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

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Next we write <strong>the</strong> Auslander-Reiten quiver <strong>of</strong> D b (modΓ). In this case, Γ = End A (T ) 0 is isomorphic to 2 × 2 upper triangular matrix algebra over K. We put P 1 := (KK), P 2 := (0K) and I 1 := (K0). It is known that <strong>the</strong> set {P 1 , P 2 , I 1 } is a complete set <strong>of</strong> indecomposable Γ-modules, and <strong>the</strong> Auslander-Reiten quiver <strong>of</strong> D b (modΓ) is as follows. I 1 [−1] ❄ P 1 P 2 [1] I 1 [1] P 1 [2] ❄ ❄❄❄❄❄ ❄ ❄ ❄❄❄❄❄ ❄ ❄❄❄❄❄ ❄❄❄❄❄ · · · · · · · · · · · · 88888 P 1 8 8 88888 [−1] P 2 8 88888 I 1 8 88888 P 1 8 [1] P 2 8 888888 [2] Here dotted arrows mean <strong>the</strong> Auslander-Reiten translation in D b (modΓ). From shape <strong>of</strong> <strong>the</strong> above Auslander-Reiten quivers, one can see that mod Z A and D b (modΓ) should be equivalent to each o<strong>the</strong>r. In fact, we gave a triangle-equivalence between those. References [1] R. O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings, preprint. [2] Xiao-Wu Chen, Graded self-injective algebras ”are” trivial extensions, J. Algebra 322 (2009), no. 7, 2601–2606. [3] R. Gordon and E. L . Green, Graded Artin algebras, J. Algebra 76 (1982), no. 1, 111–137. [4] , Representation <strong>the</strong>ory <strong>of</strong> graded Artin algebras, J. Algebra 76 (1982), no. 1, 138–152. [5] D. Happel, Triangulated categories in <strong>the</strong> representation <strong>the</strong>ory <strong>of</strong> finite-dimensional algebras, London Ma<strong>the</strong>matical Society Lecture Note Series, 119. Cambridge University Press, Cambridge, 1988. [6] , Auslander-Reiten triangles in derived categories <strong>of</strong> finite-dimensional algebras, Proc. Amer. Math. Soc. 112 (1991), no. 3, 641–648. [7] B. Keller, Deriving DG categories, Ann. Sci. Ecole Norm. Sup. (4) 27 (1994), no. 1, 63–102. [8] B. Keller, D. Vossieck, Sous les catégories dérivées, (French), C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 6, 225–228. [9] J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), no. 3, 303–317. Graduate School <strong>of</strong> Ma<strong>the</strong>matics NagoyaUniversity Frocho, Chikusaku, Nagoya 464-8602 Japan E-mail address: m07052d@math.nagoya-u.ac.jp –255–
ALGEBRAIC STRATIFICATIONS OF DERIVED MODULE CATEGORIES AND DERIVED SIMPLE ALGEBRAS DONG YANG Abstract. In this note I will survey on some recent progress in <strong>the</strong> study <strong>of</strong> recollements <strong>of</strong> derived module categories. Key Words: Recollement, Algebraic stratification, Derived simple algebra. 2010 Ma<strong>the</strong>matics Subject Classification: 16E35. The notion <strong>of</strong> recollement <strong>of</strong> triangulated categories was introduced in [5] as an analogue <strong>of</strong> short exact sequence <strong>of</strong> modules or groups. In representation <strong>the</strong>ory <strong>of</strong> algebras it provides us with reduction techniques, which have proved very useful, for example, in • proving conjectures on homological dimensions, see [9]; • computing homological invariants, see [11, 12]; • classifying t-structures, see [14]. In this note I will survey on some recent progress in <strong>the</strong> study <strong>of</strong> recollements <strong>of</strong> derived module categories. 1. Recollements Let k be a field. For a k-algebra A denote by D(A) = D(Mod A) <strong>the</strong> (unbounded) derived category <strong>of</strong> <strong>the</strong> category Mod A <strong>of</strong> right A-modules. The objects <strong>of</strong> D(A) are complexes <strong>of</strong> right A-modules. The category D(A) is triangulated with shift functor Σ being <strong>the</strong> shift <strong>of</strong> complexes. See [10] for a nice introduction on derived categories. A recollement <strong>of</strong> derived module categories is a diagram <strong>of</strong> derived module categories and triangle functors (1.1) i ∗ j ! D(B) i ∗ =i ! D(A) j ! =j ∗ D(C), i ! j ∗ where A, B and C are k-algebras, such that (1) (i ∗ , i ∗ = i ! , i ! ) and (j ! , j ! = j ∗ , j ∗ ) are adjoint triples; (2) j ! , i ∗ and j ∗ are fully faithful; (3) j ∗ i ∗ = 0; The detailed /final/ version <strong>of</strong> this paper will be /has been/ submitted for publication elsewhere. –256–
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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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Page 223 and 224: ON A GENERALIZATION OF COSTABLE TOR
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Page 231 and 232: GRADED FROBENIUS ALGEBRAS AND QUANT
Page 233 and 234: Definition 2. A graded algebra A is
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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
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Page 249 and 250: 4.1. Fourier Transform. Gleason’s
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Page 259 and 260: ψ : ε(A) → Â. In this way, C
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Page 267 and 268: Next we consider the</stron
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Page 283 and 284: INTRODUCTION TO REPRESENTATION THEO
Page 285 and 286: is exact. Since E n ∼ = En+r , Mi
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Page 293 and 294: R : CM(R⊗ k V ) → CM(R) defined
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Page 297 and 298: SUBCATEGORIES OF EXTENSION MODULES
Page 299 and 300: Lemma 6. Let S 1 and S 2 be Serre s
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