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

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[16] W. C. Huffman and V. Pless, Fundamentals <strong>of</strong> error-correcting codes, Cambridge University Press, Cambridge, 2003. MR 1996953 (2004k:94077) [17] M. Klemm, Eine Invarianzgruppe für die vollständige Gewichtsfunktion selbstdualer Codes, Arch. Math. (Basel) 53 (1989), no. 4, 332–336. MR 1015996 (91a:94032) [18] V. L. Kurakin, A. S. Kuzmin, V. T. Markov, A. V. Mikhalev, and A. A. Nechaev, Linear codes and polylinear recurrences over finite rings and modules (a survey), Applied algebra, algebraic algorithms and error-correcting codes (Honolulu, HI, 1999) (M. Fossorier, H. Imai, S. Lin, and A. Poli, eds.), Lecture Notes in Comput. Sci., vol. 1719, Springer, Berlin, 1999, pp. 365–391. MR 1846512 (2002h:94092) [19] T. Y. Lam, Lectures on modules and rings, Graduate Texts in Ma<strong>the</strong>matics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294 (99i:16001) [20] , A first course in noncommutative rings, second ed., Graduate Texts in Ma<strong>the</strong>matics, vol. 131, Springer-Verlag, New York, 2001. MR 1838439 (2002c:16001) [21] F. J. MacWilliams, Error-correcting codes for multiple-level transmission, Bell System Tech. J. 40 (1961), 281–308. MR 0141541 (25 #4945) [22] , Combinatorial properties <strong>of</strong> elementary abelian groups, Ph.D. <strong>the</strong>sis, Radcliffe College, Cambridge, Mass., 1962. [23] F. J. MacWilliams and N. J. A. Sloane, The <strong>the</strong>ory <strong>of</strong> error-correcting codes, North-Holland Publishing Co., Amsterdam, 1977, North-Holland Ma<strong>the</strong>matical Library, Vol. 16. MR 0465509 (57 #5408a) [24] G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-dual codes and invariant <strong>the</strong>ory, Algorithms and Computation in Ma<strong>the</strong>matics, vol. 17, Springer-Verlag, Berlin, 2006. MR 2209183 (2007d:94066) [25] A. A. Nechaev, Kerdock code in cyclic form, Discrete Ma<strong>the</strong>matics and Applications 1 (1991), no. 4, 365–384. [26] J.-P. Serre, Linear representations <strong>of</strong> finite groups, Springer-Verlag, New York, 1977, Translated from <strong>the</strong> second French edition by Leonard L. Scott, Graduate Texts in Ma<strong>the</strong>matics, Vol. 42. MR 0450380 (56 #8675) [27] C. E. Shannon, A ma<strong>the</strong>matical <strong>the</strong>ory <strong>of</strong> communication, Bell System Tech. J. 27 (1948), 379–423, 623–656. MR 0026286 (10,133e) [28] A. Terras, Fourier analysis on finite groups and applications, London Ma<strong>the</strong>matical Society Student Texts, vol. 43, Cambridge University Press, Cambridge, 1999. MR 1695775 (2000d:11003) [29] H. N. Ward and J. A. Wood, Characters and <strong>the</strong> equivalence <strong>of</strong> codes, J. Combin. Theory Ser. A 73 (1996), no. 2, 348–352. MR 1370137 (96i:94028) [30] E. Witt, Theorie der quadratischen Formen in beliebigen Körpern, J. Reine. Angew. Math. 176 (1937), 31–44. [31] J. A. Wood, Duality for modules over finite rings and applications to coding <strong>the</strong>ory, Amer. J. Math. 121 (1999), no. 3, 555–575. MR 1738408 (2001d:94033) [32] , Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc. 136 (2008), no. 2, 699–706. MR 2358511 (2008j:94062) [33] , Foundations <strong>of</strong> linear codes defined over finite modules: <strong>the</strong> extension <strong>the</strong>orem and <strong>the</strong> MacWilliams identities, Codes over rings (Ankara, 2008) (P. Solé, ed.), World Scientific, Singapore, 2009. [34] , Anti-isomorphisms, character modules and self-dual codes over non-commutative rings, Int. J. Information and Coding Theory 1 (2010), no. 4, 429–444, Special Issue on Algebraic and Combinatorial Coding Theory: in Honour <strong>of</strong> <strong>the</strong> Retirement <strong>of</strong> Vera Pless. Series 3. Department <strong>of</strong> Ma<strong>the</strong>matics Western Michigan University 1903 W. Michigan Ave. Kalamazoo, MI 49008–5248 USA E-mail address: jay.wood@wmich.edu –245–
REALIZING STABLE CATEGORIES AS DERIVE CATEGORIES KOTA YAMAURA Abstract. In this paper, we compare two different kinds <strong>of</strong> triangulated categories. First one is <strong>the</strong> stable category modA <strong>of</strong> <strong>the</strong> category <strong>of</strong> Z-graded modules over a positively grade self-injective algebra A. Second one is <strong>the</strong> derived category D b (modΛ) <strong>of</strong> <strong>the</strong> category <strong>of</strong> modules over an algebra Λ. Our aim is give <strong>the</strong> complete answer to <strong>the</strong> following question. For a positively graded self-injective algebra A, when is modA triangle-equivalent to D b (modΛ) for some algebra Λ ? The main result <strong>of</strong> this paper gives <strong>the</strong> following very simple answer. modA is triangle-equivalent to D b (modΛ) for some algebra Λ if and only if <strong>the</strong> 0-th subring A 0 <strong>of</strong> A has finite global dimension. 