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

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(2) ⇒ (1) We only have to prove that a subcategory (S 1 , S 2 ) is closed under extensions by Proposition 5. Let 0 → L → M → N → 0 be a short exact sequence <strong>of</strong> R-modules such that L and N are in (S 1 , S 2 ). We shall show that M is also in (S 1 , S 2 ). Since L is in (S 1 , S 2 ), <strong>the</strong>re exists a short exact sequence 0 → S → L → L/S → 0 <strong>of</strong> R-modules where S is in S 1 such that L/S is in S 2 . We consider <strong>the</strong> following push out diagram 0 0 ⏐ ⏐ ↓ ↓ S ⏐ ↓ S ⏐ ↓ 0 −−−→ L −−−→ M −−−→ N −−−→ 0 ⏐ ⏐ ↓ ↓ ‖ 0 −−−→ L/S −−−→ P −−−→ N −−−→ 0 ⏐ ⏐ ↓ ↓ 0 0 <strong>of</strong> R-modules with exact rows and columns. Next, since N is in (S 1 , S 2 ), we have a short exact sequence 0 → T → N → N/T → 0 <strong>of</strong> R-modules where T is in S 1 such that N/T is in S 2 . We consider <strong>the</strong> following pull back diagram 0 0 ⏐ ⏐ ↓ ↓ 0 −−−→ L/S −−−→ P ′ −−−→ T −−−→ 0 ⏐ ⏐ ‖ ↓ ↓ 0 −−−→ L/S −−−→ P −−−→ N −−−→ 0 ⏐ ⏐ ↓ ↓ <strong>of</strong> R-modules with exact rows and columns. N/T ⏐ ↓ N/T ⏐ ↓ 0 0 –285–
In <strong>the</strong> first row <strong>of</strong> <strong>the</strong> second diagram, since L/S is in S 2 and T is in S 1 , P ′ is in (S 2 , S 1 ). Now here, it follows from <strong>the</strong> assumption (2) that P ′ is in (S 1 , S 2 ). Next, in <strong>the</strong> middle column <strong>of</strong> <strong>the</strong> second diagram, we have <strong>the</strong> short exact sequence such that P ′ is in (S 1 , S 2 ) and N/T is in S 2 . Therefore, it follows from Lemma 6 that P is in (S 1 , S 2 ). Finally, in <strong>the</strong> middle column <strong>of</strong> <strong>the</strong> first diagram, <strong>the</strong>re exists <strong>the</strong> short exact sequence such that S is in S 1 and P is in (S 1 , S 2 ). Consequently, we see that M is in (S 1 , S 2 ) by Lemma 6. The pro<strong>of</strong> is completed. □ Corollary 8. A subcategory (S f.g. , S) is a Serre subcategory for a Serre subcategory S <strong>of</strong> R-Mod. Pro<strong>of</strong>. Let S be a Serre subcategory <strong>of</strong> R-Mod. To prove our assertion, it is enough to show that one has (S, S f.g. ) ⊆ (S f.g. , S) by Theorem 7. Let M be in (S, S f.g. ). Then <strong>the</strong>re exists a short exact sequence 0 → Y → M → M/Y → 0 <strong>of</strong> R-modules where Y is in S such that M/Y is in S f.g. . It is easy to see that <strong>the</strong>re exists a finitely generated R-submodule X <strong>of</strong> M such that M = X + Y . Since X ⊕ Y is in (S f.g. , S) and M is a homomorphic image <strong>of</strong> X ⊕ Y , M is in (S f.g. , S) by Proposition 5. □ We note that a subcategory S Artin consisting <strong>of</strong> Artinian R-modules is a Serre subcategory which is closed under injective hulls. (Also see [1, Example 2.4].) Therefore we can see that a subcategory (S, S Artin ) is also Serre subcategory for a Serre subcategory <strong>of</strong> R-Mod by <strong>the</strong> following assertion. Corollary 9. Let S 2 be a Serre subcategory <strong>of</strong> R-Mod which is closed under injective hulls. Then a subcategory (S 1 , S 2 ) is a Serre subcategory for a Serre subcategory S 1 <strong>of</strong> R-Mod. Pro<strong>of</strong>. By Theorem 7, it is enough to show that one has (S 2 , S 1 ) ⊆ (S 1 , S 2 ). We assume that M is in (S 2 , S 1 ) and shall show that M is in (S 1 , S 2 ). Then <strong>the</strong>re exists a short exact sequence 0 → Y → M → X → 0 <strong>of</strong> R-modules where X is in S 1 and Y is in S 2 . Since S 2 is closed under injective hulls, we note that <strong>the</strong> injective hull E R (Y ) <strong>of</strong> Y is also in S 2 . We consider a push out diagram 0 −−−→ Y −−−→ M −−−→ X −−−→ 0 ⏐ ⏐ ↓ ↓ ‖ 0 −−−→ E R (Y ) −−−→ T −−−→ X −−−→ 0 <strong>of</strong> R-modules with exact rows and injective vertical maps. The second exact sequence splits, and we have an injective homomorphism M → X ⊕ E R (Y ). Since <strong>the</strong>re is a short exact sequence 0 → X → X ⊕ E R (Y ) → E R (Y ) → 0 <strong>of</strong> R-modules, <strong>the</strong> R-module X ⊕ E R (Y ) is in (S 1 , S 2 ). Consequently, we see that M is also in (S 1 , S 2 ) by Proposition 5. The pro<strong>of</strong> is completed. □ –286–
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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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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
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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
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weight wt(x) = d(x, 0) of</
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Page 259 and 260: ψ : ε(A) → Â. In this way, C
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Page 269 and 270: (2) There exists a triangle-equival
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 297 and 298: SUBCATEGORIES OF EXTENSION MODULES
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show all

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