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

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11. , On degenerations <strong>of</strong> Cohen-Macaulay modules, Journal <strong>of</strong> Algebra 248 (2002), 272–290. 12. , On degenerations <strong>of</strong> modules, Journal <strong>of</strong> Algebra 278 (2004), 217–226. 13. , Degeneration and G-dimension <strong>of</strong> modules, Lecture Notes in Pure and Applied Ma<strong>the</strong>matics vol. 244, ’Commutative Algebra’ Chapman and Hall/CRC (2006), 259 – 265. 14. , Universal lifts <strong>of</strong> chain complexes over non-commutative parameter algebras, J. Math. Kyoto Univ. 48 (2008), no. 4, 793–845. 15. , Stable degenerations <strong>of</strong> Cohen-Macaulay modules, J. Algebra 332 (2011), 500–521. 16. G. Zwara, A degeneration-like order for modules, Arch. Math. 71 (1998), 437–444. 17. , Degenerations <strong>of</strong> finite dimensional modules are given by extensions, Compositio Math. 121 (2000), 205–218. Department <strong>of</strong> Ma<strong>the</strong>matics Graduate School <strong>of</strong> Natural Science and Techonology Okayama University Okayama, 700-8530 JAPAN E-mail address: yoshino@math.okayama-u.ac.jp –281–
SUBCATEGORIES OF EXTENSION MODULES RELATED TO SERRE SUBCATEGORIES TAKESHI YOSHIZAWA Abstract. We consider subcategories consisting <strong>of</strong> <strong>the</strong> extensions <strong>of</strong> modules in two given Serre subcategories to find a method <strong>of</strong> constructing Serre subcategories <strong>of</strong> <strong>the</strong> module category. We shall give a criterion for this subcategory to be a Serre subcategory. 1. Introduction Let R be a commutative Noe<strong>the</strong>rian ring. We denote by R-Mod <strong>the</strong> category <strong>of</strong> R- modules and by R-mod <strong>the</strong> full subcategory consisting <strong>of</strong> finitely generated R-modules. In [2], P. Gabriel showed that one has lattice isomorphisms between <strong>the</strong> set <strong>of</strong> Serre subcategories <strong>of</strong> R-mod, <strong>the</strong> set <strong>of</strong> Serre subcategories <strong>of</strong> R-Mod which are closed under arbitrary direct sums and <strong>the</strong> set <strong>of</strong> specialization closed subsets <strong>of</strong> Spec (R). By this result, Serre subcategories <strong>of</strong> R-mod are classified. However, it has not yet classified Serre subcategories <strong>of</strong> R-Mod. In this paper, we shall give a way <strong>of</strong> constructing Serre subcategories <strong>of</strong> R-Mod by considering subcategories <strong>of</strong> extension modules related to Serre subcategories. 2. The definition <strong>of</strong> a subcategory <strong>of</strong> extension modules by Serre subcategories We assume that all full subcategories <strong>of</strong> R-Mod are closed under isomorphisms. We recall that a subcategory S <strong>of</strong> R-Mod is said to be Serre subcategory if <strong>the</strong> following condition is satisfied: For any short exact sequence 0 → L → M → N → 0 <strong>of</strong> R-modules, it holds that M is in S if and only if L and N are in S. In o<strong>the</strong>r words, S is called a Serre subcategory if it is closed under submodules, quotient modules and extensions. We give <strong>the</strong> definition <strong>of</strong> a subcategory <strong>of</strong> extension modules by Serre subcategories. Definition 1. Let S 1 and S 2 be Serre subcategories <strong>of</strong> R-Mod. We denote by (S 1 , S 2 ) a subcategory consisting <strong>of</strong> R-modules M with a short exact sequence 0 → X → M → Y → 0 <strong>of</strong> R-modules where X is in S 1 and Y is in S 2 , that is ⎧ ∣ ⎨ ∣∣∣∣∣ <strong>the</strong>re are X ∈ S 1 and Y ∈ S 2 such that (S 1 , S 2 ) = ⎩ M ∈ R-Mod 0 → X → M → Y → 0 is a short exact sequence. The detailed version <strong>of</strong> this paper has been submitted for publication elsewhere. –282– ⎫ ⎬ ⎭ .
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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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Page 271 and 272: ALGEBRAIC STRATIFICATIONS OF DERIVE
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Page 283 and 284: INTRODUCTION TO REPRESENTATION THEO
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