1. Main Result There are two kinds <strong>of</strong> triangulated categories which are important for representation <strong>the</strong>ory for algebras. First one is <strong>the</strong> derived category D b (modΛ) <strong>of</strong> <strong>the</strong> category modA <strong>of</strong> modules over an algebra Λ. Second one is algebraic triangulated categories, that is <strong>the</strong> stable categories <strong>of</strong> Frobenius categories (cf. [5]). A typical example is <strong>the</strong> stable category modA <strong>of</strong> <strong>the</strong> category modA <strong>of</strong> modules over a self-injective algebra A. In this paper, our aim is to compare derived categories <strong>of</strong> algebras and <strong>the</strong> stable categories <strong>of</strong> self-injective algebras, and find a ”nice” relationship between <strong>the</strong>m. If we find it, <strong>the</strong>n those triangulated categories can be investigated from mutual viewpoints. There several method to compare derived categories <strong>of</strong> algebras and <strong>the</strong> stable categories <strong>of</strong> self-injective algebras. We focus on <strong>the</strong> following Happel’s result. For any algebra Λ, one can associate a self-injective algebra A which is called <strong>the</strong> trivial extension <strong>of</strong> Λ. A admits a natural positively grading such that A 0 = Λ where A 0 is <strong>the</strong> 0-th subring. Therefore A is a positively graded self-injective algebra. So <strong>the</strong> stable category mod Z A <strong>of</strong> <strong>the</strong> category mod Z A <strong>of</strong> Z-graded A-modules has <strong>the</strong> structure <strong>of</strong> triangulated category. In this setting, D. Happel [6] showed that Λ has finite global dimension if and only if <strong>the</strong>re exists a triangle-eqiuvalence (1.1) mod Z A ≃ D b (modΛ). This equivalence gives a ”nice” relationship between derived category D b (modΛ) and <strong>the</strong> stable categories mod Z A. The above result asserts that sometimes representation <strong>the</strong>ory <strong>of</strong> Λ and that <strong>of</strong> A are deeply related. We consider <strong>the</strong> drastic generalization <strong>of</strong> <strong>the</strong> above Happel’s result. Happel started from an algebra Λ, and constructed <strong>the</strong> special positively graded self-injective algebra <strong>of</strong> A. In contrast, we start from a positively graded self-injective algebra A = ⊕ i≥0 A i, and suggest <strong>the</strong> following question. The detailed version <strong>of</strong> this paper will be submitted for publication elsewhere. The author is supported by JSPS Fellowships for Young Scientists No.22-5801. –246–
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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
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X ∈ mod Π Q S 1 S 2 S 3 1 ❁
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The previous proposition has a dual
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Computing inductively all t
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Example 23. We have ⎛ ⎞ ⎛ ⎞
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classification of
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quiver Q over K, and let D b (H) <s
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Then Q ∈ Q 17 . Suppose char K
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Moreover the Hochs
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DERIVED AUTOEQUIVALENCES AND BRAID
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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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Page 211 and 212: MATRIX FACTORIZATIONS, ORBIFOLD CUR
Page 213 and 214: ( ∂f Jac(f t t ) := C[x 1 , . . .
Page 215 and 216: Definition 14. CM L f (R f ) Ob(C
Page 217 and 218: 4. Dolgachev Proposition 24 ([1]
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Page 221 and 222: (f, G f ) Dolgachev A Gf f Berg
Page 223 and 224: ON A GENERALIZATION OF COSTABLE TOR
Page 225 and 226: (2) P/t(P ) is σ-projective for an
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Page 231 and 232: GRADED FROBENIUS ALGEBRAS AND QUANT
Page 233 and 234: Definition 2. A graded algebra A is
Page 235 and 236: In this case, ν ∈ Aut k E induce
Page 237 and 238: References [1] X. W. Chen, Graded s
Page 239 and 240: The topics covered are ring-<strong
Page 241 and 242: (2.3) soc( R R) ∼ = k⊕ s i T i
Page 243 and 244: principal ideal Rr ⊂ ker ϖ. Thus
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Page 249 and 250: 4.1. Fourier Transform. Gleason’s
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Page 271 and 272: ALGEBRAIC STRATIFICATIONS OF DERIVE
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Page 275 and 276: Step 2: Let A be a finite-dimension
Page 277 and 278: RECOLLEMENTS GENERATED BY IDEMPOTEN
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Page 283 and 284: INTRODUCTION TO REPRESENTATION THEO
Page 285 and 286: is exact. Since E n ∼ = En+r , Mi
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Page 293 and 294: R : CM(R⊗ k V ) → CM(R) defined
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Page 297 and 298: SUBCATEGORIES OF EXTENSION MODULES
Page 299 and 300: Lemma 6. Let S 1 and S 2 be Serre s
Page 301 and 302: In the first row <
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