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

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a Γ 0 = (I, Ω 0 ) V a (d 0 ) = ( C ← C ← {0} ← {0} ← {0}) ⊕ ({0} ← C ← C ← C ← {0}) ⊕ ({0} ← {0} ← C ← {0} ← {0}) ⊕ ({0} ← {0} ← {0} ← C ← C ) ( d 0 := (1, 2, 2, 2, 1) = −wt(a)). B a = (B τ1 , B τ2 , B τ3 , B τ4 ) B τ1 = ( 1 0 ) ( ) ( ) ( 0 0 1 0 0 , B τ2 = , B 1 0 τ3 = , B 0 0 τ4 = (5.5.1) 1) B a E Ω0 (d 0 ) G(d 0 )-orbit O a.Ω0 Λ a = π −1 Λ(d 0 ),Ω 0 (O a,Ω0 ) = G(d 0 ) · π −1 Λ(d 0 ),Ω 0 (B a ) Λ a rigid ⇔ ⇔ G(d 0 ) · π −1 Λ(d 0 ),Ω 0 (B a ) dense G(d 0 )-orbit π −1 Λ(d 0 ),Ω 0 (B a ) dense G(d 0 ) Ba -orbit G(d 0 ) Ba := {g ∈ G(d 0 ) | g · B a = B a } (B a stabilizer) π −1 Λ(d 0 ),Ω 0 (B a ) G(d 0 ) Ba ⎧ ⎨ π −1 Λ(d 0 ),Ω 0 (B a ) = ⎩ (B τ 1 , B τ 2 , B τ 3 , B τ 4 ) ∣ B τ1 B τ 1 = 0, B τ 1 B τ1 = B τ2 B τ 2 , B τ 2 B τ2 = B τ3 B τ 3 , B τ 3 B τ3 = B τ4 B τ 4 , B τ 4 B τ4 = 0 ⎫ ⎬ ⎭ . B τ i (i = 1, 2, 3, 4) C B τ ←−− 1 C 2 B τ ←−− 2 C 2 B τ ←−− 3 C 2 B τ ←−− 4 C preprojective relations (5.5.1) {(( ) ( ) ( ) 0 s 0 0 0 π −1 Λ(d 0 ),Ω 0 (B a ) = , , , ( u 0 )) ∣ } ∣∣ s, t, u, v ∈ C ∼ = C 4 s t 0 u v stabilizer G(d 0 ) Ba = { g = (g 1 , g 2 , g 3 , g 4 , g 5 ) ∈ GL 1 (C) × (GL 2 (C)) 3 × GL 1 (C) ∣ } g · Ba = B a { ( ( ) ( ) ( ) ∣ } a 0 c 0 c 0 ∣∣∣ = g = a, , , , f) a, c, d, f ∈ C b c 0 d e f × , b, e ∈ C –193–
(( ( ) ( ) 0 s 0 0 0 , , , s) ( u 0 )) t 0 u v (( ) ( ) ( ) g 0 a ↦−→ a −1 , −1 cs 0 0 0 cs a −1 , dt 0 c −1 fu d −1 , ( c fv −1 fu 0 )) b e “” a, c, d, f ( C ×) 4 ( C × ) 4 C 4 . s, t, u, v generic 0 su tv su tv g ↦−→ a−1 cs · c −1 fu a −1 dt · d −1fv = su tv . G(d 0 ) Ba π −1 Λ(d 0 ),Ω 0 (B a ) ∼ = C 4 C 4 su tv = const orbit dense orbit 29 0 2 0 0 0 0 0 2 0 2 0 0 0 2 0 , 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 rigid Example 27 a 2a k ka Λ a non-rigid ⇒ Λ ka (k ∈ Z >0 ) non-rigid k primitive rigid component References [1] I. Assem, D. Simson and A. Skowroński, Elements <strong>of</strong> <strong>the</strong> representation <strong>the</strong>ory <strong>of</strong> associative algebras. Vol. 1. Techniques <strong>of</strong> representation <strong>the</strong>ory, London. Math. Soc. Student Texts 65 (2006), Cambridge. [2] P. Baumann and J. Kamnitzer Preprojective algebras and MV polytopes, arXiv:1009.2469. [3] Berenstein, Fomin and A. Zelevinsky, Parametrizations <strong>of</strong> canonical bases and totally positive matrices, Adv. Math. 122 (1996), 49-149. [4] C. Geiss, B. Leclerc and J. Schöler, Semicanonical bases and preprojective algebras, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 2, 193–253. 29 π −1 Λ(d 0 ),Ω 0 (B a ) ∼ = C 4 orbit –194–
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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
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 and 194: Example 15. Γ loop quiver λ ∈
Page 195 and 196: P (Γ)-module i-th radical B →
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: “Λ(d) G(d)-” Definition 25.
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
Page 245 and 246: By Example 3, the
Page 247 and 248: weight wt(x) = d(x, 0) of</
Page 249 and 250: 4.1. Fourier Transform. Gleason’s
Page 251 and 252: 5. The Extension Problem In this se
Page 253 and 254: Because R is assumed to be Frobeniu
Page 255 and 256: is injective for every finite left
Page 257 and 258: 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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