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

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symmetric Kac-Moody Lie algebra 14 Lie algebra Γ Dyn A, D, E Dynkin case Definition 8. Γ = (I, Ω) loop quiver (Z I ) ⊗ Q = Q I {s(i)|i ∈ I}, bilinear form (·, ·) ( Q I , {s(i)}, (·, ·) ) Dynkin diagram Γ Dyn root datum (·, ·) Z I bilinear form Q I Remark 9. Lie <strong>the</strong>ory root datum 15 Lie <strong>the</strong>ory root datum Cartan matrix symmetric quiver Cartan matrix symmetric Cartan matrix symmetric 4.2. Crystal . crystal crystal “” root datum ( Q I , {s(i)}, (·, ·) ) fix {s(i)} Q I Z-submodule Q := ⊕ Zs(i) i∈I root lattice Q ∼ = Z I P : (a) P Q I rank n = |I| Z-submodule (b) (z, Q) ⊂ Z for every z ∈ P. (c) s(i) ∈ P for every i ∈ I. (c) Q ⊂ P Remark 10. (1) bilinear form (·, ·) Γ A, D, E (a), (b), (c) canonical P P := {z ∈ Q I | (z, Q) ⊂ Z} lattice Q dual lattice (·, ·) Γ extended Dynkin P Z (a) canonical P choice 14 symmetric A(Γ Dyn ) Kac-Moody Lie algebra Cartan matrix 15 Lie <strong>the</strong>ory –175–
(2) P Lie <strong>the</strong>ory weight lattice crystal 16 Definition 11. B wt : B → P, ε i : B → Z ⊔ {−∞}, ϕ i : B → Z ⊔ {−∞}, ẽ i : B → B ⊔ {0}, ˜fi : B → B ⊔ {0} (i ∈ I) ( B; wt, ε i , ϕ i , ẽ i , ˜f i ) root datum ( Q I , {s(i)}, (·, ·) ) crystal (C1) i ∈ I, b ∈ B (C2) b ∈ B ẽ i b ∈ B ϕ i (b) = ε i (b) + (s(i), wt(b)). wt(ẽ i b) = wt(b) + s(i), ε i (ẽ i b) = ε i (b) − 1, ϕ i (ẽ i b) = ϕ i (b) + 1. (C2)’ b ∈ B ˜f i b ∈ B wt( ˜f i b) = wt(b) − s(i), ε i ( ˜f i b) = ε i (b) + 1, ϕ i ( ˜f i b) = ϕ i (b) − 1. (C3) b, b ′ ∈ B b ′ = ẽ i b ⇔ b = ˜f i b ′ . (C4) b ∈ B ϕ i (b) = −∞ ẽ i b = ˜f i b = 0. “−∞” “0” Z extra B extra −∞ Z “” (−∞) + a = a + (−∞) = −∞ (a ∈ Z) 17 ϕ i (b) = −∞ ⇔ ε i (b) = −∞. b ∈ B wt(b) ∈ P (s(i), wt(b)) ∈ Z (C1) “” ϕ i (b) = −∞ ε i (b) = −∞ crystal ( B; wt, ε i , ϕ i , ẽ i , ˜f i ) crystal B crystal crystal graph 16 “” root datum quiver root datum 17 −∞ –176–
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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
Page 139 and 140: References [1] Apostol, T. M., The
Page 141 and 142: 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: Definition 6. n × n A(Γ Dyn ) =
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 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
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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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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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