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

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Notice that P 2 = Q 2 = I. Let G be <strong>the</strong> group generated by P and Q (inside GL 2 (C)). Define an action <strong>of</strong> G on <strong>the</strong> polynomial ring C[X, Y ] by linear substitution: (fS)(X, Y ) = f((X, Y )S) for S ∈ G. The paragraphs above prove <strong>the</strong> following Proposition 38. Let C be a self-dual binary code. Then its Hamming weight enumerator W C (X, Y ) is invariant under <strong>the</strong> action <strong>of</strong> <strong>the</strong> group G. That is, W C (X, Y ) ∈ C[X, Y ] G , <strong>the</strong> ring <strong>of</strong> G-invariant polynomials. Much more is true, in fact. Let C 2 ⊂ F 2 2 be <strong>the</strong> linear code C 2 = {00, 11}. Then C 2 is self-dual, and W C2 (X, Y ) = X 2 + Y 2 . Let E 8 ⊂ F 8 2 be <strong>the</strong> linear code generated by <strong>the</strong> rows <strong>of</strong> <strong>the</strong> following binary matrix ⎛ ⎞ 1 1 1 1 0 0 0 0 ⎜0 0 1 1 1 1 0 0 ⎟ ⎝0 0 0 0 1 1 1 1⎠ . 1 0 1 0 1 0 1 0 Then E 8 is also self-dual, with W E8 (X, Y ) = X 8 + 14X 4 Y 4 + Y 8 . Theorem 39 (Gleason (1970)). The ring <strong>of</strong> G-invariant polynomials is generated as an algebra by W C2 and W E8 . That is, C[X, Y ] G = C[X 2 + Y 2 , X 8 + 14X 4 Y 4 + Y 8 ]. Gleason proved similar statements in several o<strong>the</strong>r contexts (doubly-even self-dual binary codes, self-dual ternary codes, Hermitian self-dual quaternary codes) [9]. The results all have this form: for linear codes <strong>of</strong> a certain type (e.g., binary self-dual), <strong>the</strong>ir Hamming weight enumerators are invariant under a certain finite matrix group G, and <strong>the</strong> ring <strong>of</strong> G-invariant polynomials is generated as an algebra by <strong>the</strong> weight enumerators <strong>of</strong> two explicit linear codes <strong>of</strong> <strong>the</strong> given type. Gleason’s Theorem has been generalized greatly by Nebe, Rains, and Sloane [24]. Those authors have a general definition <strong>of</strong> <strong>the</strong> type <strong>of</strong> a self-dual linear code defined over an alphabet A, where A is a finite left R-module. Associated to every type is a finite group G, called <strong>the</strong> Clifford-Weil group, and <strong>the</strong> (complete) weight enumerator <strong>of</strong> every selfdual linear code <strong>of</strong> <strong>the</strong> given type is G-invariant. Finally, <strong>the</strong> authors show (under certain hypo<strong>the</strong>ses on <strong>the</strong> ring R) that <strong>the</strong> ring <strong>of</strong> all G-invariant polynomials is spanned by weight enumerators <strong>of</strong> self-dual codes <strong>of</strong> <strong>the</strong> given type. In order to define self-dual codes over non-commutative rings, Nebe, Rains, and Sloane must cope with <strong>the</strong> difficulty that <strong>the</strong> dual code <strong>of</strong> a left linear code C in A n is a right linear code <strong>of</strong> <strong>the</strong> form (Ân : C) ⊂ Ân (cf., <strong>the</strong> pro<strong>of</strong> <strong>of</strong> Theorem 23 in subsection 4.2). This difficulty can be addressed first by assuming that <strong>the</strong> ring R admits an anti-isomorphism ε, i.e., an isomorphism ε : R → R <strong>of</strong> <strong>the</strong> additive group, with ε(rs) = ε(s)ε(r), for r, s ∈ R. Then every left (resp., right) R-module M defines a right (resp., left) R-module ε(M). The additive group <strong>of</strong> ε(M) is <strong>the</strong> same as that <strong>of</strong> M, and <strong>the</strong> right scalar multiplication on ε(M) is mr := ε(r)m, m ∈ M, r ∈ R, where ε(r)m uses <strong>the</strong> left scalar multiplication <strong>of</strong> M. (And similarly for right modules.) Secondly, in order to identify <strong>the</strong> character-<strong>the</strong>oretic annihilator (Ân : C) ⊂ Ân with a submodule in A n , Nebe, Rains, and Sloane assume <strong>the</strong> existence <strong>of</strong> an isomorphism –243–
