University July 24-28, 2017 Happy
More informationd-calabi-yau algebras and d-cluster tilting subcategories
d-calabi-yau algebras and d-cluster tilting subcategories Osamu Iyama Recently, the the concept of cluster tilting object played an important role in representation theory [BMRRT]. The concept of d-cluster
More informationSERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN
SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN Abstract. Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free O X -bimodule of rank
More informationContributions to the Stable Derived Categories of Gorenstein Rings. Vincent Gélinas
Contributions to the Stable Derived Categories of Gorenstein Rings by Vincent Gélinas A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Mathematics
More informationNoncommutative motives and their applications
MSRI 2013 The classical theory of pure motives (Grothendieck) V k category of smooth projective varieties over a field k; morphisms of varieties (Pure) Motives over k: linearization and idempotent completion
More informationHochschild and cyclic homology of a family of Auslander algebras
Hochschild and cyclic homology of a family of Auslander algebras Rachel Taillefer Abstract In this paper, we compute the Hochschild and cyclic homologies of the Auslander algebras of the Taft algebras
More informationALGEBRAS OF DERIVED DIMENSION ZERO
Communications in Algebra, 36: 1 10, 2008 Copyright Taylor & Francis Group, LLC ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870701649184 Key Words: algebra. ALGEBRAS OF DERIVED DIMENSION ZERO
More informationSingularities of Noncommutative Surfaces. Simon Philip Crawford
This thesis has been submitted in fulfilment of the requirements for a postgraduate degree (e.g. PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following terms and conditions
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 informationarxiv: v2 [math.ra] 3 Oct 2017
MCKAY CORRESPONDENCE FOR SEMISIMPLE HOPF ACTIONS ON REGULAR GRADED ALGEBRAS, II arxiv:1610.01220v2 [math.ra] 3 Oct 2017 K. CHAN, E. KIRKMAN, C. WALTON AND J. J. ZHANG Abstract. We continue our study of
More informationCanonical singularities of orders over surfaces
Canonical singularities of orders over surfaces Daniel Chan, Paul Hacking, and Colin Ingalls July 4, 007 Abstract We define and study canonical singularities of orders over surfaces. These are noncommutative
More informationPeriodicity of selfinjective algebras of polynomial growth
Periodicity of selfinjective algebras of polynomial growth 0 Periodicity of selfinjective algebras of polynomial growth Jerzy Białkowski (Nagoya, November 2013) joint work with Karin Erdmann and Andrzej
More informationRECOLLEMENTS GENERATED BY IDEMPOTENTS AND APPLICATION TO SINGULARITY CATEGORIES
RECOLLEMENTS GENERATED BY IDEMPOTENTS AND APPLICATION TO SINGULARITY CATEGORIES DONG YANG Abstract. In this note I report on an ongoing work joint with Martin Kalck, which generalises and improves a construction
More informationDimensions of Triangulated Categories, joint work with M. Ballard and L. Katzarkov
Dimensions of Triangulated Categories, joint work with M. Ballard and L. Katzarkov David Favero University of Miami January 21, 2010 The Dimension of a Triangulated Category The Dimension of a Triangulated
More informationHOMOLOGICAL PROPERTIES OF QUANTIZED COORDINATE RINGS OF SEMISIMPLE GROUPS
HOMOLOGICAL PROPERTIES OF QUANTIZED COORDINATE RINGS OF SEMISIMPLE GROUPS K.R. GOODEARL AND J.J. ZHANG Abstract. We prove that the generic quantized coordinate ring O q(g) is Auslander-regular, Cohen-Macaulay,
More informationON THE GEOMETRY OF ORBIT CLOSURES FOR REPRESENTATION-INFINITE ALGEBRAS
