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

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<strong>the</strong> ring has finite global dimension (respectively, <strong>the</strong> scheme is regular) [14]. It has turned out by work <strong>of</strong> Oppermann and Šťovíček [11] that over a Noe<strong>the</strong>rian algebra (respectively, a projective scheme) all proper thick subcategories <strong>of</strong> <strong>the</strong> bounded derived category <strong>of</strong> finitely generated modules (respectively, coherent sheaves) containing perfect complexes have infinite dimension. However, <strong>the</strong>se do not apply for <strong>the</strong> finiteness <strong>of</strong> <strong>the</strong> derived dimension <strong>of</strong> a non-regular Noe<strong>the</strong>rian ring <strong>of</strong> positive Krull dimension. As a main result <strong>of</strong> <strong>the</strong> paper [14], Rouquier gave <strong>the</strong> following <strong>the</strong>orem. Theorem 1 (Rouquier). Let X be a separated scheme <strong>of</strong> finite type over a perfect field. Then <strong>the</strong> bounded derived category <strong>of</strong> coherent sheaves on X has finite dimension. Applying this <strong>the</strong>orem to an affine scheme, one obtains: Corollary 2. Let R be a commutative ring which is essentially <strong>of</strong> finite type over a perfect field k. Then <strong>the</strong> bounded derived category D b (mod R) <strong>of</strong> finitely generated R-modules has finite dimension, and so does <strong>the</strong> singularity category D Sg (R) <strong>of</strong> R. The main purpose <strong>of</strong> this paper is to study <strong>the</strong> dimension and generators <strong>of</strong> <strong>the</strong> bounded derived category <strong>of</strong> finitely generated modules over a commutative Noe<strong>the</strong>rian ring. We will give lower bounds <strong>of</strong> <strong>the</strong> dimensions over general rings under some mild assumptions, and over some special rings we will also give upper bounds and explicit generators. The main result <strong>of</strong> this paper is <strong>the</strong> following <strong>the</strong>orem. (See Definition 5 for <strong>the</strong> notation.) Main Theorem. Let R be ei<strong>the</strong>r a complete local ring containing a field with perfect residue field or a ring that is essentially <strong>of</strong> finite type over a perfect field. Then <strong>the</strong>re exist a finite number <strong>of</strong> prime ideals p 1 , . . . , p n <strong>of</strong> R and an integer m ≥ 1 such that D b (mod R) = 〈R/p 1 ⊕ · · · ⊕ R/p n 〉 m . In particular, D b (mod R) and D Sg (R) have finite dimension. In Rouquier’s result stated above, <strong>the</strong> essential role is played, in <strong>the</strong> affine case, by <strong>the</strong> Noe<strong>the</strong>rian property <strong>of</strong> <strong>the</strong> enveloping algebra R ⊗ k R. The result does not apply to a complete local ring, since it is in general far from being (essentially) <strong>of</strong> finite type and <strong>the</strong>refore <strong>the</strong> enveloping algebra is non-Noe<strong>the</strong>rian. Our methods not only show finiteness <strong>of</strong> dimensions over a complete local ring but also give a ring-<strong>the</strong>oretic pro<strong>of</strong> <strong>of</strong> Corollary 2. 2. Preliminaries This section is devoted to stating our convention, giving some basic notation and recalling <strong>the</strong> definition <strong>of</strong> <strong>the</strong> dimension <strong>of</strong> a triangulated category. We assume <strong>the</strong> following throughout this paper. Convention 3. (1) All subcategories are full and closed under isomorphisms. (2) All rings are associative and with identities. (3) A Noe<strong>the</strong>rian ring, an Artinian ring and a module mean a right Noe<strong>the</strong>rian ring, a right Artinian ring and a right module, respectively. (4) All complexes are cochain complexes. We use <strong>the</strong> following notation. –7–
Notation 4. (1) Let A be an abelian category. (a) For a subcategory X <strong>of</strong> A, <strong>the</strong> smallest subcategory <strong>of</strong> A containing X which is closed under finite direct sums and direct summands is denoted by add A X . (b) We denote by C(A) <strong>the</strong> category <strong>of</strong> complexes <strong>of</strong> objects <strong>of</strong> A. The derived category <strong>of</strong> A is denoted by D(A). The left bounded, <strong>the</strong> right bounded and <strong>the</strong> bounded derived categories <strong>of</strong> A are denoted by D + (A), D − (A) and D b (A), respectively. We set D ∅ (A) = D(A), and <strong>of</strong>ten write D ⋆ (A) with ⋆ ∈ {∅, +, −, b} to mean D ∅ (A), D + (A), D − (A) and D b (A). (2) Let R be a ring. We denote by Mod R and mod R <strong>the</strong> category <strong>of</strong> R-modules and <strong>the</strong> category <strong>of</strong> finitely generated R-modules, respectively. For a subcategory X <strong>of</strong> mod R (when R is Noe<strong>the</strong>rian), we put add R X = add mod R X . The concept <strong>of</strong> <strong>the</strong> dimension <strong>of</strong> a triangulated category has been introduced by Rouquier [14]. Now we recall its definition. Definition 5. Let T be a triangulated category. (1) A triangulated subcategory <strong>of</strong> T is called thick if it is closed under direct summands. (2) Let X , Y be two subcategories <strong>of</strong> T . We denote by X ∗ Y <strong>the</strong> subcategory <strong>of</strong> T consisting <strong>of</strong> all objects M that admit exact triangles X → M → Y → ΣX with X ∈ X and Y ∈ Y. We denote by 〈X 〉 <strong>the</strong> smallest subcategory <strong>of</strong> T containing X which is closed under finite direct sums, direct summands and shifts. For a non-negative integer n, we define <strong>the</strong> subcategory 〈X 〉 n <strong>of</strong> T by ⎧ ⎪⎨ {0} (n = 0), 〈X 〉 n = 〈X 〉 (n = 1), ⎪⎩ 〈〈X 〉 ∗ 〈X 〉 n−1 〉 (2 ≤ n < ∞). Put 〈X 〉 ∞ = ∪ n≥0 〈X 〉 n . When <strong>the</strong> ground category T should be specified, we write 〈X 〉 T n instead <strong>of</strong> 〈X 〉 n . For a ring R and a subcategory X <strong>of</strong> D(Mod R), we put 〈X 〉 R R) n = 〈X 〉D(Mod n . (3) The dimension <strong>of</strong> T , denoted by dim T , is <strong>the</strong> infimum <strong>of</strong> <strong>the</strong> integers d such that <strong>the</strong>re exists an object M ∈ T with 〈M〉 d+1 = T . 3. Upper bounds The aim <strong>of</strong> this section is to find explicit generators and upper bounds <strong>of</strong> dimensions for derived categories in several cases. We observe that <strong>the</strong> dimensions <strong>of</strong> <strong>the</strong> bounded derived categories <strong>of</strong> finitely generated modules over quotient singularities are at most <strong>the</strong>ir (Krull) dimensions, particularly that <strong>the</strong>y are finite. Proposition 6. Let S be ei<strong>the</strong>r <strong>the</strong> polynomial ring k[x 1 , . . . , x n ] or <strong>the</strong> formal power series ring k[[x 1 , . . . , x n ]] over a field k. Let G be a finite subgroup <strong>of</strong> <strong>the</strong> general linear –8–
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: DIMENSIONS OF DERIVED CATEGORIES TA
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 and 70: We now specialise to the</s
Page 71 and 72: 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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✤ ✤ ✤ ✤ ✤ ✤ 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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