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

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Then by Theorem 10 Gr(X) is presented by <strong>the</strong> quiver ⎛ 21 2α (a, 1 1) 22 ♣ ♦ ♦ ♥ ♥ ♠ ♠ ❧ ❦ ❦ ❥ ❥ ❚ ❚ ❙ ❙ ❘ ❘ ◗ (b, 2 1) (a, 1 2) 11 51 Q ′ = 1α ♣ ♦ ♦ ♥ ♥ ♠ ♠ ❧ ❦ ❦ ❥ ❥ ❬ ❬ ❩ ❩ ❨ ❖ ❨ ❳ ❳ ❖ ❲ ◆ (b, 2 2) ❲ ❱ ❯ ▼ ❯ ▼ ❚ ❚▲ ❬ ▲ ❬ ❩ ❩ ❨ ❨ ❳ (c, 1 1) ❢ ❢ ▼ ▼ ❳ ❲ ❲ (d, 3 1) ❢ ❢ ❢ ◆ ❱ ❢ ❢ ❢ ❖ ❯ ❯ ❢ ❢ ❢ ❖ (c,1 2) ❚ ❚ ❢ ❢ ❢ 12 ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❢ ❢ ❢ ▲ 31 52 ▼ ◗ ▼ ❘ (f, 4 1) ◆ (e, 1 1) ❘ ❙ ❖ ❙ ❚ ❖ ❚ ♣ ♦ ♦ ♥ ♥ ♠ ♠ ❧ ❦ ❦ ❥ ❥ 41 ◗ (e, ⎜ 1 2) ❘ ❘ (f, 4 2) 4α ⎝ ❙ ❙ ❚ ❚ ❥ 42 ❥ ❦ ❦ ❧ ♠ ♠ ♥ ♥ ♦ ♦ ♣ with relations R ′ = {π(ba, 1 1) − π(dc, 1 1), π(ba, 1 2) − π(dc, 1 2)} ∪{(a, i y) i α − j (aα)(a, i x) | a : i → j ∈ Q 1 , α : x → y ∈ Q (i) 1 }, where <strong>the</strong> new arrows are presented by broken arrows. a Example 14. Let Q = ( 1 2 ) and I := 〈Q〉. Define functors X, X ′ : I → k-Cat by <strong>the</strong> k-linearizations <strong>of</strong> <strong>the</strong> following quivers in frames and <strong>the</strong> k-functors induced by <strong>the</strong> vertex maps expressed by dotted arrows between <strong>the</strong>m: 5α ⎞ ⎟ ⎠ X : 1 ✼ 2 ✼✼✼✼✼✼✼✼ X(a) α β ✞ ✞✞✞✞✞✞✞✞ 3 X(a) X(1) X ′ : 1 ✼ ✼✼✼✼✼✼✼✼ α β ✞ ✞✞✞✞✞✞✞✞ 2 X ′ (a) 3 X ′ (1) X(a) 1 X(2), X ′ (a) 1 X ′ (a) X ′ (2). Then by Theorem 10 Gr(X) is given by <strong>the</strong> following quiver with no relations ⎛ ⎞ ⎛ ⎞ 11 ❅ 12 11 ❅❅❅❅❅❅ ❅ 12 1α 1β ❅❅❅❅❅❅ 1α 1β 7 777777 (a, 13 1 1) (a, 1 2) , (a, 1 3) 1 α − (a, 1 1), (a, 1 3) 1 β − (a, 1 2) ∼ 7 777777 = 13 , ⎜ ⎝ (a, 1 ⎟ ⎜ 3) ⎠ ⎝ (a, 1 ⎟ 3) ⎠ 21 –21– 21
and Gr(X ′ ) is given by <strong>the</strong> following quiver with a commutativity relation ⎛ ⎞ ⎛ ⎞ 11 ❅ 11 ❅❅❅❅❅❅ ❅ 1α 1β ❅❅❅❅❅❅ 1α 1β 7 777777 12 (a, 1 1) 13 , (a, 1 2) 1 α − (a, 1 1), (a, 1 3) 1 β − (a, 1 1) ∼ 7 777777 = 12 13 . ⎜ ⎟ ⎜ ⎟ ⎝ (a, 1 2) (a, 1 ⎠ ⎝ 3) (a, 1 2) (a, 1 ⎠ 3) 21 21 By using <strong>the</strong> main <strong>the</strong>orem in [3] derived equivalences between X(1) and X ′ (1) and between X(2) and X ′ (2) are glued toge<strong>the</strong>r to have a derived equivalence between Gr(X) and Gr(X ′ ). References [1] Asashiba, H.: A generalization <strong>of</strong> Gabriel’s Galois covering functors and derived equivalences, J. Algebra 334 (2011), 109–149. [2] : Derived equivalences <strong>of</strong> actions <strong>of</strong> a category, arXiv:1111.2239. [3] : Gluing derived equivalences toge<strong>the</strong>r, in preparation. [4] Revêtements étales et groupe fondamental, Springer-Verlag, Berlin, 1971. Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Dirigé par Alexandre Gro<strong>the</strong>ndieck. Augmenté de deux exposés de M. Raynaud, Lecture Notes in Ma<strong>the</strong>matics, Vol. 224. [5] Howie, J. M.: Fundamentals <strong>of</strong> Semigroup Theory, London Ma<strong>the</strong>matical Society Monographs New Series, 12. