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

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Now we keep <strong>the</strong> notation as above and put Γ := End A (T ) 0 . <strong>the</strong> endomorphism algebra <strong>of</strong> T in mod Z A. homological property if so does A 0 . Theorem 13. If A 0 has finite global dimension, <strong>the</strong>n so does Γ. Now we ready to prove Theorem 1 (1) ⇒ (2). This endomorphism algebra Γ has a nice Theorem 14. Under <strong>the</strong> above setting, <strong>the</strong> following assertions hold. (1) There exists a triangle-equivalence thickT −→ K b (projΓ). (2) If A 0 has finite global dimension, <strong>the</strong>n <strong>the</strong>re exists a triangle-equivalence mod Z A −→ D b (modΓ). Pro<strong>of</strong>. (1) By Theorem 10 and Theorem 12 (1), we have <strong>the</strong> triangle-equivalence thickT −→ K b (projΓ). (2) We assume that A 0 has finite global dimension. First by Theorem 10 and Theorem 12 (2), we have <strong>the</strong> triangle-equivalence mod Z A −→ K b (projΓ). Next by Theorem 13, <strong>the</strong> natural triangle-functor K b (projΛ) −→ D b (modΓ) is an equivalence. Finally by composing <strong>the</strong>se equivalences, we have a triangle-equivalence mod Z A −→ D b (modΓ). In <strong>the</strong> above pro<strong>of</strong>, <strong>the</strong> triangle-equivalence mod Z A → D b (modΓ) was given by <strong>the</strong> existence <strong>of</strong> tilting object T in mod Z A and Keller’s Theorem 10 automatically. In <strong>the</strong> rest <strong>of</strong> this section, we construct a triangle-equivalence D b (modΓ) → mod Z A by derived tensor functor directly. To construct <strong>the</strong> triangle-equivalence, first we want to consider <strong>the</strong> derived tensor functor − L ⊗ Γ T : D b (modΓ) → D b (mod Z A). However Γ does not act on T naturally since Γ is defined by <strong>the</strong> morphism space in <strong>the</strong> stable category mod Z A. To solve this problem, we give <strong>the</strong> description <strong>of</strong> T in mod Z A below. The description allow us to realize Γ as <strong>the</strong> morphism space in <strong>the</strong> category mod Z A. Proposition 15. T is decomposed as T = T ⊕ P where T is a direct sum <strong>of</strong> all indecomposable non-projective direct summand <strong>of</strong> T . Then <strong>the</strong> following assertions hold. (1) T is in mod Z A. (2) T and T are isomorphic to each o<strong>the</strong>r in mod Z A. (3) There exists an algebra isomorphism Γ ≃ End A (T ) 0 . Let T = T ⊕P be <strong>the</strong> decomposition which was given in Proposition 15. By Proposition 15 (3), T is regarded as a Z-graded Γ op ⊗ K A-module naturally. So we have <strong>the</strong> left derived tensor functor − L ⊗ Γ T : D b (modΓ) → D b (mod Z A). –251– □
Next we consider <strong>the</strong> quotient category D b (mod Z A)/K b (proj Z A) <strong>of</strong> D b (mod Z A), and <strong>the</strong> quotient functor D b (mod Z A) −→ D b (mod Z A)/K b (proj Z A). The following triangle-equivalence is <strong>the</strong> realization <strong>of</strong> mod Z A as <strong>the</strong> quotient category D b (mod Z A)/K b (proj Z A). The ungraded version <strong>of</strong> this realization was studied by several authors [1], [8] and [9]. Theorem 16. [9, Theorem 2.1.] The natural embedding mod Z A → D b (mod Z A) induces a triangle-equivalence mod Z A −→ D b (mod Z A)/K b (proj Z A) Now we consider <strong>the</strong> following composition <strong>of</strong> <strong>the</strong> above three functors G : D b (modΓ) − L ⊗ Γ T −−−−→ D b (mod Z A) −→ D b (mod Z A)/K b (proj Z A) −→ mod Z A. where <strong>the</strong> second one is <strong>the</strong> quotient functor, and <strong>the</strong> third one is a quasi-inverse <strong>of</strong> Theorem 16. This is <strong>the</strong> triangle-functor which we want. Theorem 17. Under <strong>the</strong> above setting, <strong>the</strong> following assertions hold. (1) G is fully faithful on K b (projΓ). (2) A 0 has finite global dimension if and only if G is a triangle-equivalence. Pro<strong>of</strong>. (1) It is easy to check that G(Γ) is isomorphic to T , so is isomorphic to T . Moreover by Theorem 12 (1), G induces an isomorphism Hom D b (modΓ)(Γ, Γ[i]) ≃ Hom A (G(Γ), G(Γ)[i]) 0 for any i ∈ Z. By this and thickΓ = K b (projΓ), G is fully faithful on K b (projΓ). Thus G induces a triangle-equivalence K b (projΓ) → thickT . (2) We assume that A 0 has finite global dimension. Then Γ has finite global dimension by Theorem 13. Thus we have thickΓ = D b (modΓ), and so G is fully faithful. Again since A 0 has finite global dimension, we have thickT = mod Z A by Theorem 12 (2). Thus G is dense. We omit <strong>the</strong> pro<strong>of</strong> <strong>of</strong> converse. □ 4. Examples In this section, we show some examples and applications <strong>of</strong> results which was shown in previous section. First example is famous Happel’s result [6], which gives a relationship between representation <strong>the</strong>ory <strong>of</strong> algebras and that <strong>of</strong> <strong>the</strong> trivial extensions. We show it as an application <strong>of</strong> Theorem 1. Example 18. If an algebra is given, <strong>the</strong>n we can always construct a positively graded selfinjective algebra called trivial extension, which contains original algebra as a subalgebra. Let us recall <strong>the</strong> definition <strong>of</strong> trivial extensions. Let Λ be an algebra. The trivial extension A <strong>of</strong> Λ is defined as follows. • A := Λ ⊕ DΛ as an abelian group. –252–
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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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Page 217 and 218: 4. Dolgachev Proposition 24 ([1]
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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
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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 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 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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