Proceedings of the 44th Symposium on Ring Theory and ...

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[22] A. Skowroński and J. Waschbüsch, Representation-finite biserial algebras, J. Reine und Angew. Math. 345 (1983), 172–181. [23] N. Snashall, Support varieties and <strong>the</strong> Hochschild ohomology ring modulo nilpotence <strong>Proceedings</strong> <strong>of</strong> <strong>the</strong> 41st <strong>Symposium</strong> on Ring Theory and Representation Theory, 68–82, Symp. Ring Theory Represent. Theory Organ. Comm., Tsukuba, 2009. [24] D. F. Vatne, The mutation class <strong>of</strong> D n quivers, Comm. Algebra 38 (2010), 1137–1146. [25] B. B. Venkov, Cohomology algebras for some classifying spaces, Dokl. Akad. Nauk SSSR 127 (1959), 943–944. [26] F. Xu, Hochschild and ordinary cohomology rings <strong>of</strong> small categories, Adv. Math. 219 (2008), 1872– 1893. Takahiko Furuya Department <strong>of</strong> Ma<strong>the</strong>matics Tokyo University <strong>of</strong> Science 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601 JAPAN E-mail address: furuya@ma.kagu.tus.ac.jp Takao Hayami Faculty <strong>of</strong> Engineering Hokkai-Gakuen University 4-1-40, Asahi-machi, Toyohira-ku, Sapporo, JAPAN E-mail address: hayami@ma.kagu.tus.ac.jp –49–
DERIVED AUTOEQUIVALENCES AND BRAID RELATIONS JOSEPH GRANT Abstract. We will consider braid relations between autoequivalences <strong>of</strong> derived categories <strong>of</strong> symmetric algebras. We first recall <strong>the</strong> construction <strong>of</strong> spherical twists for symmetric algebras and <strong>the</strong> braid relations that <strong>the</strong>y satisfy, as illustrated by Brauer tree algebras. Then we explain <strong>the</strong> construction <strong>of</strong> periodic twists, which generalise spherical twists for symmetric algebras. Finally, we explain a lifting <strong>the</strong>orem for periodic twists, and show how this gives a new interpretation <strong>of</strong> <strong>the</strong> action on <strong>the</strong> derived Picard group <strong>of</strong> lifts <strong>of</strong> longest elements <strong>of</strong> <strong>the</strong> symmetric group to <strong>the</strong> braid group. 1. Preliminaries Let k be an algebraically closed field. All algebras we consider will be finite-dimensional k-algebras, and for simplicity we will also assume that A is basic. We will denote <strong>the</strong> category <strong>of</strong> finite-dimensional left A-modules by A -mod, and <strong>of</strong> finite-dimensional right A-modules by mod- A. Given an algebra A and a left (or right) A-module M, we have a right (or left, respectively) A-module M ∗ = Hom k (M, k) with A-action fa(m) = f(am) for m ∈ M, f ∈ M ∗ , and a ∈ A. This gives a duality (−) ∗ : A -mod ∼ → mod- A. Similarly, if M is an A-B-bimodule for algebras A and B, <strong>the</strong>n M ∗ is a B-A-bimodule. There is ano<strong>the</strong>r way to construct a right module from M ∈ A -mod: we set M ∨ = Hom A (M, A), where <strong>the</strong> action is given here by fa(m) = f(m)a for m ∈ M, f ∈ M ∨ , and a ∈ A. This defines a functor (−) ∨ : A -mod → mod- A. but in general this is not an equivalence. However, in <strong>the</strong> cases we consider below this will be an equivalence. Any algebra A has a natural structure <strong>of</strong> an A-A-bimodule given by <strong>the</strong> multiplication. We say that A is a symmetric algebra if <strong>the</strong>re exists an isomorphism <strong>of</strong> A-A-bimodules A ∼ → A ∗ . Symmetric algebras have various equivalent definitions: one is that (−) ∗ and (−) ∨ are natually isomorphic functors, and ano<strong>the</strong>r is a Calabi-Yau type condition on <strong>the</strong> derived category. For more information on this, we refer <strong>the</strong> reader to [Ric2, Section 3]. We will be interested in bounded derived categories <strong>of</strong> module categories over algebras A, which we will denote D b (A). We refer <strong>the</strong> reader to [Wei, Chapter 10] for <strong>the</strong>ir definition and basic properties. In partuicular, we will study autoequivalences <strong>of</strong> D b (A). Clearly <strong>the</strong> autoequivalences form a group, but in fact we can restrict ourselves to a particular subset. One way to define an end<strong>of</strong>unctor <strong>of</strong> D b (A) is to take <strong>the</strong> derived tensor product with The detailed version <strong>of</strong> this paper will be submitted for publication elsewhere. –50–
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Proceedings <stron
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Organizing Committee of</st
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Polycyclic codes and sequential cod
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Preface The 44th <
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8:40-9:30 Dan ZachariaSyracuse Univ
Page 13 and 14: The 44th S
Page 15 and 16: Tuesday September 27 8:40-9:30 Atsu
Page 17 and 18: Conversely, let (T , F) be a stable
