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

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[4] D. Happel, Relative invariants and subgeneric orbits <strong>of</strong> quivers <strong>of</strong> finite and tame type, J. Algebra 78 (1982), 445–459. MR 680371 (84a:16052) [5] T. Kimura, Introduction to prehomogeneous vector spaces, Translations <strong>of</strong> Ma<strong>the</strong>matical Monographs 215, American Ma<strong>the</strong>matical Society, 2003, Translated from <strong>the</strong> 1998 Japanese original by Makoto Nagura and Tsuyoshi Niitani and revised by <strong>the</strong> author. MR 1944442 (2003k:11180) [6] A. Sch<strong>of</strong>ield, Semi-invariants <strong>of</strong> quivers, J. London Math. Soc. (2) 43 (1991), 385–395. MR 1113382 (92g:16019) [7] U. Seidel, Exceptional sequences for quivers <strong>of</strong> Dynkin type, Comm. Algebra 29 (2001), 1373–1386. MR 1842420 (2002j:16016) [8] D. A. Shmelkin, Locally semi-simple representations <strong>of</strong> quivers, Transform. Groups 12 (2007), 153– 173. MR 2308034 (2008b:16017) Tokyo Gakugei University 4-1-1, Nukuikitamachi, Koganei Tokyo 184-8501 JAPAN E-mail address: nagase@u-gakugei.ac.jp Nara National College <strong>of</strong> Technology Yamato-Koriyama Nara 639-1080 JAPAN E-mail address: nagura@libe.nara-k.ac.jp –131–
THE NOETHERIAN PROPERTIES OF THE RINGS OF DIFFERENTIAL OPERATORS ON CENTRAL 2-ARRANGEMENTS NORIHIRO NAKASHIMA Abstract. P. Holm began to study <strong>the</strong> ring <strong>of</strong> differential operators <strong>of</strong> <strong>the</strong> coordinate ring <strong>of</strong> a hyperplane arrangement. In this paper, we introduce Noe<strong>the</strong>rian properties <strong>of</strong> <strong>the</strong> ring differential operators <strong>of</strong> <strong>the</strong> coordinate ring <strong>of</strong> a central 2-arrangement and its graded ring associated to <strong>the</strong> order filtration. Ring <strong>of</strong> differential operators, Noe<strong>the</strong>rian property, Hyperplane ar- Key Words: rangement. 2010 Ma<strong>the</strong>matics Subject Classification: Primary 13N10; Secondary 32S22. 1. Introduction For a commutative algebra R over a field K <strong>of</strong> characteristic zero, define vector spaces inductively by D 0 (R) := {θ ∈ End K (R) | a ∈ R, θa − aθ = 0}, D m (R) := { θ ∈ End K (R) | a ∈ R, θa − aθ ∈ D m−1 (R) } (m ≥ 1). We define <strong>the</strong> ring D(R) := ∪ m≥0 D m (R) <strong>of</strong> differential operators <strong>of</strong> R. Let S := K[x 1 , . . . , x n ] be <strong>the</strong> polynomial ring. The ring D(S) is <strong>the</strong> n-th Weyl algebra K[x 1 , . . . , x n ]〈∂ 1 , . . . , ∂ n 〉 where ∂ i := ∂ ∂x i (see for example [3]). We use <strong>the</strong> multi-index notetions, for example, ∂ α := ∂ α 1 1 · · · ∂ α n n and |α| := α 1 + · · · + α n for α = (α 1 , . . . , α n ) ∈ N n . Define D (m) (S) := ⊕ |α|=m . Then D(S) = ⊕ m≥0 D (m) (S). It is well known D(R) that D(R) is Noe<strong>the</strong>rian, if R is a regular domain (see [3]). Holm [2] showed that D(R) is finitely generated as a K-algebra when R is a coordinate ring <strong>of</strong> a generic hyperplane arrangement. Holm [1] also proved that <strong>the</strong> ring <strong>of</strong> differential operators <strong>of</strong> a central 2-arrangement is a free S-module, and gave a basis <strong>of</strong> it. We can write any ellement in D(R) as a linearly combination <strong>of</strong> this basis ellements. In this paper, we introduce <strong>the</strong> Noe<strong>the</strong>rian property <strong>of</strong> D(R) when R is <strong>the</strong> coordinate ring <strong>of</strong> a central arrangement. In particular, <strong>the</strong> case n = 2, D(R) is a Noe<strong>the</strong>rian ring . We give an example <strong>of</strong> a finitely generated ideal in <strong>the</strong> end <strong>of</strong> this paper. The details <strong>of</strong> this note are in [4]. 2. Hyperplane arrangement In this section, we fix some notation, and we introduce some properties <strong>of</strong> <strong>the</strong> ring <strong>of</strong> differential operators <strong>of</strong> a central arrangement. Let A = {H i | i = 1, . . . , r} be a central (hyperplane) arrangement (i.e., every hyperplane in A contains <strong>the</strong> origin) in K n . Fix a The detailed version <strong>of</strong> this paper will be submitted for publication elsewhere. –132–
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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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Page 97 and 98: then we obtain a c
Page 99 and 100: Proposition 16. Let C s be a stable
Page 101 and 102: 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: is equal to ( n 2s−1 ); hence we
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 and 194: Example 15. Γ loop quiver λ ∈
Page 195 and 196: P (Γ)-module i-th radical B →
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B ∈ Λ(d) ε ∗ i (B) := dim C
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I-graded vector space V (d) (B τ )
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A ∗(i) l (a) = ∑n+1 t=l+1 (a i,
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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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