ON THE GEOMETRY OF ORBIT CLOSURES FOR REPRESENTATION-INFINITE ALGEBRAS CALIN CHINDRIS ABSTRACT. For the Kronecker algebra, Zwara found in [13] an example of a module whose orbit closure is neither unibranch
More informationIsotropic Schur roots
Isotropic Schur roots Charles Paquette University of Connecticut November 21 st, 2016 joint with Jerzy Weyman Outline Describe the perpendicular category of an isotropic Schur root. Describe the ring of
More informationTHE GL(2, C) MCKAY CORRESPONDENCE
THE GL(2, C) MCKAY CORRESPONDENCE MICHAEL WEMYSS Abstract. In this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially
More informationCLUSTER-TILTED ALGEBRAS OF FINITE REPRESENTATION TYPE
CLUSTER-TILTED ALGEBRAS OF FINITE REPRESENTATION TYPE ASLAK BAKKE BUAN, ROBERT J. MARSH, AND IDUN REITEN Abstract. We investigate the cluster-tilted algebras of finite representation type over an algebraically
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 informationTwisted rings and moduli stacks of fat point modules in non-commutative projective geometry
Twisted rings and moduli stacks of fat point modules in non-commutative projective geometry DANIEL CHAN 1 University of New South Wales e-mail address:danielc@unsw.edu.au Abstract The Hilbert scheme of
More informationREPRESENTATION DIMENSION OF ARTIN ALGEBRAS
REPRESENTATION DIMENSION OF ARTIN ALGEBRAS STEFFEN OPPERMANN In 1971, Auslander [1] has introduced the notion of representation dimension of an artin algebra. His definition is as follows (see Section
More informationarxiv: v1 [math.rt] 12 Jan 2016
THE MCM-APPROXIMATION OF THE TRIVIAL MODULE OVER A CATEGORY ALGEBRA REN WANG arxiv:1601.02737v1 [math.rt] 12 Jan 2016 Abstract. For a finite free EI category, we construct an explicit module over its category
More informationON THE DERIVED DIMENSION OF ABELIAN CATEGORIES
ON THE DERIVED DIMENSION OF ABELIAN CATEGORIES JAVAD ASADOLLAHI AND RASOOL HAFEZI Abstract. We give an upper bound on the dimension of the bounded derived category of an abelian category. We show that
More informationarxiv: v1 [math.ra] 23 Jul 2016
MCKAY CORRESPONDENCE FOR SEMISIMPLE HOPF ACTIONS ON REGULAR GRADED ALGEBRAS, I arxiv:1607.06977v1 [math.ra] 23 Jul 2016 K. CHAN, E. KIRKMAN, C. WALTON, AND J.J. ZHANG Abstract. In establishing a more general
More informationCATEGORICAL ASPECTS OF ALGEBRAIC GEOMETRY IN MIRROR SYMMETRY ABSTRACTS
CATEGORICAL ASPECTS OF ALGEBRAIC GEOMETRY IN MIRROR SYMMETRY Alexei Bondal (Steklov/RIMS) Derived categories of complex-analytic manifolds Alexender Kuznetsov (Steklov) Categorical resolutions of singularities
More informationTRIVIAL MAXIMAL 1-ORTHOGONAL SUBCATEGORIES FOR AUSLANDER 1-GORENSTEIN ALGEBRAS
J. Aust. Math. Soc. 94 (2013), 133 144 doi:10.1017/s1446788712000420 TRIVIAL MAXIMAL 1-ORTHOGONAL SUBCATEGORIES FOR AUSLANDER 1-GORENSTEIN ALGEBRAS ZHAOYONG HUANG and XIAOJIN ZHANG (Received 25 February
More informationRepresentation Theory of Orders over Cohen- Macaulay Rings
Syracuse University SURFACE Dissertations - ALL SURFACE June 2017 Representation Theory of Orders over Cohen- Macaulay Rings Josh John Stangle Syracuse University Follow this and additional works at: https://surface.syr.edu/etd
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 informationarxiv: v1 [math.kt] 27 Jan 2015
INTRODUCTION TO DERIVED CATEGORIES AMNON YEKUTIELI arxiv:1501.06731v1 [math.kt] 27 Jan 2015 Abstract. Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the old
More informationDimer models and cluster categories of Grassmannians
Dimer models and cluster categories of Grassmannians Karin Baur University of Graz Rome, October 18, 2016 1/17 Motivation Cluster algebra structure of Grassmannians Construction of cluster categories (k,n)
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 information6. Dynkin quivers, Euclidean quivers, wild quivers.
