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

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[4] Lidia Angeleri Hügel, Steffen Koenig, Qunhua Liu, and Dong Yang, Derived simple algebras and restrictions <strong>of</strong> recollements <strong>of</strong> derived module categories, preprint, 2012. [5] Alexander A. Beilinson, Joseph Bernstein, and Pierre Deligne, Analyse et topologie sur les espaces singuliers, Astérisque, vol. 100, Soc. Math. France, 1982 (French). [6] Hongxing Chen and Changchang Xi, Good tilting modules and recollements <strong>of</strong> derived module categories, Proc. Lon. Math. Soc, to appear. Also arXiv:1012.2176. [7] , Stratifications <strong>of</strong> derived categories from tilting modules over tame hereditary algebras, preprint, 2011, arXiv:1107.0444. [8] Dieter Happel, A family <strong>of</strong> algebras with two simple modules and Fibonacci numbers, Arch. Math. (Basel) 57 (1991), no. 2, 133–139. [9] , Reduction techniques for homological conjectures, Tsukuba J. Math. 17 (1993), no. 1, 115– 130. [10] Bernhard Keller, Introduction to abelian and derived categories, Representations <strong>of</strong> reductive groups, Publ. Newton Inst., Cambridge Univ. Press, Cambridge, 1998, pp. 41–61. [11] , Invariance and localization for cyclic homology <strong>of</strong> DG algebras, J. Pure Appl. Algebra 123 (1998), no. 1-3, 223–273. [12] Steffen Koenig and Hiroshi Nagase, Hochschild cohomologies and stratifying ideals, J. Pure Appl. Algebra 213 (2009), no. 4, 886–891. [13] Steffen König, Tilting complexes, perpendicular categories and recollements <strong>of</strong> derived module categories <strong>of</strong> rings, J. Pure Appl. Algebra 73 (1991), no. 3, 211–232. [14] Qunhua Liu and Jorge Vitória, t-structures via recollements for piecewise hereditary algebras, J. Pure Appl. Algebra 216 (2012), 837–849. [15] Qunhua Liu and Dong Yang, Blocks <strong>of</strong> group algebras are derived simple, in press, doi:10.1007/s00209-011-0963-y. Also arXiv:1104.0500. [16] Pedro Nicolás and Manuel Saorín, Parametrizing recollement data for triangulated categories, J. Algebra 322 (2009), no. 4, 1220–1250. [17] Alfred Wiedemann, On stratifications <strong>of</strong> derived module categories, Canad. Math. Bull. 34 (1991), no. 2, 275–280. Universität Stuttgart Institut für Algebra und Zahlen<strong>the</strong>orie Pfaffenwaldring 57, D-70569 Stuttgart, Germany E-mail address: dongyang2002@gmail.com –261–
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 <strong>of</strong> Thanh<strong>of</strong>fer de Völcsey and Van den Bergh. Key Words: Recollement, Singularity category, Non-commutative resolution. 2010 Ma<strong>the</strong>matics Subject Classification: 16E35, 16E45, 16G50. In [15] Thanh<strong>of</strong>fer de Völcsey and Van den Bergh showed that <strong>the</strong> stable category <strong>of</strong> maximal Cohen–Macaulay modules over a local complete commutative Gorenstein algebra with isolated singularity can be realized as <strong>the</strong> triangle quotient <strong>of</strong> <strong>the</strong> perfect derived category by <strong>the</strong> finite-dimensional category <strong>of</strong> a certain nice dg algebra constructed from <strong>the</strong> given Gorenstein algebra. We generalises and improves <strong>the</strong>ir construction by studying recollements <strong>of</strong> derived categories generated by idempotents. 1. Recollements generated by idempotents Let k be a field, let A be a k-algebra and e ∈ A be an idempotent. Let D(A) denote <strong>the</strong> (unbounded) derived category <strong>of</strong> <strong>the</strong> category <strong>of</strong> right modules over A. This is a triangulated category with shift functor Σ being <strong>the</strong> shift <strong>of</strong> complexes. Consider <strong>the</strong> following standard diagram (1.1) i ∗ j ! D(A/AeA) i ∗ =i ! D(A) j ! =j ∗ D(eAe) where i ! i ∗ =? ⊗ L A A/AeA, j ! =? ⊗ L eAe eA, i ∗ = RHom A/AeA (A/AeA, ?), j ! = RHom A (eA, ?), i ! =? ⊗ L A/AeA A/AeA, j ∗ =? ⊗ L A Ae, i ! = RHom A (A/AeA, ?), j ∗ = RHom eAe (Ae, ?). One asks when this diagram is a recollement ([3]), i.e. <strong>the</strong> following conditions hold (1) (i ∗ , i ∗ = i ! , i ! ) and (j ! , j ! = j ∗ , j ∗ ) are adjoint triples; (2r) j ! and j ∗ are fully faithful; (2l) i ∗ = i ! is fully faithful; The detailed version <strong>of</strong> this paper will be submitted for publication elsewhere. –262– j ∗
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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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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
Page 239 and 240: The topics covered are ring-<strong
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
Page 261 and 262: REALIZING STABLE CATEGORIES AS DERI
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Page 267 and 268: Next we consider the</stron
Page 269 and 270: (2) There exists a triangle-equival
Page 271 and 272: ALGEBRAIC STRATIFICATIONS OF DERIVE
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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
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