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

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One can prove in a similar fashion that for a selfinjective algebra, a component C contains a simple (projective) module, if and only if <strong>the</strong> component ΩC contains a projective (respectively simple) module. Remark 11. Let C be an Auslander-Reiten component having a boundary, that is, a component containing indecomposable modules whose Auslander-Reiten sequences have indecomposable middle terms. Assume that C is not a tube, and let C be an indecomposable module lying on <strong>the</strong> boundary <strong>of</strong> C. Without loss <strong>of</strong> generality we may assume that nei<strong>the</strong>r C nor ΩC has a simple module in <strong>the</strong> positive direction <strong>of</strong> <strong>the</strong>ir τ-orbit. This means that if 0 → τC → B → C → 0 is <strong>the</strong> Auslander-Reiten sequence ending at C, <strong>the</strong>n both maps τC → B and B → C are Ω-perfect and so C is an Ω-perfect module. So we see that nonperiodic components with boundaries, always contain Ω-perfect maps and Ω-perfect modules. As we will see soon, this need not happen in components <strong>of</strong> <strong>the</strong> type ZA ∞ ∞. Lemma 12. Let g : B → C be an irreducible map that is not eventually Ω-perfect, where nei<strong>the</strong>r B nor C has a nonzero projective summand. Then, <strong>the</strong>re exists a positive integer α such that for each i ≥ 0 we have |l(Ω i B) − l(Ω i C)| ≤ α. Pro<strong>of</strong>. By taking enough powers <strong>of</strong> <strong>the</strong> Auslander-Reiten translate τ, we may assume without loss <strong>of</strong> generality that g is onto, and that its kernel S is a simple periodic module. Note that by applying Ω we obtain an induced exact sequence 0 → ΩB −→ Ωg ΩC → S → 0. If <strong>the</strong> induced map Ω 2 g is again a monomorphism, <strong>the</strong>n we get <strong>the</strong> commutative exact diagram 0 0 0 0 Ω 2 B Ω 2 g Ω 2 C L 0 0 P ΩB 0 ΩB Ωg P ΩC ΩC Q S 0 0 0 0 0 hence <strong>the</strong> two modules L and ΩS are isomorphic, and we have a short exact sequence 0 → Ω 2 B → Ω 2 C → ΩS → 0. If on <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong> map Ω 2 g is an epimorphism, –81–
<strong>the</strong>n we obtain a commutative diagram 0 0 L Ω 2 B 0 L P ΩB 0 ΩB 0 Ω 2 g Ω 2 C Ωg P ΩC ΩC 0 0 0 S 0 S 0 and <strong>the</strong>refore we obtain a short exact sequence 0 → Ω 2 S → Ω 2 B → Ω 2 C → 0. Continuing in this fashion we see that for each integer i ≥ 0 we get ei<strong>the</strong>r short exact sequences 0 → Ω i S → Ω i B → Ω i C → 0, or <strong>of</strong> <strong>the</strong> form 0 → Ω i+1 B → Ω i+1 C → Ω i S → 0. By letting α = max i∈N {l(Ω i S)}, our result follows, since <strong>the</strong> simple module S is periodic. □ The following (most probably) well-known lemma will be used to characterize Ω-perfect maps in terms <strong>of</strong> <strong>the</strong> “τ-perfect” property. As usual, if M is an indecomposable nonprojective module, α(M) denotes <strong>the</strong> number <strong>of</strong> non-projective indecomposable direct summands <strong>of</strong> <strong>the</strong> middle term <strong>of</strong> <strong>the</strong> Auslander-Reiten sequence ending in M. Lemma 13. Let Λ be a selfinjective artin algebra and let M be an indecomposable non projective and non simple Λ-module with α(M) = 2 with n = l(M) = l(τM). Assume also that <strong>the</strong>re exists an irreducible map E → M where E is indecomposable and that l(E) = l(M) − 1. (a) The middle term <strong>of</strong> <strong>the</strong> Auslander-Reiten sequence ending at M has no nonzero projective summand. (b) If E, M and τM are uniserial, <strong>the</strong>n <strong>the</strong> remaining summand F <strong>of</strong> <strong>the</strong> Auslander- Reiten sequence ending at M is uniserial too and its length is l(F ) = l(M) + 1. Pro<strong>of</strong>. Let 0 → τM → E ⊕ F ⊕ P → M → 0 be <strong>the</strong> Auslander-Reiten sequence ending at M, where F is indecomposable non projective, and P is a nonzero projective module. Note first that P must be indecomposable since <strong>the</strong> algebra is selfinjective. Using now <strong>the</strong> fact that τM = rP , a length argument shows that l(F ) = 2n − l(E) − l(P ) = 2n − (n − 1) − (n + 1) = 0 contradicting our assumption. This proves <strong>the</strong> first part <strong>of</strong> <strong>the</strong> lemma. For part (b), note first that <strong>the</strong> Auslander-Reiten sequence ending at M has <strong>the</strong> form 0 → τM → E ⊕ F → M → 0 where E and F are both indecomposable and l(F ) = n + 1. Hence τM is a maximal submodule <strong>of</strong> F . To prove <strong>the</strong> uniseriality <strong>of</strong> F , it suffices to show that τM = rF . It is folklore (see also [17], Proposition 2.5.) that, since τM is not simple, we have an induced exact sequence 0 → rτM → rE ⊕ rF → rM → 0. Counting lengths, we get l(rF ) = 2(n − 1) − (n − 2) = n. Since <strong>the</strong> image <strong>of</strong> τM in F contains <strong>the</strong> radical <strong>of</strong> F , it follows that τM ∼ = rF , and F is also an uniserial module. □ –82–
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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
Page 45 and 46: EXAMPLE OF CATEGORIFICATION OF A CL
Page 47 and 48: The aim of <strong
Page 49 and 50: X ∈ mod Π Q S 1 S 2 S 3 1 ❁
Page 51 and 52: The previous proposition has a dual
Page 53 and 54: Computing inductively all t
Page 55 and 56: Example 23. We have ⎛ ⎞ ⎛ ⎞
Page 57 and 58: classification of
Page 59 and 60: quiver Q over K, and let D b (H) <s
Page 61 and 62: Then Q ∈ Q 17 . Suppose char K
Page 63 and 64: Moreover the Hochs
Page 65 and 66: DERIVED AUTOEQUIVALENCES AND BRAID
Page 67 and 68: 3. Periodic Twists We now describe
Page 69 and 70: We now specialise to the</s
Page 71 and 72: and τ − n := Ext n Λ(DΛ, −)
Page 73 and 74: where r v ij := a i a j −a j a i
Page 75 and 76: ON A DEGENERATION PROBLEM FOR COHEN
Page 77 and 78: We make several othe</stron
Page 79 and 80: Let A be a commutative Gorenstein r
Page 81 and 82: 3. Extended orders In the</
Page 83 and 84: WEAK GORENSTEIN DIMENSION FOR MODUL
Page 85 and 86: 1.2. Introduction. The notion <stro
Page 87 and 88: e 2 A z 1 −→ X → 0 in mod-A,
Page 89 and 90: 5. Gorenstein algebras In this sect
Page 91 and 92: ON Ω-PERFECT MODULES AND SEQUENCES
Page 93 and 94: when R is a Nakayama algebra <stron
Page 95: says that components of</st
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 and 146: 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
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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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