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

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Question. When is mod Z A triangle-equivalent to <strong>the</strong> derived category D b (modΛ) for some algebra Λ ? The following result is main <strong>the</strong>orem <strong>of</strong> this paper which gives <strong>the</strong> complete answer to our question. Theorem 1. Let A be a positively graded self-injective algebra. Then <strong>the</strong> following are equivalent. (1) The global dimension <strong>of</strong> A 0 is finite. (2) There exists an algebra Λ, and a triangle-equivalence (1.2) mod Z A ≃ D b (modΛ). The aim <strong>of</strong> <strong>the</strong> rest <strong>of</strong> this paper is to give an explanation <strong>of</strong> <strong>the</strong> pro<strong>of</strong> <strong>of</strong> Theorem 1, and some examples. Our plan is as follows. In Section 2, we give two preliminaries. First we recall that mod Z A for a positively graded algebra A is a Frobenius category, and so its stable category mod Z A is an algebraic triangulated category. Secondly we give an explanation <strong>of</strong> Keller’s tilting <strong>the</strong>orem. Our approach to <strong>the</strong> question is using Keller’s titling <strong>the</strong>orem for algebraic triangulated categories. B. Keller [7] introduced and investigated differential graded categories and its derived categories. In his work, it was determine when is an algebraic triangulated category triangle-equivalent to <strong>the</strong> derived category <strong>of</strong> some algebra by <strong>the</strong> existence <strong>of</strong> tilting objects (tilting <strong>the</strong>orem). In Section 3, we apply Keller’s tilting <strong>the</strong>orem to our study. In Section 3, we give an outline <strong>of</strong> <strong>the</strong> pro<strong>of</strong> <strong>of</strong> Theorem 1. We omit <strong>the</strong> pro<strong>of</strong> <strong>of</strong> (2) ⇒ (1). We give pro<strong>of</strong>s (1) ⇒ (2). We start from finding a concrete tilting object in mod Z A which has ”good” properties. After finding it, we show two ways to prove (1) ⇒ (2). The first pro<strong>of</strong> is based on Keller’s tilting <strong>the</strong>orem, namely we entrust with constructing <strong>the</strong> triangle-equivalence (1.2). The second pro<strong>of</strong> is direct more than <strong>the</strong> first one, namely we construct <strong>the</strong> triangle-equivalence (1.2) explicitly. In Section 4, we give some examples <strong>of</strong> Theorem 1. In particular as an application <strong>of</strong> our main <strong>the</strong>orem, we show Happel’s result, and its generalization shown by X-W Chen [2]. Throughout this paper, let K be an algebraically closed field. An algebra means a finite dimensional associative algebra over K. We always deal with finitely generated right modules over algebras. For an algebra Λ, we denote by modΛ <strong>the</strong> category <strong>of</strong> Λ- modules, projΛ <strong>the</strong> category <strong>of</strong> projective Λ-modules. The same notations is used for graded case. For an additive category A, we denote by K b (A) <strong>the</strong> homotopy category <strong>of</strong> bounded complexes <strong>of</strong> A. For an abelian category A, we denote by D b (A) <strong>the</strong> bounded derived category <strong>of</strong> A. 2. Preliminaries In this section, we recall basic facts about representation <strong>the</strong>ory <strong>of</strong> a positively graded algebras, and tilting <strong>the</strong>orem for algebraic triangulated categories for <strong>the</strong> readers convenient. –247–
2.1. Positively graded self-injective algebras. In this subsection, our aim is to recall that <strong>the</strong> stable category <strong>of</strong> Z-graded modules over positively graded self-injective algebras are algebraic triangulated categories. Most <strong>of</strong> results stated here are due to Gordon-Green [3, 4]. In details, readers should refer to [3, 4]. We start with setting notations. Let A = ⊕ i≥0 A i be a positively graded self-injective algebra. We say that an A-module is Z-gradable if it can be regarded as a Z-graded A-module. For a Z-graded A-module X, we write X i <strong>the</strong> i-degree part <strong>of</strong> X. We denote by mod Z A <strong>the</strong> category <strong>of</strong> Z-graded A-modules. For Z-graded A-modules X and Y , we write Hom A (X, Y ) 0 <strong>the</strong> morphism space in mod Z A from X to Y . We recall that mod Z A has two important functors. The first one is <strong>the</strong> grading shift functor. For i ∈ Z, we denote by (i) : mod Z A → mod Z A <strong>the</strong> grading shift functor, that is defined as follows. For a Z-graded A-module X, • X(i) := X as an A-module, • Z-grading on X(i) is defined by X(i) j := X j+i for any j ∈ Z. This is an aut<strong>of</strong>unctor on mod Z A whose inverse is (−i). The second one is <strong>the</strong> K-dual. It is already known that <strong>the</strong>re is <strong>the</strong> standard duality D := Hom K (−, K) : modA → modA op . This functor induces <strong>the</strong> following duality. For a Z-graded A-module X, we regard DX as a Z-graded A op -module by defining (DX) i := D(X −i ) for any i ∈ Z. By this observation, we have <strong>the</strong> duality D : mod Z A → mod Z A op . Next we recall a few important facts about objects and morphism spaces in mod Z A. The following results are two <strong>of</strong> <strong>the</strong> most basic categorical properties <strong>of</strong> mod Z A. Proposition 2. mod Z A is a Hom-finite Krull-Schmidt category Proposition 3. [3, Theorem 3.2. Theorem 3.3.] The following assertions hold. (1) A Z-graded A-module is indecomposable in mod Z A if and only if it is an indecompsable A-module. (2) Any direct summand <strong>of</strong> a Z-gradable A-module is also Z-gradable. (3) Let X and Y be indecomposable Z-graded A-modules. If X and Y are isomorphic to each o<strong>the</strong>r in modA, <strong>the</strong>n <strong>the</strong>re exists i ∈ Z such that X and Y (i) are isomorphic to each o<strong>the</strong>r in mod Z A. Next we recall what are projective objects and injective objects in mod Z A. A is naturally regarded as a Z-graded A-module. By Proposition 3 (2), any projective A-modules are Z-gradable. Moreover it is easy to check that all projective object in mod Z A is given by projective A-modules. By <strong>the</strong> standard duality, <strong>the</strong> same argument hold for injective objects in mod Z A. Proposition 4. A complete list <strong>of</strong> indecomposable projective objects in mod Z A is given by {P (i) | i ∈ Z, P is an indecomposable projective A-module}. –248–
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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
Page 211 and 212: MATRIX FACTORIZATIONS, ORBIFOLD CUR
Page 213 and 214: ( ∂f Jac(f t t ) := C[x 1 , . . .
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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 285 and 286: is exact. Since E n ∼ = En+r , Mi
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Page 299 and 300: Lemma 6. Let S 1 and S 2 be Serre s
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