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

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[7] , Endomorphism algebras and representation <strong>the</strong>ory, Algebraic groups and <strong>the</strong>ir representations (Cambridge, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 517, Kluwer Acad. Publ., Dordrecht, 1998, pp. 131–149. [8] , Stratifying endomorphism algebras, Mem. Amer. Math. Soc. 124 (1996), no. 591. [9] Lingyan Guo, Cluster tilting objects in generalized higher cluster categories, J. Pure Appl. Algebra 215 (2011), no. 9, 2055–2071. [10] Osamu Iyama, Auslander correspondence, Adv. Math. 210 (2007), no. 1, 51–82. [11] Peter Jørgensen, Recollement for differential graded algebras, J. Algebra 299 (2006), no. 2, 589–601. [12] Martin Kalck and Dong Yang, work in preparation. [13] , Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102. [14] Steffen Koenig and Hiroshi Nagase, Hochschild cohomologies and stratifying ideals, J. Pure Appl. Algebra 213 (2009), no. 4, 886–891. [15] Louis de Thanh<strong>of</strong>fer de Völcsey and Michel Van den Bergh, Explicit models for some stable categories <strong>of</strong> maximal Cohen-Macaulay modules, arXiv:1006.2021. [16] Michel Van den Bergh, Non-commutative crepant resolutions, The legacy <strong>of</strong> Niels Henrik Abel, Springer, Berlin, 2004, pp. 749–770. Universität Stuttgart Institut für Algebra und Zahlen<strong>the</strong>orie Pfaffenwaldring 57, D-70569 Stuttgart, Germany E-mail address: dongyang2002@gmail.com –267–
INTRODUCTION TO REPRESENTATION THEORY OF COHEN-MACAULAY MODULES AND THEIR DEGENERATIONS YUJI YOSHINO Abstract. This is a quick introduction to <strong>the</strong> <strong>the</strong>ory <strong>of</strong> representation <strong>the</strong>ory <strong>of</strong> Cohen- Macaulay modules and <strong>the</strong>ir degenerations. 1. Representation <strong>the</strong>ory <strong>of</strong> Cohen-Macaulay modules. Let k be a field and let R be commutative noe<strong>the</strong>rian complete local k-algebra with unique maximal ideal m. We assume k ∼ = R/m naturally. Then it is known that <strong>the</strong>re is a regular local k-subalgebra T <strong>of</strong> R such that R is a module-finite T -algebra. (Cohen’s structure <strong>the</strong>orem for complete local rings.) Note that T is isomorphic to a formal power series ring over k. Definition 1. (1) R is called a Cohen-Macaulay ring (a CM ring for short) if R is free as a T -module. (2) A finitely generated R-module M is called a Cohen-Macaulay module over R, or a maximal Cohen-Macaulay module (a CM module or an MCM module for short) if M is free as a T -module. Given a CM module M, since M ∼ = T n for some n ≥ 0, we have a k-algebra homomorphism R → End T (M) ∼ = T n×n , which is a matrix-representation <strong>of</strong> R over T . In <strong>the</strong> following we always assume that R is a CM complete local k-algebra. We denote by mod(R) (res. CM(R) ) <strong>the</strong> category <strong>of</strong> finitely generated R-modules (resp. CM modules over R ) and R-homomorphisms. CM(R) := { CM modules over R} ⊆ mod(R) := {finitely generated R-modules } Since R is complete, mod(R) and CM(R) are Krull-Schmidt categories. Note that CM(R) is a resolving subcategory <strong>of</strong> mod(R) in <strong>the</strong> following sense: Suppose <strong>the</strong>re is an exact sequence 0 → L → M → N → 0 in mod(R). (i) If L, N ∈ CM(R) <strong>the</strong>n M ∈ CM(R). (ii) If M, N ∈ CM(R) <strong>the</strong>n L ∈ CM(R). Let d be <strong>the</strong> Krull-dimension <strong>of</strong> <strong>the</strong> ring R (so that we can take T = k[[t 1 , . . . , t d ]] on which R is finite). If d = 1 and if R is reduced, <strong>the</strong>n CM modules are just torsion-free modules. If d = 2 and if R is normal, <strong>the</strong>n CM modules are nothing but reflexive modules. In general, if d ≥ 3 and if R is normal, <strong>the</strong>n CM(R) ⊆ {reflexive modules} but this is not necessarily an equality. If R is regular (i.e. gl-dimR < ∞) <strong>the</strong>n all CM modules over R are free. Let K R := Hom T (R, T ) and call it <strong>the</strong> canonical module <strong>of</strong> R. Since R is a CM ring, K R ∈ CM(R). For any X ∈ mod(R), we have a natural isomorphism Hom R (X, K R ) ∼ = –268–
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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 , ·
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Let B = A/U(a d−1 ). We t
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• quiver Γ = (I, Ω) I Ω
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B = (B τ ) relations ρ 1 , · ·
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Definition 1. double quiver ˜Γ p
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◦ side Γ = (I, Ω) cycle (ac
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Definition 6. n × n A(Γ Dyn ) =
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(2) P Lie theory
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Example 15. Γ loop quiver λ ∈
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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
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;
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
Page 265 and 266: By the above resul
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
Page 273 and 274: The leaves of <str
Page 275 and 276: Step 2: Let A be a finite-dimension
Page 277 and 278: RECOLLEMENTS GENERATED BY IDEMPOTEN
Page 279 and 280: (a) the adjoint tr
Page 281: Cluster-tilting the</strong
Page 285 and 286: is exact. Since E n ∼ = En+r , Mi
Page 287 and 288: field of V . We de
Page 289 and 290: (Proof) ( ) p 0 0
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Page 293 and 294: R : CM(R⊗ k V ) → CM(R) defined
Page 295 and 296: P = P ′ t by t is a projective R
Page 297 and 298: SUBCATEGORIES OF EXTENSION MODULES
Page 299 and 300: Lemma 6. Let S 1 and S 2 be Serre s
Page 301 and 302: In the first row <
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