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

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V ′′ i-th radical I-graded vector space V ′′ ∼ = V (d ′′ ) d ′′ := d − ls(i) I-graded vector space V (d ′′ ) P (Γ)-module structure 0 → V (d ′′ ) −→ φ′′ V (d) −→ φ′ S(i) ⊕l → 0 (4.3.1) P (Γ)-module P (Γ)-module V (d ′′ ) Λ(d ′′ ) B ′′ Λ(d) ∋ B → B ′′ ∈ Λ(d ′′ ) V (d) i-th radical B → B ′′ V ′′ ∼ = V (d ′′ ) i-th radical V (d ′′ ) P (Γ)-module structure B ′′ ∈ Λ(d ′′ ) (4.3.1) I-graded vector space V (d) P (Γ)-module structure B ∈ Λ(d) V (d ′′ ) P (Γ)-module structure B ′′ ∈ Λ(d ′′ ) φ ′′ : V (d ′′ ) → V (d) P (Γ)-module P (Γ)-module Imφ ′′ B-stable 18 V (d)/V (d ′′ ) ∼ = S(i) ⊕l P (Γ)-module φ ′ : V (d) → S(i) ⊕l P (Γ)-module (B, φ ′ , φ ′′ ) B ∈ Λ(d) Imφ ′′ B- stable (B, φ ′ , φ ′′ ) Λ(d; d ′′ ) Λ(d; d ′′ ) Λ(d)Λ(d ′′ ) q 1 : Λ(d; d ′′ ) ∋ (B, φ ′ , φ ′′ ) ↦→ B ∈ Λ(d), q 2 : Λ(d; d ′′ ) ∋ (B, φ ′ , φ ′′ ) ↦→ B ′′ ∈ Λ(d ′′ ) Λ(d ′′ ) q 2 ←− Λ(d; d ′′ ) q 1 −→ Λ(d) (4.3.2) q 1 , q 2 B → B ′′ Propositi<strong>on</strong> 16 (Kashiwara-S [10]). (4.3.2) ( ) ∼ IrrΛ(d) → ( IrrΛ(d ′′ ) ) i,0 i,l 18 φ ′′ I-graded vector ( space Imφ ′′ V (d) I-graded vector subspace Imφ ′′ = ⊕ ) i∈I Imφ ′′ τ ∈ H i Imφ ′′ B-stable B τ ( Imφ ′′ ) out(τ) ⊂ ( Imφ ′′) in(τ) –179–
P (Γ)-module i-th radical B → B ′′ well-defined well-defined q 1 , q 2 ([10]) ẽ max i : ( IrrΛ(d) ) ∼ → ( IrrΛ(d ′′ ) ) i,l i,0 ẽ i ẽ i { (Λ(d) ) ẽ max i −→ ( Λ(d − ls(i)) ) (ẽ max i ) −→ −1 ( Λ(d − s(i)) ) if l > 0, ẽ i : ( ) i,l i,0 i,l−1 Λ(d) −→ {0} if l = 0. i,0 ẽ i “” ẽ i P (Γ)-module i-th top generic 19 i-th top 0 l = 0 0 B = ⊔ d∈Z IrrΛ(d) comp<strong>on</strong>ent ẽ I i ≥0 ẽ i : B → B ⊔ {0} well-defined ◦ ˜f i ẽ max i ˜f i : ( Λ(d) ) i,l ẽ max i −→ ( Λ(d − ls(i)) ) i,0 (ẽ max i ) −1 −→ ( Λ(d + s(i)) ) i,l+1 . l 0 ˜f i “” P (Γ)-module i-th top generic l ˜f i ˜f i : B → B 19 P (Γ)-module i-th top generic ẽ i ( ) ∼ IrrΛ(d) → ( IrrΛ(d − s(i)) ) i,l−1 i,l ( IrrΛ(d ′′ ) ) i,0 –180–
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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
Page 143 and 144: Now we will show that the</
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Page 147 and 148: THE NOETHERIAN PROPERTIES OF THE RI
Page 149 and 150: Proposition 4 (Paper III, Propositi
Page 151 and 152: HOCHSCHILD COHOMOLOGY OF QUIVER ALG
Page 153 and 154: (4) In [10], Green, Snashall and So
Page 155 and 156: 4. Quiver algebras defined by two c
Page 157 and 158: [6] L. Evens, The cohomology ring <
Page 159 and 160: Then there is an i
Page 161 and 162: • ( ˜F , ˜m) = ({ (1, 3), (2, 3
Page 163 and 164: Corollary 7 ([16, Theorem 3.4]). Th
Page 165 and 166: We have the direct
Page 167 and 168: We say a CW complex is regular, if
Page 169 and 170: SHARP BOUNDS FOR HILBERT COEFFICIEN
Page 171 and 172: Let us briefly note how this paper
Page 173 and 174: whence the set Λ
Page 175 and 176: Proof of</
Page 177 and 178: We have Q·H j m(A/(a 1 , a 2 , ·
Page 179 and 180: Let B = A/U(a d−1 ). We t
Page 181 and 182: • quiver Γ = (I, Ω) I Ω
Page 183 and 184: B = (B τ ) relations ρ 1 , · ·
Page 185 and 186: Definition 1. double quiver ˜Γ p
Page 187 and 188: ◦ side Γ = (I, Ω) cycle (ac
Page 189 and 190: Definition 6. n × n A(Γ Dyn ) =
Page 191 and 192: (2) P Lie theory
Page 193: Example 15. Γ loop quiver λ ∈
Page 197 and 198: B ∈ Λ(d) ε ∗ i (B) := dim C
Page 199 and 200: I-graded vector space V (d) (B τ )
