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

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R-module ̂R. Because | ̂R| = |R|, one <strong>of</strong> <strong>the</strong>se homomorphisms is an isomorphism if and only if it is injective if and only if it is surjective. Remark 4. The phrase generating character (“erzeugenden Charakter”) is due to Klemm [17]. Claasen and Goldbach [6] used <strong>the</strong> adjective admissible to describe <strong>the</strong> same phenomenon, although <strong>the</strong>ir use <strong>of</strong> left and right is <strong>the</strong> reverse <strong>of</strong> ours. The <strong>the</strong>orem below relates generating characters and finite Frobenius rings. While <strong>the</strong> <strong>the</strong>orem is over ten years old, we will give a new pro<strong>of</strong>. Theorem 5 ([14, Theorem 1], [31, Theorem 3.10]). Let R be a finite ring. Then <strong>the</strong> following are equivalent: (1) R is Frobenius; (2) R admits a left generating character, i.e., ̂R is a free left R-module; (3) R admits a right generating character, i.e., ̂R is a free right R-module. Moreover, when <strong>the</strong>se conditions are satisfied, every left generating character is also a right generating character, and vice versa. Example 6. Here are several examples <strong>of</strong> finite Frobenius rings and generating characters (when easy to describe). (1) Finite field F q with generating character ϑ q <strong>of</strong> Example 2. Note that ϑ p is injective, but that for q > p, ker ϑ q = ker tr q/p is a nonzero F p -linear subspace <strong>of</strong> F q . However, ker ϑ q is not an F q -linear subspace. (Compare with Proposition 7 below.) (2) Integer residue ring Z/nZ with generating character ϑ n defined by ϑ n (a) = a/n, for a ∈ Z/nZ. (3) Finite chain ring R; i.e., a finite ring all <strong>of</strong> whose left ideals form a chain under inclusion. See Corollary 15 for information about a generating character. (4) If R 1 , . . . , R n are Frobenius with generating characters ϱ 1 , . . . , ϱ n , <strong>the</strong>n <strong>the</strong>ir direct sum R = ⊕R i is Frobenius with generating character ϱ = ∑ ϱ i . Conversely, if R = ⊕R i is Frobenius with generating character ϱ, <strong>the</strong>n each R i is Frobenius, with generating character ϱ i = ϱ ◦ ι i , where ι i : R i → R is <strong>the</strong> inclusion; ϱ = ∑ ϱ i . (5) If R is Frobenius with generating character ϱ, <strong>the</strong>n <strong>the</strong> matrix ring M m (R) is Frobenius with generating character ϱ ◦ tr, where tr is <strong>the</strong> matrix trace. (6) If R is Frobenius with generating character ϱ and G is any finite group, <strong>the</strong>n <strong>the</strong> group ring R[G] is Frobenius with generating character ϱ ◦ pr e , where pr e : R[G] → R is <strong>the</strong> projection that associates to every element a = ∑ a g g ∈ R[G] <strong>the</strong> coefficient a e <strong>of</strong> <strong>the</strong> identity element <strong>of</strong> G. In preparation for <strong>the</strong> pro<strong>of</strong> <strong>of</strong> Theorem 5, we prove several propositions concerning generating characters. Proposition 7 ([6, Corollary 3.6]). Let R be a finite ring. A character ϖ <strong>of</strong> R is a left (resp., right) generating character if and only if ker ϖ contains no nonzero left (resp., right) ideal <strong>of</strong> R. Pro<strong>of</strong>. By <strong>the</strong> definition and | ̂R| = |R|, ϖ is a left generating character if and only if <strong>the</strong> homomorphism f : R → ̂R, r ↦→ rϖ, is injective. Then r ∈ ker f if and only if <strong>the</strong> –227–
