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

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Dually a complete list <strong>of</strong> indecomposable injective objects in mod Z A is given by {I(i) | i ∈ Z, I is an indecomposable injective A-module}. If A is self-injective, <strong>the</strong>n mod Z A is a Frobenius category by Proposition 3 and Proposition 4. So in this case, <strong>the</strong> stable category mod Z A has a structure <strong>of</strong> triangulated category by [5]. Lemma 5. If A is self-injective, <strong>the</strong> following assertions hold. (1) mod Z A is a Frobenius category. (2) mod Z A has a structure <strong>of</strong> triangulated category whose shift functor [1] is given by <strong>the</strong> graded cosyzygy functor Ω −1 : mod Z A → mod Z A. 2.2. Tilting <strong>the</strong>orem for algebraic triangulated categories. In this subsection, we recall tilting <strong>the</strong>orem for algebraic triangulated categories which is due to Keller [7]. It is a <strong>the</strong>orem which provides a method for comparison <strong>of</strong> given triangulated category and homotopy category <strong>of</strong> bounded complexes <strong>of</strong> projective modules over some algebra. First let us recall <strong>the</strong> definition <strong>of</strong> algebraic triangulated categories again. Definition 6. A triangulated category T is algebraic if it is triangle-equivalent to <strong>the</strong> stable category <strong>of</strong> some Frobenius category. A class <strong>of</strong> algebraic triangulated categories contains <strong>the</strong> following important examples. Example 7. (1) Let Z be an abelian group, and A a Z-graded self-injective algebra. Then mod Z A is a Frobenius category, and <strong>the</strong> stable category mod Z A is an algebraic triangulated category (Lemma 5). (2) Let Λ be an algebra. The category C b (projΛ) <strong>of</strong> bounded complexes <strong>of</strong> projective Λ-modules can be regarded as a Frobenius category whose stable category is <strong>the</strong> homotopy category K b (projΛ) <strong>of</strong> bounded complexes <strong>of</strong> projective Λ-modules (cf. [5]). In tilting <strong>the</strong>ory, tilting objects which is defined as follows play an important role. Definition 8. Let T be a triangulated category. An object T ∈ T is called a tilting object in T if it satisfies <strong>the</strong> following conditions. (1) Hom T (T, T [i]) = 0 for i ≠ 0. (2) T = thickT . Here thickT is <strong>the</strong> smallest triangulated full subcategory <strong>of</strong> T which contains T , and is closed under direct summands. The following is a typical example <strong>of</strong> tilting objects. Example 9. Let Λ be a ring. Λ can be regarded as a complex which concentrates in degree 0. So Λ is contained in a triangulated category K b (projΛ). It is a tilting object in K b (projΛ). The following result is Keller’s tilting <strong>the</strong>orem which determine when is an algebraic triangulated category triangle-equivalent to K b (projΛ) for some algebra Λ, Theorem 10. [7, Theorem 4.3.] Let T be an algebraic triangulated category. If T has a tilting object T , <strong>the</strong>n <strong>the</strong>re exists a triangle-equivalence up to direct summands T ≃ K b (projEnd T (T )). –249–
By <strong>the</strong> above result, finding tilting objects is a basic problem for <strong>the</strong> study <strong>of</strong> a given algebraic triangulated category. We will consider this problem for Example 7 (1) in <strong>the</strong> next section (Theorem 11). 3. Triangle-equivalences between stable categories and derived categories Throughout this section, let A be a positively graded self-injective algebra. In this section, we discuss triangle-equivalences between <strong>the</strong> stable category mod Z A and derived categories <strong>of</strong> algebras. First we prove Theorem 1 in <strong>the</strong> half <strong>of</strong> this section. We omit <strong>the</strong> pro<strong>of</strong> <strong>of</strong> (2) ⇒ (1). We prove (1) ⇒ (2). We begin <strong>the</strong> pro<strong>of</strong> from giving <strong>the</strong> necessary and sufficient condition for existence <strong>of</strong> tilting objects in <strong>the</strong> stable category mod Z A. The necessary and sufficient condition is described by important homological property <strong>of</strong> <strong>the</strong> subring A 0 <strong>of</strong> A which is stated as follows. Theorem 11. mod Z A has a tilting object if and only if A 0 has finite global dimension. We omit <strong>the</strong> pro<strong>of</strong> <strong>of</strong> only if part <strong>of</strong> Theorem 11. In <strong>the</strong> following, we show <strong>the</strong> pro<strong>of</strong> <strong>of</strong> if part <strong>of</strong> Theorem 11 which is given by constructing a tilting object in mod Z A. To construct it, we consider truncation functors and (−) ≥i : modA → modA (−) ≤i : modA → modA which are defined as follows. For a Z-graded A-module X, X ≥i is a Z-graded sub A-module <strong>of</strong> X defined by { 0 (j < i) (X ≥i ) j := (j ≥ i), and X ≤i is a Z-graded factor A-module X/X ≥i+1 <strong>of</strong> X. Now we define (3.1) X j T := ⊕ i≥0 A(i) ≤0 . which is an object in Mod Z A but not an object in mod Z A. However since A(i) ≤0 = A(i) for enough large i, T can be regarded as an object in mod Z A. Then we have <strong>the</strong> following result. Theorem 12. Under <strong>the</strong> above setting, <strong>the</strong> following assertions hold. (1) T is a tilting object in thickT . (2) If A 0 has finite global dimension, <strong>the</strong>n T is a tilting object in mod Z A. It is proved that T satisfies <strong>the</strong> first condition in Definition 8 with no assumptions for A, and T satisfies <strong>the</strong> second condition in Definition 8 if A 0 has finite global dimension. Then we finish <strong>the</strong> pro<strong>of</strong> <strong>of</strong> if part <strong>of</strong> Theorem 11. □ –250–
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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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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 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 283 and 284: INTRODUCTION TO REPRESENTATION THEO
Page 285 and 286: is exact. Since E n ∼ = En+r , Mi
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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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