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

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each relation r ∈ R. Let d be <strong>the</strong> unique continuous k-linear automorphism <strong>of</strong> ̂kQ which satisfies <strong>the</strong> graded Leibniz rule and which takes ρ r to r for each relation r ∈ R. Then <strong>the</strong>re is a quasi-isomorphism from ( ̂kQ, d) to A. Example 5. Let A be as in Example 1. Let Q be <strong>the</strong> graded quiver 1 α β 2 ρ where α and β are in degree 0 and ρ is in degree −1. Let d be <strong>the</strong> unique continuous k-linear automorphism <strong>of</strong> ̂kQ which satisfies <strong>the</strong> graded Leibniz rule and which takes ρ to αβ. Then <strong>the</strong> obvious map from ( ̂kQ, d) to A is a quasi-isomorphism. Let e = e 1 . The associated dg algebra B as in Theorem 3 is (quasi-isomorphic to) <strong>the</strong> dg algebra k[ρ] with ρ in degree −1 and with vanishing differential. 2. Application to singularity categories Let k be a field, and let R be a Iwanaga–Gorenstein k-algebra, i.e. R is left and right noe<strong>the</strong>rian as a ring and R has finite injective dimension both as left R-module and as right R-module. Let mod R denote <strong>the</strong> category <strong>of</strong> finitely generated right R-modules. On <strong>the</strong> one hand, one defines <strong>the</strong> singularity category D sg (R) := D b (mod R)/K b (proj R), which measures <strong>the</strong> complexity <strong>of</strong> <strong>the</strong> singularity <strong>of</strong> R. (K b (proj R) is considered as <strong>the</strong> smooth part.) On <strong>the</strong> o<strong>the</strong>r hand, one defines <strong>the</strong> category MCM(R) <strong>of</strong> maximal Cohen– Macaulay R-modules MCM(R) := {M ∈ mod R | Ext i R(M, R) = 0 for any i > 0}. The following nice result <strong>of</strong> Buchweitz relates <strong>the</strong>se categories. Theorem 6. ([4]) MCM(R) is a Frobenius category whose full subcategory <strong>of</strong> projectiveinjective objects is precisely proj R. Moreover, <strong>the</strong> embedding MCM(R) → mod R induces a triangle equivalence from <strong>the</strong> stable category MCM(R) to <strong>the</strong> singularity category D sg (R). Let M 1 , . . . , M r ∈ MCM(R) be pairwise non-isomorphic non-projective R-modules and let M = R ⊕ M 1 ⊕ . . . ⊕ M r . Let A = End R (M) and e = id R considered as an element <strong>of</strong> A. Then R = eAe and A/AeA = End MCM(R) (M). For example, <strong>the</strong> ring R = k[x]/x 2 has a unique simple module S, and letting M = R ⊕ S we obtain that A = End R (M) is <strong>the</strong> algebra given in Example 1. There is always an embedding <strong>of</strong> K b (proj R) into K b (proj A) with essential image being thick(eA). If <strong>the</strong> following condition is satisfied (c1) A has finite global dimension, <strong>the</strong>n A becomes a non-commutative/categorical resolution <strong>of</strong> R. The condition (c1) has an interesting consequence: <strong>the</strong> object M generates MCM(R) as a triangulated category. –265–
Cluster-tilting <strong>the</strong>ory comes into <strong>the</strong> story because cluster-tilting objects are closely related to Van den Bergh’s non-commutative crepant resolutions [16], see [10]. The triangle quotient K b (proj A)/ thick(eA) measures <strong>the</strong> difference between <strong>the</strong> resolution and <strong>the</strong> smooth part <strong>of</strong> <strong>the</strong> singularity, see [5]. So K b (proj A)/ thick(eA) is in some sense a ‘categorical exceptional locus’. A natural question is: how is K b (proj A)/ thick(eA) related to D sg (R)? Consider <strong>the</strong> following condition (c2) MCM(R) is Hom-finite. Theorem 7. ([12]) Keep <strong>the</strong> above notations and assume that (c1) and (c2) hold. There is a dg algebra B with a morphism f : A → B such that f induces a triangle equivalence per(B) ∼ = (K b (proj A)/ thick(eA)) ω . Moreover, B satisfies <strong>the</strong> following properties: (a) B i = 0 for any i > 0, (b) H 0 (B) ∼ = A/AeA, (c) D fd (B) ⊆ per(B), (d) per(B) is Hom-finite, (e) <strong>the</strong>re is a triangle equivalence D sg (R) ω ∼ = (per(B)/Dfd (B)) ω . Theorem 7 (a–d) are obtained by applying Theorem 3, and part (e) needs more work. This <strong>the</strong>orem was proved by Thanh<strong>of</strong>fer de Völcsey and Van den Bergh in [15] for R being a local complete commutative Gorenstein k-algebra with isolated singularity. As an application, <strong>the</strong>y proved <strong>the</strong> following result, which was independently proved by Amiot, Iyama and Reiten. Theorem 8. ([2, 15]) Let d ∈ N. Let G ⊂ SL d (k) be a finite subgroup, acting naturally on S = k [x 1 , . . . , x d ] and let R = S G be <strong>the</strong> ring <strong>of</strong> invariants. Then MCM(R) is a generalized (d − 1)-cluster category in <strong>the</strong> sense <strong>of</strong> Amiot [1] and Guo [9]. References [1] Claire Amiot, Cluster categories for algebras <strong>of</strong> global dimension 2 and quivers with potential, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2525–2590. [2] Claire Amiot, Osamu Iyama, and Idun Reiten, Stable categories <strong>of</strong> Cohen-Macauley modules and cluster categories, arXiv:1104.3658. [3] Alexander A. Beilinson, Joseph Bernstein, and Pierre Deligne, Analyse et topologie sur les espaces singuliers, Astérisque, vol. 100, Soc. Math. France, 1982 (French). [4] Ragnar-Olaf Buchweitz, Maximal Cohen-Macaulay modules and Tate-Cohomology over Gorenstein rings, preprint 1987. [5] Igor Burban and Martin Kalck, Singularity category <strong>of</strong> a non-commutative resolution <strong>of</strong> singularities, arXiv:1103.3936. [6] Edward Cline, Brian Parshall, and Leonard L. Scott, Finite-dimensional algebras and highest weight categories, J. reine ang. Math. 391 (1988), 85–99. –266–