6 Dynkin quivers, Euclidean quivers, wild quivers This last section is more sketchy, its aim is, on the one hand, to provide a short survey concerning the difference between the Dynkin quivers, the Euclidean
More informationRepresentation type, boxes, and Schur algebras
10.03.2015 Notation k algebraically closed field char k = p 0 A finite dimensional k-algebra mod A category of finite dimensional (left) A-modules M mod A [M], the isomorphism class of M ind A = {[M] M
More informationA note on standard equivalences
Bull. London Math. Soc. 48 (2016) 797 801 C 2016 London Mathematical Society doi:10.1112/blms/bdw038 A note on standard equivalences Xiao-Wu Chen Abstract We prove that any derived equivalence between
More informationTHE NEGATIVE SIDE OF COHOMOLOGY FOR CALABI-YAU CATEGORIES
THE NEGATIVE SIDE OF COHOMOLOGY FOR CALABI-YAU CATEGORIES PETTER ANDREAS BERGH, DAVID A. JORGENSEN & STEFFEN OPPERMANN Abstract. We study Z-graded cohomology rings defined over Calabi-Yau categories. We
More informationOn root categories of finite-dimensional algebras
On root categories of finite-dimensional algebras Changjian Department of Mathematics, Sichuan University Chengdu August 2012, Bielefeld Ringel-Hall algebra for finitary abelian catgories Ringel-Hall Lie
More informationA = A(f) = k x 1,...,x n /(f = f ij x i x j )
Noncommutative Algebraic Geometry Shanghai September 12-16, 211 Calabi-Yau algebras linked to Poisson algebras Roland Berger (Saint-Étienne, France (jointly Anne Pichereau Calabi-Yau algebras viewed as
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 informationThe homotopy categories of injective modules of derived discrete algebras
Dissertation zur Erlangung des Doktorgrades der Mathematik (Dr.Math.) der Universität Bielefeld The homotopy categories of injective modules of derived discrete algebras Zhe Han April 2013 ii Gedruckt
More informationRepresentations of Quivers
MINGLE 2012 Simon Peacock 4th October, 2012 Outline 1 Quivers Representations 2 Path Algebra Modules 3 Modules Representations Quiver A quiver, Q, is a directed graph. Quiver A quiver, Q, is a directed
More informationDERIVED EQUIVALENCES AND GORENSTEIN PROJECTIVE DIMENSION
DERIVED EQUIVALENCES AND GORENSTEIN PROJECTIVE DIMENSION HIROTAKA KOGA Abstract. In this note, we introduce the notion of complexes of finite Gorenstein projective dimension and show that a derived equivalence
More informationElementary (super) groups
Elementary (super) groups Julia Pevtsova University of Washington, Seattle Auslander Days 2018 Woods Hole 2 / 35 DETECTION QUESTIONS Let G be some algebraic object so that Rep G, H (G) make sense. Question
More informationSTABILITY OF FROBENIUS ALGEBRAS WITH POSITIVE GALOIS COVERINGS 1. Kunio Yamagata 2
STABILITY OF FROBENIUS ALGEBRAS WITH POSITIVE GALOIS COVERINGS 1 Kunio Yamagata 2 Abstract. A finite dimensional self-injective algebra will be determined when it is stably equivalent to a positive self-injective
More informationTopological equivalences of differential graded algebras (Joint work with D. Dugger)
Topological equivalences of differential graded algebras (Joint work with D. Dugger) Abelian groups up to homotopy spectra generalized cohomology theories Examples: 1. Ordinary cohomology: For A any abelian
More informationNOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE
NOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE GUFANG ZHAO Contents 1. Introduction 1 2. What is a Procesi bundle 2 3. Derived equivalences from exceptional objects 4 4. Splitting of the
More informationTHE EXT ALGEBRA AND A NEW GENERALISATION OF D-KOSZUL ALGEBRAS
THE EXT ALGEBRA AND A NEW GENERALISATION OF D-KOSZUL ALGEBRAS JOANNE LEADER AND NICOLE SNASHALL Abstract. We generalise Koszul and D-Koszul algebras by introducing a class of graded algebras called (D,
More informationarxiv: v1 [math.ag] 18 Nov 2017
KOSZUL DUALITY BETWEEN BETTI AND COHOMOLOGY NUMBERS IN CALABI-YAU CASE ALEXANDER PAVLOV arxiv:1711.06931v1 [math.ag] 18 Nov 2017 Abstract. Let X be a smooth projective Calabi-Yau variety and L a Koszul
More informationarxiv:math/ v1 [math.ra] 30 Sep 2001
arxiv:math/0110008v1 [math.ra] 30 Sep 2001 DUALIZING COMPLEXES AND TILTING COMPLEXES OVER SIMPLE RINGS AMNON YEKUTIELI AND JAMES J. ZHANG 0. Introduction Simple rings, like fields, are literally simple
More informationAlgebra Qualifying Exam Solutions January 18, 2008 Nick Gurski 0 A B C 0
1. Show that if B, C are flat and Algebra Qualifying Exam Solutions January 18, 2008 Nick Gurski 0 A B C 0 is exact, then A is flat as well. Show that the same holds for projectivity, but not for injectivity.
