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

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n-REPRESENTATION INFINITE ALGEBRAS MARTIN HERSCHEND Abstract. We introduce <strong>the</strong> class <strong>of</strong> n-representation infinite algebras and discuss some <strong>of</strong> <strong>the</strong>ir homological properties. We also present <strong>the</strong> family <strong>of</strong> n-representation infinite algebras <strong>of</strong> type Ã. 1. Introduction This brief survey contains <strong>the</strong> results from my presentation at <strong>the</strong> <strong>44th</strong> <strong>Symposium</strong> on Ring Theory and Representation Theory in Okayama. It is based on joint work Osamu Iyama and Steffen Oppermann. A detailed final version will be published elsewhere. The class <strong>of</strong> hereditary finite dimensional algebras is one <strong>of</strong> <strong>the</strong> best understood in terms <strong>of</strong> representation <strong>the</strong>ory, especially in <strong>the</strong> context <strong>of</strong> Auslander-Reiten <strong>the</strong>ory. This applies in particular to representation finite hereditary algebras. In higher dimensional Auslander- Reiten <strong>the</strong>ory an analogue <strong>of</strong> <strong>the</strong>se algebras is given by <strong>the</strong> class <strong>of</strong> n-representation finite algebras [1, 2]. Recall that a finite dimensional algebra is called n-representation finite if it has global dimension at most n and admits an n-cluster tilting module. Since a 1-cluster tilting module is <strong>the</strong> same as an additive generator <strong>of</strong> <strong>the</strong> module category, 1-representation finite means precisely hereditary and representation finite. The aim <strong>of</strong> this report is to define <strong>the</strong> class <strong>of</strong> n-representation infinite algebras, that will in a similar way be a higher dimensional analogue <strong>of</strong> representation infinite hereditary algebras. To do this we begin by recalling some properties <strong>of</strong> n-representation finite algebras. Let K be a field and Λ a finite dimensional K-algebra with gl.dim Λ ≤ n. We always assume that Λ is ring indecomposable. Denote by mod Λ <strong>the</strong> category <strong>of</strong> finite dimensional left Λ-modules and by D b (Λ) <strong>the</strong> bounded derived category <strong>of</strong> mod Λ. Combining <strong>the</strong> K- dual D := Hom K (−, K) with <strong>the</strong> Λ-dual we obtain <strong>the</strong> Nakayama functor ν := DRHom(−, Λ) : D b (Λ) → D b (Λ). It is a Serre functor in <strong>the</strong> sense that <strong>the</strong>re is a functorial ismorphism Hom D b (Λ)(X, Y ) ≃ D Hom D b (Λ)(Y, ν(X)). We combine ν with <strong>the</strong> shift functor on D b (Λ) to obtain <strong>the</strong> autoequivalence ν n := ν ◦ [−n] : D b (Λ) → D b (Λ). It plays <strong>the</strong> role <strong>of</strong> <strong>the</strong> higher Auslander-Reiten translation in D b (Λ). More precisely, define τ n := D Ext n Λ(−, Λ) : mod Λ → mod Λ The detailed version <strong>of</strong> this paper will be submitted for publication elsewhere. –55–
and τ − n := Ext n Λ(DΛ, −) : mod Λ → mod Λ. Then τ n = H 0 (ν n −) and τn − = H 0 (νn −1 −). Using <strong>the</strong>se functors we can capture <strong>the</strong> notion <strong>of</strong> n-representation finiteness in <strong>the</strong> following way. Proposition 1. [3] Let Λ be a finite dimensional K-algebra with gl.dim Λ ≤ n. Then <strong>the</strong> following conditions are equivalent. (a) Λ is n-representation finite. (b) For every indecomposable projective Λ-module P , <strong>the</strong>re is a non-negative integer l P such that ν −l P n P is an indecomposable injective Λ-module. We remark that if condition (b) is satisfied <strong>the</strong>n νn −i P ≃ τn −i P for all 0 ≤ i ≤ l P and ⊕ P l P ⊕ i=0 τn −i P = ⊕ P l P ⊕ i=0 ν −i n P is an n-cluster tilting Λ-module [1]. Fur<strong>the</strong>rmore, since ν −1 sends injectives to projectives we have ν −(l P +1) n P = ν −1 (ν −l P n P )[n] = P ′ [n] ∈ mod Λ[n] for some indecomposable projective P ′ . We conclude that knowing <strong>the</strong> τn − -orbits <strong>of</strong> <strong>the</strong> indecomposable projectives in mod Λ is enough to determine <strong>the</strong>ir νn −1 -orbits. Comparing this to <strong>the</strong> classical case n = 1 gives us a hint how to define n-representation infinite algebras. 2. n-representation infinite algebras Recall that if n = 1 and Λ is representation infinite, <strong>the</strong>n τ −i P is never injective for an indecomposable projective Λ-module P . In fact ν1 −i P = τ −i P ∈ mod Λ for all i ≥ 0. Inspired by this we make <strong>the</strong> following definition. Definition 2. Let Λ be a finite dimensional K-algebra with gl.dim Λ ≤ n. We say that Λ is n-representation infinite if ν −i n Λ ∈ mod Λ for all i ≥ 0. We remark that this condition is equivalent to ν i n(DΛ) ∈ mod Λ for all i ≥ 0. In <strong>the</strong> classical setting <strong>of</strong> n = 1 every indecomposable module is ei<strong>the</strong>r preprojective, preinjective or regular. We define higher analogues <strong>of</strong> <strong>the</strong>se classes <strong>of</strong> modules as follows. Definition 3. Let Λ be an n-representation infinite algebra. The full subcategories <strong>of</strong> n-preprojective, n-preinjective and n-regular modules are defined as respectively. P := add{ν −i n Λ | i ≥ 0}, I := add{ν i n(DΛ) | i ≥ 0}, R := {X ∈ mod Λ | Ext i Λ(P, X) = 0 = Ext i Λ(X, I) for all i ≥ 0}, –56–
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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
Page 19 and 20: Then T = gen(X) and X is Ext-projec
Page 21 and 22: DIMENSIONS OF DERIVED CATEGORIES TA
Page 23 and 24: Notation 4. (1) Let A be an abelian
Page 25 and 26: of finitely genera
Page 27 and 28: is the map given b
Page 29 and 30: (2) Let R be a commutative ring whi
Page 31 and 32: QUIVER PRESENTATIONS OF GROTHENDIEC
Page 33 and 34: Proposition 3. Let C be a category,
Page 35 and 36: Example 11. Let Q be the</s
Page 37 and 38: and Gr(X ′ ) is given by
Page 39 and 40: particular, the de
Page 41 and 42: Proposition 1 and 2 leave us with d
Page 43 and 44: 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: We now specialise to the</s
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
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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
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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
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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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