principal ideal Rr ⊂ ker ϖ. Thus, ker f = 0 if and only if ker ϖ contains no nonzero left ideals. The pro<strong>of</strong> for right generating characters is similar. □ Proposition 8 ([31, Theorem 4.3]). A character ϱ <strong>of</strong> a finite ring R is a left generating character if and only if it is a right generating character. Pro<strong>of</strong>. Suppose ϱ is a left generating character, and suppose that I ⊂ ker ϱ is a right ideal. Then for every r ∈ R, Ir ⊂ ker ϱ, so that I ⊂ ker(rϱ), for all r ∈ R. But every character <strong>of</strong> R is <strong>of</strong> <strong>the</strong> form rϱ, because ϱ is a left generating character. Thus <strong>the</strong> annihilator ( ̂R : I) = ̂R, and it follows from (2.5) that I = 0. By Proposition 7, ϱ is a right generating character. □ Proposition 9 ([33, Proposition 3.3]). Let A be a finite left R-module. Then soc(Â) ∼ = (A/ rad(R)A)̂. Pro<strong>of</strong>. There is a short exact sequence <strong>of</strong> left R-modules 0 → rad(R)A → A → A/ rad(R)A → 0. Taking character modules, as in (2.4), yields 0 → (A/ rad(R)A)̂ → Â → (rad(R)A)̂ → 0. Because A/ rad(R)A is a sum <strong>of</strong> simple modules, <strong>the</strong> same is true for (A/ rad(R)A)̂ ∼ = (Â : rad(R)A). Thus (Â : rad(R)A) ⊂ soc(Â). Conversely, soc(Â) rad(R) = 0, because <strong>the</strong> radical annihilates simple modules [7, Exercise 25.4]. Thus soc(Â) ⊂ (Â : rad(R)A), and we have <strong>the</strong> equality soc(Â) = (Â : rad(R)A). Now remember that (Â : rad(R)A) ∼ = (A/ rad(R)A)̂. □ Using Proposition 7 as a model, we extend <strong>the</strong> definition <strong>of</strong> a generating character to modules. Let A be a finite left (resp., right) R-module. A character ϖ <strong>of</strong> A is a generating character <strong>of</strong> A if ker ϖ contains no nonzero left (resp., right) R-submodules <strong>of</strong> A. Lemma 10 (†). Let A be a finite left R-module, and let B ⊂ A be a submodule. If A admits a left generating character, <strong>the</strong>n B admits a left generating character. Pro<strong>of</strong>. Simply restrict a generating character <strong>of</strong> A to B. Any submodule <strong>of</strong> B inside <strong>the</strong> kernel <strong>of</strong> <strong>the</strong> restriction will also be a submodule <strong>of</strong> A inside <strong>the</strong> kernel <strong>of</strong> <strong>the</strong> original generating character. □ Lemma 11 (†). Let R be any finite ring. Define ϱ : ̂R → Q/Z by ϱ(ϖ) = ϖ(1), evaluation at 1 ∈ R, for ϖ ∈ ̂R. Then ϱ is a left and right generating character <strong>of</strong> ̂R. Pro<strong>of</strong>. Suppose ϖ 0 ≠ 0 has <strong>the</strong> property that Rϖ 0 ⊂ ker ϱ. This means that for every r ∈ R, 0 = ϱ(rϖ 0 ) = (rϖ 0 )(1) = ϖ 0 (r), so that ϖ 0 = 0. Thus ϱ is a left generating character by definition. Similarly for ϱ being a right generating character. □ Proposition 12 (†). Let A be a finite left R-module. Then A admits a left generating character if and only if A can be embedded in ̂R. –228–
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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 ) =
Page 191 and 192: (2) P Lie theory
Page 193 and 194: Example 15. Γ loop quiver λ ∈
Page 195 and 196: P (Γ)-module i-th radical B →
Page 197 and 198: B ∈ Λ(d) ε ∗ i (B) := dim C
Page 199 and 200: I-graded vector space V (d) (B τ )
Page 201 and 202: A ∗(i) l (a) = ∑n+1 t=l+1 (a i,
Page 203 and 204: wt(a) = (−d 1 , −d 2 , −d 3 )
Page 205 and 206: Ω Q Ω ( B Ω ; wt, ε i , ϕ
Page 207 and 208: “Λ(d) G(d)-” Definition 25.
Page 209 and 210: (( ( ) ( ) 0 s 0 0 0 , , , s) ( u 0
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
Page 227 and 228: (3) P is a maximal σ-coessential e
Page 229 and 230: (2)→(1): Let σ be epi-preserving
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: (2.3) soc( R R) ∼ = k⊕ s i T i
Page 245 and 246: By Example 3, the
Page 247 and 248: weight wt(x) = d(x, 0) of</
Page 249 and 250: 4.1. Fourier Transform. Gleason’s
Page 251 and 252: 5. The Extension Problem In this se
Page 253 and 254: Because R is assumed to be Frobeniu
Page 255 and 256: is injective for every finite left
Page 257 and 258: linear code defined by η;
Page 259 and 260: ψ : ε(A) → Â. In this way, C
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
Page 269 and 270: (2) There exists a triangle-equival
Page 271 and 272: ALGEBRAIC STRATIFICATIONS OF DERIVE
Page 273 and 274: The leaves of <str
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 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 289 and 290: (Proof) ( ) p 0 0
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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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