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

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Pro<strong>of</strong>. Observe first that for each indecomposable non projective R-module M, we have l(Ω 2 M) = l(τM). Now, applying <strong>the</strong> previous Lemma 3.3, we obtain for each i ≥ 0 that l(Ω i B) ≤ l(Ω i C) + α/2, and l(Ω i+2 C) ≤ l(Ω i B) + α/2 for some positive number α. Hence, for each i ≥ 0 we have l(Ω i+2 C) − l(Ω i C) ≤ α. In particular, for each n ≥ 0 we have l(Ω 2n C) − l(C) ≤ nα. This means that <strong>the</strong> complexity <strong>of</strong> C is bounded by 2. If cx C = 1, <strong>the</strong>n, since it is not τ periodic, <strong>the</strong> module C must lie in a ZA ∞ -component by [17], but for <strong>the</strong>se components every irreducible map is eventually Ω-perfect. Hence cx C = 2. □ We obtain <strong>the</strong> following immediate consequence: assume that we have a component C, whose stable part C s is <strong>of</strong> <strong>the</strong> form ZA ∞ ∞, and assume also that <strong>the</strong>re exists an Auslander- Reiten sequence 0 → τC → E ⊕ F → C → 0 where E and F are indecomposable, and nei<strong>the</strong>r E → C nor F → C is eventually Ω-perfect. Observe also that in this case, no irreducible map in C s between indecomposable modules is eventually Ω-perfect. It follows immediately from <strong>the</strong> previous proposition that every non projective module in C has complexity 2. This situation can actually occur. The following example is due to Ringel. Example 18. Let R be <strong>the</strong> finite dimensional selfinjective string algebra given by <strong>the</strong> quiver α β 1 2 3 γ modulo <strong>the</strong> relations αβ = 0, δγ = 0, γαγα = βδβδ and αγαγα = δβδβδ = 0. There exists a ZA ∞ ∞ component where none <strong>of</strong> <strong>the</strong> irreducible maps between <strong>the</strong> indecomposable modules is eventually Ω-perfect, (or even τ-perfect). For instance, consider <strong>the</strong> string module M = r 3 P 2 . It is easy to see that M is not eventually Ω-perfect, that α(M) = 2, and that no irreducible map from an indecomposable module to M is eventually Ω-perfect. Moreover, by [6], M lies in a component consisting entirely <strong>of</strong> string modules. But <strong>the</strong> only string modules lying on <strong>the</strong> boundary <strong>of</strong> an Auslander-Reiten component can lie on tubes (see [12], II.6.4), so this module belongs to a ZA ∞ ∞ component. Note also that <strong>the</strong> simple modules S 1 and S 3 are Ω-periodic <strong>of</strong> period 6, and that <strong>the</strong>y both lie on tubes <strong>of</strong> rank 3. Example 19. Following Erdmann [12], for each positive integer m, we denote by Λ m <strong>the</strong> local symmetric string algebra over a field K, Λ m = K〈x, y〉/〈x 2 , (xy) m+1 − (yx) m+1 , x 2 − (yx) m y, x 3 〉 If <strong>the</strong> characteristic <strong>of</strong> K is 2, and m+1 = 2 n ≥ 4, <strong>the</strong>n <strong>the</strong> algebra Λ m modulo its socle is isomorphic to <strong>the</strong> group algebra <strong>of</strong> <strong>the</strong> semidihedral group <strong>of</strong> order 2 n+2 modulo its socle. Motivated by this fact, Erdmann calls this algebra semidihedral. She proves that Λ m has infinitely many stable components <strong>of</strong> type ZA ∞ ∞ and ZD ∞ ([12], Propositions II,10.1 and II,10.2), and that <strong>the</strong> o<strong>the</strong>r stable components are tubes <strong>of</strong> rank 1 and 2. Moreover, she shows that <strong>the</strong> unique simple module lies in a component <strong>of</strong> type ZD ∞ so it is not periodic. Therefore, every indecomposable non projective Λ m -module is eventually Ω-perfect by [18]. Note that in <strong>the</strong> same book, Erdmann generalizes <strong>the</strong> notion <strong>of</strong> semidihedral algebra to that <strong>of</strong> algebras <strong>of</strong> semidihedral type and one also obtains interesting examples for <strong>the</strong> non local case ([12], Lemma VIII. 2.1.). –85– δ
