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

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THE LOEWY LENGTH OF TENSOR PRODUCTS FOR DIHEDRAL TWO-GROUPS ERIK DARPÖ AND CHRISTOPHER C. GILL Abstract. The indecomposable modules <strong>of</strong> a dihedral 2-group over a field <strong>of</strong> characteristic 2 were classified by Ringel over 30 years ago. However, relatively little is known about <strong>the</strong> tensor products <strong>of</strong> such modules, except in certain special cases. We describe here <strong>the</strong> main result <strong>of</strong> our recent work determining <strong>the</strong> Loewy length <strong>of</strong> a tensor product <strong>of</strong> modules for a dihedral 2-group. As a consequence <strong>of</strong> this result, we can determine precisely when a tensor product has a projective direct summand. 1. Introduction Let k be a field <strong>of</strong> positive characteristic p and let G be a finite group. The group algebra kG is a Hopf algebra with coproduct and co-unit given by ∆( ∑ g∈G r gg) = ∑ g∈G r gg ⊗ g and ɛ( ∑ g∈G r gg) = ∑ g∈G r g for r g ∈ k. Thus, <strong>the</strong>re is a tensor product operation on <strong>the</strong> category <strong>of</strong> kG-modules. If M and N are kG-modules, <strong>the</strong>n <strong>the</strong> tensor product <strong>of</strong> M and N is <strong>the</strong> module with underlying vector space M ⊗ k N and module structure given by g(m ⊗ n) = ∆(g)(m ⊗ n) = gm ⊗ gn for g ∈ G, m ∈ M, n ∈ N. The tensor product is a frequently used tool in <strong>the</strong> representation <strong>the</strong>ory <strong>of</strong> finite groups. However, <strong>the</strong> problem <strong>of</strong> determining <strong>the</strong> decomposition <strong>of</strong> a tensor product <strong>of</strong> two modules <strong>of</strong> a finite group G – <strong>the</strong> Clebsch-Gordan problem – can be extremely difficult. One approach to understanding tensor products <strong>of</strong> kG-modules goes via <strong>the</strong> representation ring, or Green ring, <strong>of</strong> kG. The isomorphism classes <strong>of</strong> finite-dimensional kG-modules form a semiring, with addition given by <strong>the</strong> direct sum, and multiplication by <strong>the</strong> tensor product <strong>of</strong> kG-modules. The Green ring, A(kG), is <strong>the</strong> Groe<strong>the</strong>ndieck ring <strong>of</strong> this semiring, i.e., <strong>the</strong> ring obtained by formally adjoining additive inverses to all elements in <strong>the</strong> semiring. Research in this direction was pioneered by J. A. Green [6], who proved <strong>the</strong> Green ring <strong>of</strong> a cyclic p-group is semisimple. The question <strong>of</strong> semisimplicity <strong>of</strong> <strong>the</strong> Green ring for o<strong>the</strong>r finite groups has been studied by several authors since. Benson and Carlson [2] gave a method to produce nilpotent elements in a Green ring, and determined a quotient <strong>of</strong> <strong>the</strong> Green ring which has no nilpotent elements. This so-called Benson–Carlson quotient <strong>of</strong> <strong>the</strong> Green ring was studied by Archer [1] in <strong>the</strong> case <strong>of</strong> <strong>the</strong> dihedral 2-groups, who realised it as an integral group ring <strong>of</strong> an abelian, infinitely generated, torsion-free group. Archer gave a precise statement relating <strong>the</strong> multiplication <strong>of</strong> two elements in this infinite group to <strong>the</strong> Auslander–Reiten quiver <strong>of</strong> kD 4q . The Green ring <strong>of</strong> <strong>the</strong> Klein four group V 4 was completely determined by Conlon [4]; a summary <strong>of</strong> this result can be found in [1]. For <strong>the</strong> dihedral 2-groups D 4q , <strong>the</strong> indecomposable modules, over fields <strong>of</strong> characteristic 2, were classified by Ringel [7] over 30 years ago. However, very little progress has been made towards understanding <strong>the</strong> behaviour <strong>of</strong> <strong>the</strong> tensor product <strong>of</strong> <strong>the</strong> kD 4q -modules. In –23–
