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

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[7] Y. Takehana, On generalization <strong>of</strong> CQF-3’ modules and cohereditary torsion <strong>the</strong>ories, Math. J. Okayama Univ. 54 (2012), 65-76. [8] Y. Takehana, On a generalization <strong>of</strong> stable torsion <strong>the</strong>ory, Proc. <strong>of</strong> <strong>the</strong> 43rd <strong>Symposium</strong> on Ring <strong>the</strong>ory and Representation Theory, 2011, 71-78. [9] L. E. T. Wu and J. P. Jans, On Quasi Projectives, Illinois J. Math. Volume 11, Issue 3 (1967), 439-448. GENERAL EDUCATION HAKODATE NATIONAL COLLEGE OF TECHNOLOGY 14-1 TOKURA-CHO HAKODATE-SI HOKKAIDO, 042-8501 JAPAN E-mail address: takehana@hakodate-ct.ac.jp –215–
GRADED FROBENIUS ALGEBRAS AND QUANTUM BEILINSON ALGEBRAS KENTA UEYAMA Abstract. Frobenius algebras are one <strong>of</strong> <strong>the</strong> important classes <strong>of</strong> algebras studied in representation <strong>the</strong>ory <strong>of</strong> finite dimensional algebras. In this article, we will study when given graded Frobenius Koszul algebras are graded Morita equivalent. As applications, we apply our results to quantum Beilinson algebras. Key Words: equivalence. Frobenius Koszul algebras, quantum Beilinson algebras, graded Morita 2010 Ma<strong>the</strong>matics Subject Classification: Primary 16W50, 16S37; Secondary 16D90. 1. Introduction This is based on a joint work with Izuru Mori. Classification <strong>of</strong> Frobenius algebras is an active project in representation <strong>the</strong>ory <strong>of</strong> finite dimensional algebras. This article tries to answer <strong>the</strong> question when given graded Frobenius Koszul algebras are graded Morita equivalent, that is, <strong>the</strong>y have equivalent graded module categories. This problem is related to classification <strong>of</strong> quasi-Fano algebras. It is known that every finite dimensional algebra <strong>of</strong> global dimension 1 is a path algebra <strong>of</strong> a finite acyclic quiver up to Morita equivalence, so such algebras can be classified in terms <strong>of</strong> quivers. As an obvious next step, it is interesting to classify finite dimensional algebras <strong>of</strong> global dimension 2 or higher. Recently, Minamoto introduced a nice class <strong>of</strong> finite dimensional algebras <strong>of</strong> finite global dimension, called (quasi-)Fano algebras [2], which are a very interesting class <strong>of</strong> algebras to study and classify. It was shown that, for a graded Frobenius Koszul algebra A, we can define ano<strong>the</strong>r algebra ∇A, called <strong>the</strong> quantum Beilinson algebra associated to A, and with some additional assumptions, ∇A turns out to be a quasi-Fano algebra. Moreover, it was shown that two graded Frobenius algebras A, A ′ are graded Morita equivalent if and only if ∇A, ∇A ′ are isomorphic as algebras, so classifying graded Frobenius (Koszul) algebra up to graded Morita equivalence is related to classifying quasi-Fano algebras up to isomorphism (see [3] for details). In addition, this problem is related to <strong>the</strong> study <strong>of</strong> AS-regular algebras which are <strong>the</strong> most important class <strong>of</strong> algebras in noncommutative algebraic geometry (see [8]). Our main <strong>the</strong>orem (Theorem 9) is as follows. For every co-geometric Frobenius Koszul algebra A, we define ano<strong>the</strong>r graded algebra A, and see that if two co-geometric Frobenius Koszul algebras A, A ′ are graded Morita equivalent, <strong>the</strong>n A, A ′ are isomorphic as graded algebras. Unfortunately, <strong>the</strong> converse does not hold in general. On <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong> converse is also true for many co-geometric Frobenius Koszul algebras <strong>of</strong> Gorenstein parameter −3. The detailed version <strong>of</strong> this paper will be submitted for publication elsewhere. –216–
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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 , ·
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 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
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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;
Page 259 and 260: ψ : ε(A) → Â. In this way, C
Page 261 and 262: REALIZING STABLE CATEGORIES AS DERI
Page 263 and 264: 2.1. Positively graded self-injecti
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Page 267 and 268: Next we consider the</stron
Page 269 and 270: (2) There exists a triangle-equival
Page 271 and 272: ALGEBRAIC STRATIFICATIONS OF DERIVE
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Page 275 and 276: Step 2: Let A be a finite-dimension
Page 277 and 278: RECOLLEMENTS GENERATED BY IDEMPOTEN
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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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