3.1. ZD ∞ -components. We assume for <strong>the</strong> remainder <strong>of</strong> this section that C is a connected Auslander-Reiten component whose stable part is <strong>of</strong> <strong>the</strong> form ZD ∞ . Let C be an indecomposable module lying on <strong>the</strong> boundary <strong>of</strong> C. Then, without loss <strong>of</strong> generality we may assume that C is Ω-perfect, by Remark 11. In this context we have <strong>the</strong> following: Lemma 20. Let A and B be two indecomposable modules lying on <strong>the</strong> boundary <strong>of</strong> C with Auslander-Reiten sequences 0 → τA f 1 −→ M g 1 −→ A → 0 and 0 → τB f 2 −→ M g 2 −→ B → 0. Then <strong>the</strong> irreducible map [g 1 , g 2 ] t : M → A ⊕ B is an epimorphism if and only if <strong>the</strong> map [f 1 , f 2 ]: τA ⊕ τB → M is also an epimorphism. Pro<strong>of</strong>. Counting lengths, we have l(τA)+l(A)+l(τB)+l(B) = 2l(M). This means that l(A) + l(B) < l(M) if and only if l(τA) + l(τB) > l(M). The result follows, since an irreducible map is ei<strong>the</strong>r a monomorphism, or an epimorphism. □ Keeping <strong>the</strong> notation from <strong>the</strong> lemma, we may clearly assume that <strong>the</strong> modules A and B lying on <strong>the</strong> boundary <strong>of</strong> <strong>the</strong> component are Ω-perfect, and that <strong>the</strong> Auslander-Reiten sequence ending at M is 0 → τM → τA ⊕ τB ⊕ τX → M → 0 for some indecomposable module X. Since <strong>the</strong> irreducible epimorphisms M → A and M → B are Ω-perfect, <strong>the</strong>n <strong>the</strong> irreducible epimorphisms τA ⊕ τX → M and τB ⊕ τX → M are also Ω-perfect being “parallel” to Ω-perfect epimorphisms. Similarly, <strong>the</strong> irreducible monomorphisms τM → τX ⊕ τA and τM → τX ⊕ τB are also Ω-perfect. Putting toge<strong>the</strong>r our remarks, we have: Proposition 21. Let C be an Auslander-Reiten component whose stable part is <strong>of</strong> type ZD ∞ . Assume that <strong>the</strong>re is an irreducible map between indecomposable non projective modules X → Y that is not eventually Ω-perfect. Then each non projective module in C has complexity 2. Pro<strong>of</strong>. From <strong>the</strong> shape <strong>of</strong> our component, it follows by looking at “parallel” maps one at a time, that we may assume that <strong>the</strong>re exists an irreducible map <strong>of</strong> <strong>the</strong> form τM → τA⊕τB or τA ⊕ τB → M that is not eventually Ω-perfect, where A and B are indecomposable modules lying on <strong>the</strong> boundary <strong>of</strong> C, and M is an indecomposable module. Observe that, nei<strong>the</strong>r τM → τA ⊕ τB nor τA ⊕ τB → M can be eventually Ω-perfect by Lemma 20. Being <strong>of</strong> type ZD ∞ means also that C cannot contain modules <strong>of</strong> complexity 1 by [17]. We apply now 17. and <strong>the</strong> result follows. □ We would like to propose <strong>the</strong> following questions summarizing <strong>the</strong> discussion in <strong>the</strong> first three sections. The first one has been around for some time and is due to Rickard [25]. Questions 22. Let R be a selfinjective algebra. (1) Assume that <strong>the</strong>re exists an indecomposable R-module <strong>of</strong> complexity greater than 2. Is R is <strong>of</strong> wild representation type? (2) Assume that R has stable components <strong>of</strong> type ZD ∞ or ZA ∞ ∞. Is R <strong>of</strong> tame representation type? Must <strong>the</strong>se components have complexity 2? (3) Assume that R has a stable component <strong>of</strong> type ZA ∞ . Is R necessarily <strong>of</strong> wild representation type? The answer to <strong>the</strong> first question is known to be yes if R admits a <strong>the</strong>ory <strong>of</strong> support varieties, for instance in <strong>the</strong> group algebras case. See also [14]. The answer to <strong>the</strong> second –86–
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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
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 and 96: says that components of</st
Page 97 and 98: then we obtain a c
Page 99: Proposition 16. Let C s be a stable
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
Page 147 and 148: THE NOETHERIAN PROPERTIES OF THE RI
Page 149 and 150: 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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