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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
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The 44th S
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Tuesday September 27 8:40-9:30 Atsu
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Conversely, let (T , F) be a stable
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Then T = gen(X) and X is Ext-projec
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DIMENSIONS OF DERIVED CATEGORIES TA
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Notation 4. (1) Let A be an abelian
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of finitely genera
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is the map given b
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(2) Let R be a commutative ring whi
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QUIVER PRESENTATIONS OF GROTHENDIEC
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Proposition 3. Let C be a category,
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Example 11. Let Q be the</s
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and Gr(X ′ ) is given by
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particular, the de
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Proposition 1 and 2 leave us with d
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4. Examples Example 6. Let M = M(A
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EXAMPLE OF CATEGORIFICATION OF A CL
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The aim of <strong
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X ∈ mod Π Q S 1 S 2 S 3 1 ❁
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The previous proposition has a dual
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Computing inductively all t
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Example 23. We have ⎛ ⎞ ⎛ ⎞
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classification of
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quiver Q over K, and let D b (H) <s
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Then Q ∈ Q 17 . Suppose char K
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Moreover the Hochs
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DERIVED AUTOEQUIVALENCES AND BRAID
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3. Periodic Twists We now describe
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We now specialise to the</s
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and τ − n := Ext n Λ(DΛ, −)
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where r v ij := a i a j −a j a i
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ON A DEGENERATION PROBLEM FOR COHEN
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We make several othe</stron
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Let A be a commutative Gorenstein r
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3. Extended orders In the</
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WEAK GORENSTEIN DIMENSION FOR MODUL
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1.2. Introduction. The notion <stro
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e 2 A z 1 −→ X → 0 in mod-A,
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5. Gorenstein algebras In this sect
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ON Ω-PERFECT MODULES AND SEQUENCES
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when R is a Nakayama algebra <stron
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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
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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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Page 231 and 232: GRADED FROBENIUS ALGEBRAS AND QUANT
Page 233 and 234: Definition 2. A graded algebra A is
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Page 237 and 238: References [1] X. W. Chen, Graded s
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Page 241 and 242: (2.3) soc( R R) ∼ = k⊕ s i T i
Page 243 and 244: principal ideal Rr ⊂ ker ϖ. Thus
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Page 249 and 250: 4.1. Fourier Transform. Gleason’s
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Page 259 and 260: ψ : ε(A) → Â. In this way, C
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Page 271 and 272: ALGEBRAIC STRATIFICATIONS OF DERIVE
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Page 277 and 278: RECOLLEMENTS GENERATED BY IDEMPOTEN
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Page 283 and 284: INTRODUCTION TO REPRESENTATION THEO
Page 285 and 286: is exact. Since E n ∼ = En+r , Mi
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Page 293 and 294: R : CM(R⊗ k V ) → CM(R) defined
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Page 297 and 298: SUBCATEGORIES OF EXTENSION MODULES
Page 299 and 300: Lemma 6. Let S 1 and S 2 be Serre s
Page 301 and 302: In the first row <
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