ψ : ε(A) → Â. In this way, C⊥ := ε −1 ψ −1 (Ân : C) can be viewed as <strong>the</strong> dual code <strong>of</strong> C; C ⊥ is a left linear code in A n if C is. With one additional hypo<strong>the</strong>sis on ψ, C ⊥ satisfies all <strong>the</strong> properties one would want from a dual code, such as (C ⊥ ) ⊥ = C and <strong>the</strong> MacWilliams identities. (See [34] for an exposition.) There are several questions that arise immediately from <strong>the</strong> work <strong>of</strong> Nebe, Rains, and Sloane that may be <strong>of</strong> interest to ring <strong>the</strong>orists. (1) Which finite rings admit anti-isomorphisms? Involutions? (2) Assume a finite ring R admits an anti-isomorphism ε. Which finite left R-modules A admit an isomorphism ψ : ε(A) → Â? (3) Even in <strong>the</strong> absence <strong>of</strong> complete answers to <strong>the</strong> preceding, are <strong>the</strong>re good sources <strong>of</strong> examples? There are a few results in [34], but much more is needed. Progress on <strong>the</strong>se questions may prove helpful in understanding <strong>the</strong> limits and <strong>the</strong> proper setting for <strong>the</strong> work <strong>of</strong> Nebe, Rains, and Sloane. References [1] Č. Arf, Untersuchungen über quadratische Formen in Körpern der Charakteristik 2. I., J. Reine. Angew. Math. 183 (1941), 148–167. MR 0008069 (4,237f) [2] E. F. Assmus, Jr. and H. F. Mattson, Jr., Error-correcting codes: An axiomatic approach, Inform. and Control 6 (1963), 315–330. MR 0178997 (31 #3251) [3] , Coding and combinatorics, SIAM Rev. 16 (1974), 349–388. MR 0359982 (50 #12432) [4] H. Bass, K-<strong>the</strong>ory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5–60. MR 0174604 (30 #4805) [5] K. Bogart, D. Goldberg, and J. Gordon, An elementary pro<strong>of</strong> <strong>of</strong> <strong>the</strong> MacWilliams <strong>the</strong>orem on equivalence <strong>of</strong> codes, Inform. and Control 37 (1978), no. 1, 19–22. MR 0479646 (57 #19067) [6] H. L. Claasen and R. W. Goldbach, A field-like property <strong>of</strong> finite rings, Indag. Math. (N.S.) 3 (1992), no. 1, 11–26. MR 1157515 (93b:16038) [7] C. W. Curtis and I. Reiner, Representation <strong>the</strong>ory <strong>of</strong> finite groups and associative algebras, Pure and Applied Ma<strong>the</strong>matics, vol. XI, Interscience Publishers, a division <strong>of</strong> John Wiley & Sons, New York, London, 1962. MR 0144979 (26 #2519) [8] H. Q. Dinh and S. R. López-Permouth, On <strong>the</strong> equivalence <strong>of</strong> codes over rings and modules, Finite Fields Appl. 10 (2004), no. 4, 615–625. MR 2094161 (2005g:94098) [9] A. M. Gleason, Weight polynomials <strong>of</strong> self-dual codes and <strong>the</strong> MacWilliams identities, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3, Gauthier-Villars, Paris, 1971, pp. 211–215. MR 0424391 (54 #12354) [10] M. Greferath, Orthogonality matrices for modules over finite Frobenius rings and MacWilliams’ equivalence <strong>the</strong>orem, Finite Fields Appl. 8 (2002), no. 3, 323–331. MR 1910395 (2003d:94107) [11] M. Greferath, A. Nechaev, and R. Wisbauer, Finite quasi-Frobenius modules and linear codes, J. Algebra Appl. 3 (2004), no. 3, 247–272. MR 2096449 (2005g:94099) [12] M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams’s equivalence <strong>the</strong>orem, J. Combin. Theory Ser. A 92 (2000), no. 1, 17–28. MR 1783936 (2001j:94045) [13] A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, The Z 4 -linearity <strong>of</strong> Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory 40 (1994), no. 2, 301–319. MR 1294046 (95k:94030) [14] Y. Hirano, On admissible rings, Indag. Math. (N.S.) 8 (1997), no. 1, 55–59. MR 1617802 (99b:16034) [15] T. Honold, Characterization <strong>of</strong> finite Frobenius rings, Arch. Math. (Basel) 76 (2001), no. 6, 406–415. MR 1831096 (2002b:16033) –244–
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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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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]
Page 219 and 220: ✤ ✤ ✤ ✤ ✤ ✤ γ 1 +γ 2
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
Page 251 and 252: 5. The Extension Problem In this se
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Page 261 and 262: REALIZING STABLE CATEGORIES AS DERI
Page 263 and 264: 2.1. Positively graded self-injecti
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Page 267 and 268: Next we consider the</stron
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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
Page 279 and 280: (a) the adjoint tr
Page 281 and 282: Cluster-tilting the</strong
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
Page 295 and 296: P = P ′ t by t is a projective R
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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