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

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• The multiplication on A is defined by (x, f) · (y, g) := (xy, xg + fy). for any x, y ∈ Λ and f, g ∈ DΛ. Here xg and fy is defined by (Λ, Λ)-bimodule structure on DΛ. This A becomes an algebra with respect to <strong>the</strong> above operations. Moreover it is known that A is self-injective. Now we introduce a positively grading on A by ⎧ ⎪⎨ Λ (i = 0), A i := DΛ (i = 1), ⎪⎩ 0 (i ≥ 2). Then obviously A = ⊕ i≥0 A i becomes a positively graded self-injective algebra. Under <strong>the</strong> above setting, we apply Theorem 17 to <strong>the</strong> trivial extension A <strong>of</strong> an algebra Λ. Then we have <strong>the</strong> following Happel’s triangle-equivalence. Theorem 19. [6, Theorem 2. 3.] Under <strong>the</strong> above setting, <strong>the</strong> following are equivalent. (1) Λ has finite global dimension. (2) There exists an triangle-equivalence mod Z A ≃ D b (modΛ). Pro<strong>of</strong>. We calculate T constructed in (3.1) for our setting. Then one can check that T = Λ, and End A (T ) 0 = End A (T ) 0 ≃ Λ. Thus <strong>the</strong> assertion follows from this and Theorem 17. □ Next example is X-W Chen’s result [2] which gives a generalization <strong>of</strong> Happel’s result. Example 20. Chen [2] studied relationship between <strong>the</strong> stable category mod Z A <strong>of</strong> a positively graded self-injective algebra A which has Gorenstein parameter and <strong>the</strong> derived category D b (modΓ) <strong>of</strong> <strong>the</strong> Beilinson algebra Γ <strong>of</strong> A. The notion <strong>of</strong> Gorenstein parameter is defined as follows. Definition 21. Let A be a positively graded self-injective algebra. We say that A has Gorenstein parameter l if SocA is contained in A l . Let A be a positively graded self-injective algebra <strong>of</strong> Gorenstein parameter l. Beilinson algebra Γ <strong>of</strong> A is defined by ⎛ ⎞ A 0 A 1 · · · A l−2 A l−1 A 0 · · · A l−3 A l−2 Γ := . ⎜ . . . . ⎟ ⎝ A 0 A ⎠ . 1 0 A 0 Then Chen showed <strong>the</strong> following result. Theorem 22. [2, Corollary 1.2.] Under <strong>the</strong> above setting, <strong>the</strong> following are equivalent. (1) A 0 has finite global dimension. –253– The
(2) There exists a triangle-equivalence mod Z A ≃ D b (modΓ). As an application <strong>of</strong> Theorem 12, we give a pro<strong>of</strong> <strong>of</strong> <strong>the</strong> above result. Let T be <strong>the</strong> object defined in (3.1), and T <strong>the</strong> direct summand <strong>of</strong> T defined in Proposition 15. We calculate T and <strong>the</strong> endomorphism algebra End A (T ) 0 . Then since A has Gorenstein parameter l, those can be represented as <strong>the</strong> following explicit form. Proposition 23. Under <strong>the</strong> above setting, <strong>the</strong> following assertions hold (1) T = ⊕ l−1 i=0 A(i) ≤0. (2) There exists an algebra isomorphism End A (T ) 0 ≃ Γ. Pro<strong>of</strong>. Since A has Gorenstein parameter l, we have T = ⊕ l−1 i=0 A(i) ≤0 by <strong>the</strong> definition <strong>of</strong> T . ( Moreover it is easy to calculate that <strong>the</strong>re is an algebra isomorphism End A (T ) 0 = ⊕l−1 ) End A i=0 A(i) ≤0 ≃ Γ. □ 0 Pro<strong>of</strong> <strong>of</strong> Theorem 22. The assertion follows from Theorem 17 and Proposition 23. Remark 24. The trivial extensions <strong>of</strong> algebras are positively graded self-injective algebras <strong>of</strong> Gorenstein parameter 1. Thus Theorem 22 contains Theorem 19. Next we show a concrete examples. Example 25. We consider A := K[x]/(x n+1 ), and define a grading on A by deg x := 1. Then A is a positively graded self-injective algebra <strong>of</strong> Gorenstein parameter n. Since <strong>the</strong> global dimension <strong>of</strong> A 0 = K is equal to zero, mod Z A has a tilting object by Theorem 11. Let T be <strong>the</strong> object in mod Z A which was defined in (3.1). Since A has a unique chain A ⊃ (x)/(x n+1 ) ⊃ (x 2 )/(x n+1 ) ⊃ · · · ⊃ (x n )/(x n+1 ) <strong>of</strong> Z-graded A-submodules <strong>of</strong> A, it is easy to calculate that <strong>the</strong> endomorphism algebra Γ := End A (T ) 0 <strong>of</strong> T is isomorphic to <strong>the</strong> n × n upper triangular matrix algebra over K. By Theorem 12, <strong>the</strong>re exists a triangle-equivalence mod Z A ≃ D b (modΓ). We observe <strong>the</strong> above triangle-equivalences by considering <strong>the</strong> case that n = 2, namely <strong>the</strong> case that A = K[x]/(x 3 ). For i = 1, 2, we put X i := (x i )/(x 3 ) <strong>the</strong> Z-graded A- submodule <strong>of</strong> A. Then we have a chain A ⊃ X 1 ⊃ X 2 <strong>of</strong> Z-graded A-submodules <strong>of</strong> A. It is known that {X i (j) | i = 1, 2, j ∈ Z} is a complete set <strong>of</strong> indecomposable non-projective Z-graded A-modules. The Auslander-Reiten quiver <strong>of</strong> mod Z A is as follows. X 1 (−2) X 1 (−1) ❄ X 1 X 1 (1) X 1 (2) ❄ ❄❄❄❄❄ ❄ ❄❄❄❄❄ ❄ ❄❄❄❄❄ ❄❄❄❄❄ · · · · · · · · · · · · 88888 X 2 8 88888 (−2) X 2 8 8 88888 (−1) X 2 8 88888 X 2 8 (1) X 2 8 888888 (2) Here dotted arrows mean <strong>the</strong> Auslander-Reiten translation in mod Z A. We can observe that <strong>the</strong> Auslander-Reiten translation coincides with <strong>the</strong> graded shift functor (−1). –254– □
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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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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
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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
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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: Next we consider the</stron
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
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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