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. [6] Mac Lane, S.: Categories for <strong>the</strong> Working Ma<strong>the</strong>matician, Graduate Texts in Ma<strong>the</strong>matics, 5.Springer-Verlag, New York,1998. [7] Tamaki, D.: The Gro<strong>the</strong>ndieck construction and gradings for enriched categories, arXiv:0907.0061. [8] Thomason, R. W.: Homotopy colimits in <strong>the</strong> category <strong>of</strong> small categories, Math. Proc. Cambridge Philos. Soc., 85(1), 91–109, 1979. [9] Welker, V., Ziegler, G. M. and Zǐvaljević,R. T.: Homotopy colimits—comparison lemmas for combinatorial applications, J. Reine Angew. Math., 509, 117–149, 1999. Department <strong>of</strong> Ma<strong>the</strong>matics, Faculty <strong>of</strong> Science, Shizuoka University, Suruga-ku, Shizuoka 422-8529 JAPAN Department <strong>of</strong> Information Science and Technology, Graduate School <strong>of</strong> Science and Technology, Shizuoka University, Suruga-ku, Shizuoka 422-8529 JAPAN E-mail address: shasash@ipc.shizuoka.ac.jp E-mail address: f5144005@ipc.shizuoka.ac.jp –22–
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: Example 11. Let Q be the</s
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 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
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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
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
Page 151 and 152:
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
Page 163 and 164:
Corollary 7 ([16, Theorem 3.4]). Th
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We have the direct
Page 167 and 168:
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
Page 231 and 232:
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
Page 247 and 248:
weight wt(x) = d(x, 0) of</
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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
Page 261 and 262:
REALIZING STABLE CATEGORIES AS DERI
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2.1. Positively graded self-injecti
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By the above resul
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
Page 273 and 274:
The leaves of <str
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Step 2: Let A be a finite-dimension
Page 277 and 278:
RECOLLEMENTS GENERATED BY IDEMPOTEN
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(a) the adjoint tr
Page 281 and 282:
Cluster-tilting the</strong
Page 283 and 284:
INTRODUCTION TO REPRESENTATION THEO
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is exact. Since E n ∼ = En+r , Mi
Page 287 and 288:
field of V . We de
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(Proof) ( ) p 0 0
Page 291 and 292:
we have a sequence of</stro
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R : CM(R⊗ k V ) → CM(R) defined
Page 295 and 296:
P = P ′ t by t is a projective R
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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