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

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[5] , Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, 589–632. [6] , Semicanonical bases and preprojective algebras. II. A multiplication formula, Compos. Math. 143 (2007), no. 5, 1313–1334. [7] , Partial flag varieties and preprojective algebras, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 3, 825–876. [8] G. Lusztig, Canonical basis arising form quantum universal enveloping algebras, J. AMS 3 (1991), 9–36. [9] , Quivers, perverse sheaves, and quantized enveloping algebras, J. AMS 4(2) (1991), 365–421. [10] M. Kashiwara and Y. Saito, Geometric construction <strong>of</strong> crystal bases, Duke Math J. 89 (1997), 9–36. [11] Y. Kimura, Quantum unipotent subgroup and dual canonical basis, arXiv:1010.4242. [12] M. Reineke, On <strong>the</strong> coloured graph structure <strong>of</strong> Lusztig s canonical basis, Math. Ann. 307 (1997), 705–723. [13] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Note in Math. 1099 (1980), Springer. [14] , Hall algebras and quantum groups, Invent. Math. 101 (1990), 583–591. [15] Y. Saito, PBW bases <strong>of</strong> quantized universal enveloping algebras, Publ. RIMS. 30(2) (1994), 209–232. [16] , , (2006), 63–117. [17] , Mirković-Vilonen polytopes and a quiver construction <strong>of</strong> crystal basis in type A, IMRN (2011), doi:10.1093/imrn/rnr173. [18] A. Savage, Geometric and combinatorial realization <strong>of</strong> crystal graphs, Alg. Rep. Theory 9 (2006), 161–199. Graduate School <strong>of</strong> Ma<strong>the</strong>matical Sciences University <strong>of</strong> Tokyo Meguro-ku, Tokyo 153-8914 JAPAN E-mail address: yosihisa@ms.u-tokyo.ac.jp –195–
MATRIX FACTORIZATIONS, ORBIFOLD CURVES AND MIRROR SYMMETRY ATSUSHI TAKAHASHI ( ) Abstract. Mirror symmetry is now understood as a categorical duality between algebraic geometry and symplectic geometry. One <strong>of</strong> our motivations is to apply some ideas <strong>of</strong> mirror symmetry to singularity <strong>the</strong>ory in order to understand various mysterious correspondences among isolated singularities, root systems, Weyl groups, Lie algebras, discrete groups, finite dimensional algebras and so on. In my talk, I explained <strong>the</strong> homological mirror symmetry conjecture between orbifold curves and cusp singularities via Orlov type semi-orthogonal decompositions. I also gave a summary <strong>of</strong> our results on categories <strong>of</strong> maximally-graded matrixfactorizations, in particular, on <strong>the</strong> existence <strong>of</strong> full strongly exceptional collections which gives triangulated equivalences to derived categories <strong>of</strong> finite dimensional modules over finite dimensional algebras. 1. . . . . 14 Arnold 3 Arnold Gablielov Gablielov 14 Arnold f(x, y, z) 0 ∈ C 3 f Milnor Lagrangian distinguish basis A ∞ - Fuk → (f) D b Fuk → (f) f D b Fuk → (f) full exceptional collection . f(x, y, z) HMF L f S (f) S := C[x, y, z]L f f HMF L f S (f) Calabi–Yau Landau–Ginzburg –196–
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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 <
Page 159 and 160: Then there is an i
Page 161 and 162: • ( ˜F , ˜m) = ({ (1, 3), (2, 3
Page 163 and 164: Corollary 7 ([16, Theorem 3.4]). Th
Page 165 and 166: We have the direct
Page 167 and 168: We say a CW complex is regular, if
Page 169 and 170: SHARP BOUNDS FOR HILBERT COEFFICIEN
Page 171 and 172: Let us briefly note how this paper
Page 173 and 174: whence the set Λ
Page 175 and 176: Proof of</
Page 177 and 178: We have Q·H j m(A/(a 1 , a 2 , ·
Page 179 and 180: Let B = A/U(a d−1 ). We t
Page 181 and 182: • quiver Γ = (I, Ω) I Ω
Page 183 and 184: B = (B τ ) relations ρ 1 , · ·
Page 185 and 186: Definition 1. double quiver ˜Γ p
Page 187 and 188: ◦ side Γ = (I, Ω) cycle (ac
Page 189 and 190: Definition 6. n × n A(Γ Dyn ) =
Page 191 and 192: (2) P Lie theory
Page 193 and 194: Example 15. Γ loop quiver λ ∈
Page 195 and 196: P (Γ)-module i-th radical B →
Page 197 and 198: B ∈ Λ(d) ε ∗ i (B) := dim C
Page 199 and 200: I-graded vector space V (d) (B τ )
Page 201 and 202: A ∗(i) l (a) = ∑n+1 t=l+1 (a i,
Page 203 and 204: wt(a) = (−d 1 , −d 2 , −d 3 )
Page 205 and 206: Ω Q Ω ( B Ω ; wt, ε i , ϕ
Page 207 and 208: “Λ(d) G(d)-” Definition 25.
Page 209: (( ( ) ( ) 0 s 0 0 0 , , , s) ( u 0
Page 213 and 214: ( ∂f Jac(f t t ) := C[x 1 , . . .
Page 215 and 216: Definition 14. CM L f (R f ) Ob(C
Page 217 and 218: 4. Dolgachev Proposition 24 ([1]
Page 219 and 220: ✤ ✤ ✤ ✤ ✤ ✤ γ 1 +γ 2
Page 221 and 222: (f, G f ) Dolgachev A Gf f Berg
Page 223 and 224: ON A GENERALIZATION OF COSTABLE TOR
Page 225 and 226: (2) P/t(P ) is σ-projective for an
Page 227 and 228: (3) P is a maximal σ-coessential e
Page 229 and 230: (2)→(1): Let σ be epi-preserving
Page 231 and 232: GRADED FROBENIUS ALGEBRAS AND QUANT
Page 233 and 234: Definition 2. A graded algebra A is
Page 235 and 236: In this case, ν ∈ Aut k E induce
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
Page 245 and 246: By Example 3, the
Page 247 and 248: weight wt(x) = d(x, 0) of</
Page 249 and 250: 4.1. Fourier Transform. Gleason’s
Page 251 and 252: 5. The Extension Problem In this se
Page 253 and 254: Because R is assumed to be Frobeniu
Page 255 and 256: is injective for every finite left
Page 257 and 258: linear code defined by η;
Page 259 and 260: 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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