Page 201 and 202: A ∗(i) l (a) = ∑n+1 t=l+1 (a i,
Page 203 and 204: wt(a) = (−d 1 , −d 2 , −d 3 )
Page 205 and 206: Ω Q Ω ( B Ω ; wt, ε i , ϕ
Page 207 and 208: “Λ(d) G(d)-” Definition 25.
Page 209 and 210: (( ( ) ( ) 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 , . . .
Page 215 and 216: Definition 14. CM L f (R f ) Ob(C
Page 217 and 218: 4. Dolgachev Proposition 24 ([1]
Page 219 and 220: ✤ ✤ ✤ ✤ ✤ ✤ γ 1 +γ 2
Page 221 and 222: (f, G f ) Dolgachev A Gf f Berg
Page 223 and 224: ON A GENERALIZATION OF COSTABLE TOR
Page 225 and 226: (2) P/t(P ) is σ-projective for an
Page 227 and 228: (3) P is a maximal σ-coessential e
Page 229 and 230: (2)→(1): Let σ be epi-preserving
Page 231 and 232: GRADED FROBENIUS ALGEBRAS AND QUANT
Page 233 and 234: Definition 2. A graded algebra A is
Page 235 and 236: In this case, ν ∈ Aut k E induce
Page 237 and 238: References [1] X. W. Chen, Graded s
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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By Example 3, the
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weight wt(x) = d(x, 0) of</
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4.1. Fourier Transform. Gleason’s
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5. The Extension Problem In this se
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Because R is assumed to be Frobeniu
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is injective for every finite left
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linear code defined by η;
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ψ : ε(A) → Â. In this way, C
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REALIZING STABLE CATEGORIES AS DERI
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2.1. Positively graded self-injecti
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By the above resul
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Next we consider the</stron
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(2) There exists a triangle-equival
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ALGEBRAIC STRATIFICATIONS OF DERIVE
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The leaves of <str
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Step 2: Let A be a finite-dimension
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RECOLLEMENTS GENERATED BY IDEMPOTEN
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(a) the adjoint tr
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Cluster-tilting the</strong
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INTRODUCTION TO REPRESENTATION THEO
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is exact. Since E n ∼ = En+r , Mi
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field of V . We de
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(Proof) ( ) p 0 0
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we have a sequence of</stro
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R : CM(R⊗ k V ) → CM(R) defined
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P = P ′ t by t is a projective R
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SUBCATEGORIES OF EXTENSION MODULES
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Lemma 6. Let S 1 and S 2 be Serre s
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In the first row <
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