particular, <strong>the</strong> decomposition <strong>of</strong> a tensor product <strong>of</strong> two indecomposable kD 4q -modules is not known, o<strong>the</strong>r than in some very special cases. One example is <strong>the</strong> work <strong>of</strong> Bessenrodt [3], classifying <strong>the</strong> endotrivial kD 4q -modules, thus determining <strong>the</strong> kD 4q -modules M for which <strong>the</strong> tensor product <strong>of</strong> M with its dual M ∗ is <strong>the</strong> direct sum <strong>of</strong> a trivial and a projective module. In recent work [5], we have continued <strong>the</strong> study <strong>of</strong> tensor products <strong>of</strong> kD 4q -modules, determining completely <strong>the</strong> Loewy length <strong>of</strong> <strong>the</strong> tensor product <strong>of</strong> any two indecomposable kD 4q -modules. This provides an additional piece <strong>of</strong> information towards <strong>the</strong> understanding <strong>of</strong> <strong>the</strong> Green rings <strong>of</strong> <strong>the</strong> dihedral 2-groups, and gives certain bounds on which modules can occur as direct summands <strong>of</strong> a tensor product. In particular, it determines precisely when a tensor product <strong>of</strong> two modules has a projective direct summand. The Loewy length l(M) <strong>of</strong> a module M is, by definition, <strong>the</strong> common length <strong>of</strong> <strong>the</strong> radical series and <strong>the</strong> socle series <strong>of</strong> M, that is, l(M) = min{t ∈ N | rad t (kD 4q )M = 0}. In <strong>the</strong> next section, we recall Ringel’s classification <strong>of</strong> <strong>the</strong> indecomposable kD 4q -modules. Section 3 gives a summary <strong>of</strong> <strong>the</strong> results in [5], and in Section 4, we give examples illuminating our results and showing how <strong>the</strong>y can be used to determine <strong>the</strong> direct sum decomposition <strong>of</strong> a tensor product in certain cases. 2. The indecomposable modules <strong>of</strong> dihedral 2-groups Let q be a 2-power, and write D 4q = 〈x, y | x 2 = y 2 = 1, (xy) q = (yx) q 〉 for <strong>the</strong> dihedral group <strong>of</strong> order 4q. There is an isomorphism <strong>of</strong> algebras kD 4q ˜→ Λ q := k〈X, Y 〉 (X 2 , Y 2 , (XY ) q − (Y X) q ) , given by x ↦→ 1 + X and y ↦→ 1 + Y . Setting ∆(X) = 1 ⊗ X + X ⊗ 1 + X ⊗ X and ∆(Y ) = 1 ⊗ Y + Y ⊗ 1 + Y ⊗ Y defines a coproduct on Λ q corresponding under this isomorphism to <strong>the</strong> Hopf algebra structure <strong>of</strong> kD 4q . Owing to <strong>the</strong> fact that Λ q is a special biserial algebra, its non-projective modules split into two classes, known as string modules and band modules. We describe both classes <strong>of</strong> modules below. Let ¯W be <strong>the</strong> set <strong>of</strong> words in letters a, b and inverse letters a −1 , b −1 such that a or a −1 are always followed by b or b −1 and b or b −1 are always followed by a or a −1 . A directed subword <strong>of</strong> a word w ∈ ¯W is a word w ′ in ei<strong>the</strong>r <strong>the</strong> letters {a, b} or {a −1 , b −1 } such that w = w 1 w ′ w 2 for some words w 1 , w 2 ∈ ¯W. Let W be <strong>the</strong> subset <strong>of</strong> ¯W consisting <strong>of</strong> words in which all directed subwords are <strong>of</strong> length strictly less than 2q. Define an equivalence relation ∼ 1 on W by w ∼ 1 w ′ if, and only if, w ′ = w or w ′ = w −1 . Given w = l 1 . . . l n ∈ W, <strong>the</strong> string module determined by w, denoted by M(w), is <strong>the</strong> n + 1-dimensional module with basis e 0 , . . . , e n and Λ q -action given by <strong>the</strong> following schema: l 1 l ke 0 ke 1 2 ke 2 . . . l n−1 l ke n−1 n ke n . If l i ∈ {a −1 , b −1 }, <strong>the</strong> corresponding arrow should be interpreted as going in <strong>the</strong> opposite direction, from ke i−1 to ke i , and having <strong>the</strong> label l −1 i . Now X maps e i to e j (j ∈ {i − 1, i + 1}) if <strong>the</strong>re is an arrow ke a i → ke j , and as zero if no arrow labelled with a starting in ke i exists. Similarly, <strong>the</strong> action <strong>of</strong> Y is given by arrows labelled with b. Two modules M(w) and M(w ′ ), w, w ′ ∈ W, are isomorphic if, and only if, w ∼ 1 w ′ . –24–
Page 1: Proceedings <stron
Page 5 and 6: Organizing Committee of</st
Page 7 and 8: Polycyclic codes and sequential cod
Page 9 and 10: Preface The 44th <
Page 11 and 12: 8:40-9:30 Dan ZachariaSyracuse Univ
Page 13 and 14: The 44th S
Page 15 and 16: Tuesday September 27 8:40-9:30 Atsu
Page 17 and 18: 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 Gr(X ′ ) is given by
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 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,
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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
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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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