Page 19 and 20: Then T = gen(X) and X is Ext-projec
Page 21 and 22: DIMENSIONS OF DERIVED CATEGORIES TA
Page 23 and 24: Notation 4. (1) Let A be an abelian
Page 25 and 26: of finitely genera
Page 27 and 28: is the map given b
Page 29 and 30: (2) Let R be a commutative ring whi
Page 31 and 32: QUIVER PRESENTATIONS OF GROTHENDIEC
Page 33 and 34: Proposition 3. Let C be a category,
Page 35 and 36: Example 11. Let Q be the</s
Page 37 and 38: and Gr(X ′ ) is given by
Page 39 and 40: particular, the de
Page 41 and 42: Proposition 1 and 2 leave us with d
Page 43 and 44: 4. Examples Example 6. Let M = M(A
Page 45 and 46: EXAMPLE OF CATEGORIFICATION OF A CL
Page 47 and 48: The aim of <strong
Page 49 and 50: X ∈ mod Π Q S 1 S 2 S 3 1 ❁
Page 51 and 52: The previous proposition has a dual
Page 53 and 54: Computing inductively all t
Page 55 and 56: Example 23. We have ⎛ ⎞ ⎛ ⎞
Page 57 and 58: classification of
Page 59 and 60: quiver Q over K, and let D b (H) <s
Page 61 and 62: Then Q ∈ Q 17 . Suppose char K
Page 63: Moreover the Hochs
Page 67 and 68: 3. Periodic Twists We now describe
Page 69 and 70: We now specialise to the</s
Page 71 and 72: and τ − n := Ext n Λ(DΛ, −)
Page 73 and 74: where r v ij := a i a j −a j a i
Page 75 and 76: ON A DEGENERATION PROBLEM FOR COHEN
Page 77 and 78: We make several othe</stron
Page 79 and 80: Let A be a commutative Gorenstein r
Page 81 and 82: 3. Extended orders In the</
Page 83 and 84: WEAK GORENSTEIN DIMENSION FOR MODUL
Page 85 and 86: 1.2. Introduction. The notion <stro
Page 87 and 88: e 2 A z 1 −→ X → 0 in mod-A,
Page 89 and 90: 5. Gorenstein algebras In this sect
Page 91 and 92: ON Ω-PERFECT MODULES AND SEQUENCES
Page 93 and 94: when R is a Nakayama algebra <stron
Page 95 and 96: says that components of</st
Page 97 and 98: then we obtain a c
Page 99 and 100: Proposition 16. Let C s be a stable
Page 101 and 102: 3.1. ZD ∞ -components. We assume
Page 103 and 104: Theorem 26. Let R be a local artini
Page 105 and 106: where each f i is an irreducible ep
Page 107 and 108: QUANTUM UNIPOTENT SUBGROUP AND DUAL
Page 109 and 110: {F i } i∈I and q-Serre relations
Page 111 and 112: Theorem 8. (1)Let U − q (w) → U
Page 113 and 114: E-mail address: ykimura@kurims.kyot
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Indeed, (1, 2, 3, 4) {1,2} −−
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By comparing this the</stro
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(a) (b) Proposition 17. Let G be a
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POLYCYCLIC CODES AND SEQUENTIAL COD
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⎛ ⎞ 0 0 c 0 1 c t D c = ⎜ 1
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References [1] D. Boucher and P. So
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2. Dimension of tr
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APR TILTING MODULES AND QUIVER MUTA
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(1) Q ′ is a quiver obtained from
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1 arrows of ˜µ L
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[11] S. Fomin, A. Zelevinsky, Clust
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Theorem. Assume r is a common prime
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References [1] Apostol, T. M., The
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e(n, s) n = 6 7 8 s = 1 36 63 120 2
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Now we will show that the</
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is equal to ( n 2s−1 ); hence we
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THE NOETHERIAN PROPERTIES OF THE RI
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Proposition 4 (Paper III, Propositi
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HOCHSCHILD COHOMOLOGY OF QUIVER ALG
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(4) In [10], Green, Snashall and So
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4. Quiver algebras defined by two c
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[6] L. Evens, The cohomology ring <
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Then there is an i
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• ( ˜F , ˜m) = ({ (1, 3), (2, 3
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Corollary 7 ([16, Theorem 3.4]). Th
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We have the direct
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We say a CW complex is regular, if
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SHARP BOUNDS FOR HILBERT COEFFICIEN
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Let us briefly note how this paper
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whence the set Λ
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Proof of</
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We have Q·H j m(A/(a 1 , a 2 , ·
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Let B = A/U(a d−1 ). We t
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• quiver Γ = (I, Ω) I Ω