More informationIwasawa algebras and duality
Iwasawa algebras and duality Romyar Sharifi University of Arizona March 6, 2013 Idea of the main result Goal of Talk (joint with Meng Fai Lim) Provide an analogue of Poitou-Tate duality which 1 takes place
More informationDEFORMATIONS OF ALGEBRAS IN NONCOMMUTATIVE ALGEBRAIC GEOMETRY EXERCISE SHEET 1
DEFORMATIONS OF ALGEBRAS IN NONCOMMUTATIVE ALGEBRAIC GEOMETRY EXERCISE SHEET 1 TRAVIS SCHEDLER Note: it is possible that the numbers referring to the notes here (e.g., Exercise 1.9, etc.,) could change
More informationPart II Galois Theory
Part II Galois Theory Definitions 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)
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 informationRepresentation type and Auslander-Reiten theory of Frobenius-Lusztig kernels
Representation type and Auslander-Reiten theory of Frobenius-Lusztig kernels Julian Külshammer University of Kiel, Germany 08.2012 Notation Denote by: k an algebraically closed field (of characteristic
More informationOne-Dimensional Line Schemes Michaela Vancliff
One-Dimensional Line Schemes Michaela Vancliff University of Texas at Arlington, USA http://www.uta.edu/math/vancliff/r vancliff@uta.edu Partial support from NSF DMS-1302050. Motivation Throughout, k =
More informationWeyl Groups and Artin Groups Associated to Weighted Projective Lines
Weyl Groups and Artin Groups Associated to Weighted Projective Lines (joint work with Yuuki Shiraishi and Kentaro Wada) Atsushi TAKAHASHI OSAKA UNIVERSITY November 15, 2013 at NAGOYA 1 / 29 Aim Want to
More informationA BRIEF INTRODUCTION TO GORENSTEIN PROJECTIVE MODULES. Shanghai , P. R. China
A BRIEF INTRODUCTION TO GORENSTEIN PROJECTIVE MODULES PU ZHANG Department of Mathematics, Shanghai 200240, P. R. China Shanghai Jiao Tong University Since Eilenberg and Moore [EM], the relative homological
More informationDERIVED CATEGORIES IN REPRESENTATION THEORY. We survey recent methods of derived categories in the representation theory of algebras.
DERIVED CATEGORIES IN REPRESENTATION THEORY JUN-ICHI MIYACHI We survey recent methods of derived categories in the representation theory of algebras. 1. Triangulated Categories and Brown Representability
More informationON THE EXCEPTIONAL FIBRES OF KLEINIAN SINGULARITIES WILLIAM CRAWLEY-BOEVEY
ON THE EXCEPTIONAL FIBRES OF KLEINIAN SINGULARITIES WILLIAM CRAWLEY-BOEVEY Abstract. We give a new proof, avoiding case-by-case analysis, of a theorem of Y. Ito and I. Nakamura which provides a module-theoretic
More informationA 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 informationProblem 1.1. Classify all groups of order 385 up to isomorphism.
Math 504: Modern Algebra, Fall Quarter 2017 Jarod Alper Midterm Solutions Problem 1.1. Classify all groups of order 385 up to isomorphism. Solution: Let G be a group of order 385. Factor 385 as 385 = 5
More informationDUAL REFLECTION GROUPS OF LOW ORDER KENT VASHAW. A Thesis Submitted to the Graduate Faculty of
DUAL REFLECTION GROUPS OF LOW ORDER BY KENT VASHAW A Thesis Submitted to the Graduate Faculty of WAKE FOREST UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES in Partial Fulfillment of the Requirements for
More informationPATH SUBCOALGEBRAS, FINITENESS PROPERTIES AND QUANTUM GROUPS
PATH SUBCOALGEBRAS, FINITENESS PROPERTIES AND QUANTUM GROUPS S.DĂSCĂLESCU1,, M.C. IOVANOV 1,2, C. NĂSTĂSESCU1 Abstract. We study subcoalgebras of path coalgebras that are spanned by paths (called path
More informationQuivers supporting graded twisted Calabi-Yau algebras
Quivers supporting graded twisted Calabi-Yau algebras Jason Gaddis Miami Unversity Joint with Dan Rogalski J. Gaddis (Miami Twisted graded CY algebras January 12, 2018 Calabi-Yau algebras Let A be an algebra
More informationProof of Langlands for GL(2), II
Proof of Langlands for GL(), II Notes by Tony Feng for a talk by Jochen Heinloth April 8, 016 1 Overview Let X/F q be a smooth, projective, geometrically connected curve. The aim is to show that if E is
More information2. Preliminaries In this section we give some basic definitions and known results, which will be useful in the sequel.
M ath. es. Lett. 4 (2007), no. 4, 589 596 c International Press 2007 KOSULITY AND THE HILBET SEIES OF PEPOJECTIVE ALGEBAS Pavel Etingof and Ching-Hwa Eu. Introduction The goal of this paper is to prove
More informationPaolo Stellari STABILITY CONDITIONS ON GENERIC K3 SURFACES
Paolo Stellari STABILITY CONDITIONS ON GENERIC K3 SURFACES Joint with D. Huybrechts and E. Macrì math.ag/0608430 Dipartimento di Matematica F. Enriques Università degli Studi di Milano CONTENTS A generic
More information