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TILTING MODULES ARISING FROM TWO-TERM TILTING COMPLEXES HIROKI ABE Abstract. We see that every two-term tilting complex over an Artin algebra has a tilting module over a certain factor algebra as a homology group. Also, we determine <strong>the</strong> endomorphism algebra <strong>of</strong> such a homology group, which is given as a certain factor algebra <strong>of</strong> <strong>the</strong> endomorphism algebra <strong>of</strong> <strong>the</strong> two-term tilting complex. Thus, every derived equivalence between Artin algebras given by a two-term tilting complex induces a derived equivalence between <strong>the</strong> corresponding factor algebras. Let A be an Artin algebra. We denote by mod-A <strong>the</strong> category <strong>of</strong> finitely generated right A-modules and by P A <strong>the</strong> full subcategory <strong>of</strong> mod-A consisting <strong>of</strong> projective modules. Definition 1. A pair (T , F) <strong>of</strong> full subcategories T , F in mod-A is said to be a torsion <strong>the</strong>ory for mod-A if <strong>the</strong> following conditions are satisfied: (1) T ∩ F = {0}; (2) T is closed under factor modules; (3) F is closed under submodules; and (4) for any X ∈ mod-A, <strong>the</strong>re exists an exact sequence 0 → X ′ → X → X ′′ → 0 with X ′ ∈ T and X ′′ ∈ F. If T is stable under <strong>the</strong> Nakayama functor ν, <strong>the</strong>n (T , F) is said to be a stable torsion <strong>the</strong>ory for mod-A. Let T • ∈ K b (P A ) be a two-term complex: T • : · · · → 0 → T −1 α → T 0 → 0 → · · · , and set <strong>the</strong> following subcategories in mod-A: T (T • ) = Ker Hom K(A) (T • [−1], −) ∩ mod-A, F(T • ) = Ker Hom K(A) (T • , −) ∩ mod-A. Proposition 2 ([1, Propositions 5.5 and 5.7]). The following are equivalent. (1) T • is a tilting complex. (2) (T (T • ), F(T • )) is a stable torsion <strong>the</strong>ory for mod-A. Fur<strong>the</strong>rmore, if <strong>the</strong>se equivalent conditions hold, <strong>the</strong>n <strong>the</strong> following hold. (1) T (T • ) = gen(H 0 (T • )), <strong>the</strong> generated class by H 0 (T • ), and H 0 (T • ) is Ext-projective in T (T • ). (2) F(T • ) = cog(H −1 (νT • )), <strong>the</strong> cogenerated class by H −1 (νT • ) and H −1 (νT • ) is Extinjective in F(T • ). The detailed version <strong>of</strong> this note has been submitted for publication elsewhere. –1–
Conversely, let (T , F) be a stable torsion <strong>the</strong>ory for mod-A. Proposition 3 ([1, Theorem 5.8]). Assume that <strong>the</strong>re exist X ∈ T and Y ∈ F satisfying <strong>the</strong> following conditions: (1) T = gen(X) and X is Ext-projective in T ; and (2) F = cog(Y ) and Y is Ext-injective in F. Let P X • be a minimal projective presentation <strong>of</strong> X and I• Y be a minimal injective presentation <strong>of</strong> Y , and set T X,Y • = P X • ⊕ ν−1 I Y • [1]. Then T X,Y • ∈ Kb (P A ) is a tilting complex such that T = T (T X,Y • ) and F = F(T X,Y • ). Let T • be a two-term tilting complex. We set a = ann A (H 0 (T • )), <strong>the</strong> annihilator <strong>of</strong> H 0 (T • ). Note that H 0 (T • ) is faithful in mod-A/a and <strong>the</strong> canonical full embedding mod-A/a ↩→ mod-A induces gen(H 0 (T • ) A/a ) = gen(H 0 (T • ) A ) which is closed under extensions. Thus, <strong>the</strong> next lemma follows from Proposition 2. Lemma 4. The following hold. (1) proj dim H 0 (T • ) A/a ≤ 1. (2) Ext 1 A/a(H 0 (T • ), H 0 (T • )) = 0. (3) There exists an exact sequence 0 → A/a → X 0 → X 1 → 0 in mod-A/a such that X 0 ∈ add(H 0 (T • ) A/a ) and X 1 ∈ gen(H 0 (T • ) A/a ) which is Ext-projective in gen(H 0 (T • ) A/a ). We set a ′ = ann A (H −1 (νT • )), <strong>the</strong> annihilator <strong>of</strong> H −1 (νT • ). The next lemma follows by <strong>the</strong> dual arguments <strong>of</strong> Lemma 4 Lemma 5. The following hold. (1) inj dim H −1 (νT • ) A/a ′ ≤ 1. (2) Ext 1 A/a ′(H−1 (νT • ), H −1 (νT • )) = 0. (3) There exists an exact sequence 0 → Y 1 → Y 0 → A/a ′ → 0 in mod-A/a ′ such that Y 0 ∈ add(H −1 (νT • ) A/a ′) and Y 1 ∈ cog(H −1 (νT • ) A/a ′) which is Ext-injective in cog(H −1 (νT • ) A/a ′). Let X be <strong>the</strong> direct sum <strong>of</strong> all indecomposable non-projective Ext-projective modules in gen(H 0 (T • )) which are not contained in add(H 0 (T • )). Then add(H 0 (T • )⊕X) coincides with <strong>the</strong> class <strong>of</strong> all Ext-projective modules in gen(H 0 (T • )). Also, since gen(H 0 (T • )) = gen(H 0 (T • ) ⊕ X), <strong>the</strong> pair (gen(H 0 (T • ) ⊕ X), cog(H −1 (νT • )) is a stable torsion <strong>the</strong>ory in mod-A. Let P • be <strong>the</strong> minimal projective presentation <strong>of</strong> H 0 (T • ) ⊕ X and I • be <strong>the</strong> minimal injective presentation <strong>of</strong> H −1 (νT • ), and set U • = P • ⊕ ν −1 I • [1]. Then U • is a tilting complex such that T (U • ) = gen(H 0 (T • ) ⊕ X) and F(U • ) = cog(H −1 (νT • )) by Proposition 3. Note that <strong>the</strong> stable torsion <strong>the</strong>ory induced by U • coincides with that <strong>of</strong> T • . From this fact, we can prove that add(H 0 (U • )) = add(H 0 (T • )). Since <strong>the</strong>re exist <strong>the</strong> inclusions add(H 0 (T • )) ⊂ add(H 0 (T • ) ⊕ X) ⊂ add(H 0 (U • )), we conclude that add(H 0 (T • )) = add(H 0 (T • ) ⊕ X). Thus, we have <strong>the</strong> next lemma. Lemma 6. For any M, N ∈ mod-A, <strong>the</strong> following hold. (1) M ∈ add(H 0 (T • )) if and only if M is Ext-projective in gen(H 0 (T • )). (2) N ∈ add(H −1 (νT • )) if and only if N is Ext-injective in cog(H −1 (νT • )). –2–
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: Tuesday September 27 8:40-9:30 Atsu
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
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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
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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 ⎛ ⎞ ⎛ ⎞
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3. Periodic Twists We now describe
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We now specialise to the</s
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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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