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B = (B τ ) relations ρ 1 , · ·
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Definition 1. double quiver ˜Γ p
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◦ side Γ = (I, Ω) cycle (ac
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Definition 6. n × n A(Γ Dyn ) =
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(2) P Lie theory
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Example 15. Γ loop quiver λ ∈
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P (Γ)-module i-th radical B →
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B ∈ Λ(d) ε ∗ i (B) := dim C
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I-graded vector space V (d) (B τ )
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A ∗(i) l (a) = ∑n+1 t=l+1 (a i,
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wt(a) = (−d 1 , −d 2 , −d 3 )
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Ω Q Ω ( B Ω ; wt, ε i , ϕ
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“Λ(d) G(d)-” Definition 25.
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(( ( ) ( ) 0 s 0 0 0 , , , s) ( u 0
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MATRIX FACTORIZATIONS, ORBIFOLD CUR
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( ∂f Jac(f t t ) := C[x 1 , . . .
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Definition 14. CM L f (R f ) Ob(C
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4. Dolgachev Proposition 24 ([1]
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✤ ✤ ✤ ✤ ✤ ✤ γ 1 +γ 2
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(f, G f ) Dolgachev A Gf f Berg
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ON A GENERALIZATION OF COSTABLE TOR
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(2) P/t(P ) is σ-projective for an
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(3) P is a maximal σ-coessential e
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(2)→(1): Let σ be epi-preserving
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GRADED FROBENIUS ALGEBRAS AND QUANT
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Definition 2. A graded algebra A is
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In this case, ν ∈ Aut k E induce
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References [1] X. W. Chen, Graded s
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The topics covered are ring-<strong
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(2.3) soc( R R) ∼ = k⊕ s i T i
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principal ideal Rr ⊂ ker ϖ. Thus
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By Example 3, the
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weight wt(x) = d(x, 0) of</
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4.1. Fourier Transform. Gleason’s
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5. The Extension Problem In this se
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Because R is assumed to be Frobeniu
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is injective for every finite left
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linear code defined by η;
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ψ : ε(A) → Â. In this way, C
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REALIZING STABLE CATEGORIES AS DERI
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2.1. Positively graded self-injecti
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By the above resul
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Next we consider the</stron
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(2) There exists a triangle-equival
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ALGEBRAIC STRATIFICATIONS OF DERIVE
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The leaves of <str
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Step 2: Let A be a finite-dimension
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RECOLLEMENTS GENERATED BY IDEMPOTEN
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(a) the adjoint tr
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Cluster-tilting the</strong
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INTRODUCTION TO REPRESENTATION THEO
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is exact. Since E n ∼ = En+r , Mi
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field of V . We de
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(Proof) ( ) p 0 0
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we have a sequence of</stro
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R : CM(R⊗ k V ) → CM(R) defined
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P = P ′ t by t is a projective R
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SUBCATEGORIES OF EXTENSION MODULES
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Lemma 6. Let S 1 and S 2 be Serre s
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In the first row <
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