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

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✦ ✦ ✦ ✦ ✦ ✦ ✦ ✦ ✦ ✦ 1 4 , 12 34 , 123 234 , 13 34 , 1 2 , 1 3 ✎ ✴ 1 4 , 12 34 , 123 234 , ✎ ✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴ ✎ ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ 13 34 , 23 34 , ✎ ✎ 1 ✎ ✎ 3❄ ✎ ❄❄❄❄❄❄❄❄ ✎ ✎ 1 4 , 12 34 , 123 234 ❞ , 13 34 , 1 2 , 3 1 4 4 , 12 34 , 123 234 ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ , 13 34 , 23 34 , 3 4 ✬ ✬✬✬✬✬✬✬✬✬ 1 4 , 12 34 , 123 234 , 12 1 24 , 1 2 , 3 4 4 , 12 34 , 123 234 , ❉ 1 ♦♦♦♦♦ ❉❉❉❉❉❉❉ 23 4 , 12 34 , 123 34 , 2 4 , 3 ❩❩❩ 4 234 , ☛ 12 24 , 2 4 , 3 4 1 ☛ ☛ ❄ ☛ ❄❄❄❄❄❄ 4 , 12 34 , 123 234 , ☛ 12 24 , 1 2 , 12 23 ❩❩❩❩❩❩ 1 ☛ ☛ 4 , 12 34 , 123 234 , ☛ ☛ 1 ❧❧❧❧❧❧❧ 1 4 , 12 34 , 123 234 , 12 4 , 12 34 , 123 234 , 23 , 1 2 , 1 3 12 23 24 , 2 4 , 12 23 34 , 2 4 , 2 3 ✚ ✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚ 1 ♦♦♦♦♦♦♦♦ 4 , 12 34 , 123 234 , 23 34 , 2 3 , 1 3 ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ✒ ✒✒✒✒✒✒✒✒✒ 1 4 , 12 34 , 123 234 , 12 23 , 2 4 , 2 3 ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ❖ 1 4 , 12 34 , 8 88888888888888888888888888888888888888888 123 234 , 12 23 , 2 3 , 1 3 Figure 1. Exchange graph <strong>of</strong> minors neighbours obtained by replacing one <strong>of</strong> its elements x k by a new one x ′ k relation related by a x k x ′ k = M 1 + M 2 where M 1 and M 2 are mutually prime monomials in {x 1 , . . . , x k−1 , x k+1 , . . . , x m }, given by precise combinatorial rules. These replacements, called mutations and denoted by µ k are involutive. For precise definitions and details about <strong>the</strong>se constructions, we refer to [4]. In <strong>the</strong> previous example, <strong>the</strong> coefficients are ∆ 1 4, ∆ 12 34 and ∆ 123 234 and <strong>the</strong> cluster variables are all <strong>the</strong> o<strong>the</strong>r non-trivial minors. The extended clusters are <strong>the</strong> sets appearing at <strong>the</strong> vertices <strong>of</strong> Figure 1. –31–
The aim <strong>of</strong> <strong>the</strong> following sections is to describe examples <strong>of</strong> additive categorifications <strong>of</strong> cluster algebras. It consists <strong>of</strong> enhancing <strong>the</strong> cluster algebra structure with an additive category, some objects <strong>of</strong> which reflect <strong>the</strong> combinatorial structure <strong>of</strong> <strong>the</strong> cluster algebra; moreover, <strong>the</strong>re is an explicit formula, <strong>the</strong> cluster character associating to <strong>the</strong>se particular objects elements <strong>of</strong> <strong>the</strong> algebra, in a way which is compatible with <strong>the</strong> combinatorial structure. The examples we develop here rely on (abelian) module categories. They are particular cases <strong>of</strong> categorifications by exact categories appearing in [6] (simply-laced case) and [2] (non simply-laced case). The study <strong>of</strong> cluster algebras and <strong>the</strong>ir categorifications has been particularly successful <strong>the</strong>se last years. For a survey on categorification by triangulated categories and a much more complete bibliography, see [7]. 2. The preprojective algebra and <strong>the</strong> cluster character Let Q be <strong>the</strong> following quiver (oriented graph): α 1 2 3 α ∗ As usual, denote by CQ <strong>the</strong> C-algebra, a basis <strong>of</strong> which is formed by <strong>the</strong> paths (including 0-length paths supported by each <strong>of</strong> <strong>the</strong> three vertices) and <strong>the</strong> multiplication <strong>of</strong> which is defined by concatenation <strong>of</strong> paths when it is possible and vanishes when paths can not be composed (we write here <strong>the</strong> composition from left to right, on <strong>the</strong> contrary to <strong>the</strong> usual composition <strong>of</strong> maps). Thus, a (right) CQ-module is naturally graded by idempotents (0-length paths) corresponding to vertices and <strong>the</strong> action <strong>of</strong> arrows seen as elements <strong>of</strong> <strong>the</strong> algebra can naturally be identified with linear maps between <strong>the</strong> corresponding homogeneous subspaces <strong>of</strong> <strong>the</strong> representation. We shall use <strong>the</strong> following right-hand side convenient notation: ⎛ ⎞ ⎛ ⎞ β β ∗ 0 0 0 ⎝ ⎠ ⎝ ⎠ −1 1 0 C C 2 C 2 ( ) ⎛ 1 0 ⎞ 1 0 ⎝ ⎠ 0 1 = 2 ✂ ✂✂ ❁ ❁❁ 1❁ 3 ❁ ✂ 2 ✂✂ ❁ ❁❁ 3 where each <strong>of</strong> <strong>the</strong> digits represents a basis vector <strong>of</strong> <strong>the</strong> representation and each arrow a non-zero scalar (1 when not specified) in <strong>the</strong> corresponding matrix entry. Let us now introduce <strong>the</strong> preprojective algebra <strong>of</strong> Q: Definition 2. The preprojective algebra <strong>of</strong> Q is defined by Π Q = −1 CQ (αα ∗ , α ∗ α + β ∗ β, ββ ∗ ) <strong>the</strong> representations <strong>of</strong> which are seen as particular representations <strong>of</strong> CQ (in o<strong>the</strong>r words, mod Π Q is a full subcategory <strong>of</strong> mod CQ). –32–
Page 1: Proceedings <stron
Page 5 and 6: Organizing Committee of</st
Page 7 and 8: Polycyclic codes and sequential cod
Page 9 and 10: Preface The 44th <
Page 11 and 12: 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: EXAMPLE OF CATEGORIFICATION OF A CL
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 and 64: Moreover the Hochs
Page 65 and 66: DERIVED AUTOEQUIVALENCES AND BRAID
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
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then we obtain a c
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Proposition 16. Let C s be a stable
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3.1. ZD ∞ -components. We assume
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Theorem 26. Let R be a local artini
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where each f i is an irreducible ep
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QUANTUM UNIPOTENT SUBGROUP AND DUAL
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{F i } i∈I and q-Serre relations
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Theorem 8. (1)Let U − q (w) → U
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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
Page 121 and 122:
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
Page 271 and 272:
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
Page 301 and 302:
In the first row <
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