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

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(b) The following are equivalent for an A-module M. (i) The variety <strong>of</strong> M is trivial. (ii) The projective dimension <strong>of</strong> M is finite. (iii) The injective dimension <strong>of</strong> M is finite. There are some papers which deal with <strong>the</strong> finiteness condition (Fg) as follows. (1) In [2], Bergh and Oppermann show that a codimension n quantum complete intersection satisfies (Fg) if and only if all <strong>the</strong> commutators q ij are roots <strong>of</strong> unity. Definition 4. Let n be integer with n ≥ 1, a i integer with a i ≥ 2 for 1 ≤ i ≤ n, and q ij a non-zero element in k for every 1 ≤ i < j ≤ n. A codimension n quantum complete intersection is defined by where I generated by k〈x 1 , . . . , x n 〉/I x a i i , x jx i − q ij x i x j for 1 ≤ i < j ≤ n. (2) In [5], Erdmann and Solberg gave <strong>the</strong> necessary and sufficient conditions on a Koszul algebra for it to satisfy (Fg). Theorem 5 ([5, Theorem 1.3]). Let A be a finite dimensional Koszul algebra over an algebraically closed field, and let E(A) = Ext ∗ A(A/rad A, A/rad A). A satisfies (Fg) if and only if Z gr (E(A)) is Noe<strong>the</strong>rian and E(A) is a finitely generated Z gr (E(A))-module. (3) In [8], Furuya and Snashall provided examples <strong>of</strong> (D, A)-stacked monomial algebras which are not self-injective but satisfy (Fg). Example 6. ([8, Example 3.2]) Let Q be <strong>the</strong> quiver and I <strong>the</strong> ideal <strong>of</strong> kQ generated by α 1 −−−→ 2 ↑ ⏐ ⏐ δ ⏐ ↓β 4 ←−−− γ 3 αβγδαβ, γδαβγδ. Then, A = kQ/I is not self-injective but satisfies (Fg). (4) In [15], Schroll and Snashall show that (Fg) hold for <strong>the</strong> principal block <strong>of</strong> <strong>the</strong> Heche algebra H q (S 5 ) with q = −1. –139–
4. Quiver algebras defined by two cycles and a quantum-like relation In this section, we consider <strong>the</strong> quiver algebras A q = kQ/I q defined by <strong>the</strong> quiver Q as follows: α a 2 3 ←− a2 b 2 · · · · · · α 3 ↙ ↖ α 1 β 1 . ↗ .. a 4 1 b t−2 . .. ↗ αs β t ↖ ↙ βt−2 β t−1 · · · · · · a s b t ←− bt−1 and <strong>the</strong> ideal I q <strong>of</strong> kQ generated by X sa , X s Y t − qY t X s , Y tb for a, b ≥ 2 where we set X:= α 1 + α 2 + · · · + α s and Y := β 1 + β 2 + · · · + β t , and q is non-zero element in k. Paths are written from right to left. In <strong>the</strong> case s = t = 1, A q is called a quantum complete intersection (cf. [1]). In this case, when a = b = 2, <strong>the</strong> Hochschild cohomology ring HH ∗ (A q ) <strong>of</strong> A q was described by Buchweitz, Green, Madsen and Solberg [3] for any q ∈ k. Moreover, in <strong>the</strong> case where s = t = 1, a, b ≥ 2, Bergh and Erdmann [1] determined HH ∗ (A q ) if q is not a root <strong>of</strong> unity. And in <strong>the</strong> same case, Bergh and Oppermann [2] show that A q satisfies (Fg) if and only if q is a root <strong>of</strong> unity. In [4], Erdmann, Holloway, Snashall, Solberg and Taillefer describe that if an algebra A satisfies (Fg) <strong>the</strong>n HH ∗ (A) is a finitely generated k-algebra. Therefore, we consider <strong>the</strong> case where s ≥ 2 or t ≥ 2. In this paper, we determine <strong>the</strong> Hochschild cohomology ring <strong>of</strong> A q modulo nilpotence HH ∗ (A q )/N and show that A q satisfies (Fg) if and only if q is a root <strong>of</strong> unity. In [13] and [14], we determined <strong>the</strong> ring structure <strong>of</strong> HH ∗ (A q ) by means <strong>of</strong> generators and Yoneda product. By this ring structure <strong>of</strong> HH ∗ (A q ), we have <strong>the</strong> following results. Theorem 7. In <strong>the</strong> case where q is a root <strong>of</strong> unity, HH ∗ (A q ) is finitely generated as a k-algebra. Theorem 8. In <strong>the</strong> case where q is a root <strong>of</strong> unity, HH ∗ (A q )/N is isomorphic to <strong>the</strong> polynomial ring <strong>of</strong> two variables. In <strong>the</strong> case s, t ≥ 2, r ≥ 1, we have ⎧ k[W0,0,0, ⎪⎨ 2r W2r,0,0] if s, t ≥ 2, ā ≠ 0, ¯b ≠ 0, HH ∗ (A q )/N ∼ k[W = 0,0,0, 2r W2,0,0] 2 if s, t ≥ 2, ā = 0, ¯b ≠ 0, k[W0,0,0, ⎪⎩ 2 W2r,0,0] if s, t ≥ 2, ā ≠ 0, ¯b = 0, k[W0,0,0, 2 W2,0,0] 2 if s, t ≥ 2, ā = ¯b = 0, where for any integer z, ¯z is <strong>the</strong> remainder when we divide z by r, and for n ≥ 1, t∑ W0,l,l 2n := ′ Xsl Y tl′ e 2n b(1) + Y j−1 X sl Y t(l′ −1)+t−j+1 e 2n b(j) for 0 ≤ l ≤ a − 1 and 0 ≤ l ′ ≤ b, W 2n 2n,l,l ′ := Xsl Y tl′ e 2n a(1) + j=2 s∑ i=2 In <strong>the</strong> case where s = 1 or t = 1, we have similar results. X s(l−1)+i−1 Y tl′ X s−i+1 e 2n a(i) for 0 ≤ l ≤ a and 0 ≤ l ′ ≤ b − 1. –140–
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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
Page 103 and 104: Theorem 26. Let R be a local artini
Page 105 and 106: where each f i is an irreducible ep
Page 107 and 108: QUANTUM UNIPOTENT SUBGROUP AND DUAL
Page 109 and 110: {F i } i∈I and q-Serre relations
Page 111 and 112: Theorem 8. (1)Let U − q (w) → U
Page 113 and 114: E-mail address: ykimura@kurims.kyot
Page 115 and 116: Indeed, (1, 2, 3, 4) {1,2} −−
Page 117 and 118: By comparing this the</stro
Page 119 and 120: (a) (b) Proposition 17. Let G be a
Page 121 and 122: POLYCYCLIC CODES AND SEQUENTIAL COD
Page 123 and 124: ⎛ ⎞ 0 0 c 0 1 c t D c = ⎜ 1
Page 125 and 126: References [1] D. Boucher and P. So
Page 127 and 128: 2. Dimension of tr
Page 129 and 130: APR TILTING MODULES AND QUIVER MUTA
Page 131 and 132: (1) Q ′ is a quiver obtained from
Page 133 and 134: 1 arrows of ˜µ L
Page 135 and 136: [11] S. Fomin, A. Zelevinsky, Clust
Page 137 and 138: 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</
Page 145 and 146: is equal to ( n 2s−1 ); hence we
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: (4) In [10], Green, Snashall and So
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 )
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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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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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