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

More documents

Recommendations

Info

The first and last points were already proved in [8]. Here’s one way to see <strong>the</strong> first point. We know from (2.3) that soc( R R) is a sum <strong>of</strong> matrix modules M mi ×s i (F qi ). If m i ≥ s i for all i, <strong>the</strong>n each <strong>of</strong> <strong>the</strong> M mi ×s i (F qi ) would admit a generating character, by Theorem 13. By adding <strong>the</strong>se generating characters, one would obtain a generating character for soc( R R) itself. Then, by Proposition 14, R would admit a generating character, and hence would be Frobenius by Theorem 5. For <strong>the</strong> third point, consider counter-examples C 1 , C 2 ⊂ A n to <strong>the</strong> extension property for <strong>the</strong> alphabet A with respect to <strong>the</strong> Hamming weight. Because A n ⊂ soc( R R) n ⊂ R R n , C 1 , C 2 can also be viewed as R-modules via (2.1). The Hamming weight <strong>of</strong> an element x <strong>of</strong> A n equals <strong>the</strong> Hamming weight <strong>of</strong> x considered as an element <strong>of</strong> R n , because <strong>the</strong> Hamming weight just depends upon <strong>the</strong> entries <strong>of</strong> x being zero or not. In this way, C 1 , C 2 will also be counter-examples to <strong>the</strong> extension property for <strong>the</strong> alphabet R with respect to <strong>the</strong> Hamming weight. Thus, <strong>the</strong> key step remaining is <strong>the</strong> second point in <strong>the</strong> strategy. An explicit construction <strong>of</strong> counter-examples to <strong>the</strong> extension property for <strong>the</strong> alphabet A = M m×k (F q ), m < k, was given in [32]. Here, we give a short existence pro<strong>of</strong>; more details are available in [32] and [33]. Let R = M m (F q ) be <strong>the</strong> ring <strong>of</strong> m × m matrices over F q . Let A = M m×k (F q ), with m < k; A is a left R-module. It is clear from Theorem 36 that A will fail to have <strong>the</strong> extension property with respect to <strong>the</strong> Hamming weight if we can find a finite left R- module M with dim Q F 0 (O ♯ , Q) > dim Q F 0 (O, Q). It turns out that this inequality will hold for any nonzero M. Because R is simple, any finite left R-module M has <strong>the</strong> form M = M m×l (F q ), for some l. First, let us determine O, which is <strong>the</strong> set <strong>of</strong> left U-orbits on M. The group U is <strong>the</strong> group <strong>of</strong> units <strong>of</strong> R, which is precisely <strong>the</strong> general linear group GL m (F q ). The left orbits <strong>of</strong> GL m (F q ) on M = M m×l (F q ) are represented by <strong>the</strong> row reduced echelon matrices 1 over F q <strong>of</strong> size m × l. Now, let us determine O ♯ , which is <strong>the</strong> set <strong>of</strong> right Aut(A)-orbits on Hom R (M, A). The automorphism group Aut(A) equals GL k (F q ), acting on A = M m×k (F q ) by right matrix multiplication. On <strong>the</strong> o<strong>the</strong>r hand, Hom R (M, A) = M l×k (F q ), again using right matrix multiplication. Thus O ♯ consists <strong>of</strong> <strong>the</strong> right orbits <strong>of</strong> GL k (F q ) acting on M l×k (F q ). These orbits are represented by <strong>the</strong> column reduced echelon matrices over F q <strong>of</strong> size l × k. Because <strong>the</strong> matrix transpose interchanges row reduced echelon matrices and column reduced echelon matrices, we see that |O ♯ | > |O| if and only if k > m (for any positive l). Finally, notice that dim Q F 0 (O ♯ , Q) = |O ♯ |−1 and dim Q F 0 (O, Q) = |O|−1. Thus, for any nonzero module M, dim Q F 0 (O ♯ , Q) > dim Q F 0 (O, Q) if and only if m < k. Consequently, if m < k, <strong>the</strong>n W fails to be injective and A fails to have <strong>the</strong> extension property with respect to Hamming weight. 5.5. Technical Remarks. Here is <strong>the</strong> technical argument regarding <strong>the</strong> zero functional needed to justify Theorem 34. Remark 37. For η ∈ F (O ♯ , N), define <strong>the</strong> length <strong>of</strong> η to be l(η) = ∑ λ∈O η(λ) and <strong>the</strong> ♯ essential length <strong>of</strong> η to be l 0 (η) = ∑ λ≠0 η(λ). The length l(η) equals <strong>the</strong> length <strong>of</strong> <strong>the</strong> 1 Pr<strong>of</strong>. Yamagata tells me that <strong>the</strong> Japanese name for this concept translates literally as “step matrices.” –241–
linear code defined by η; <strong>the</strong> reduced length l 0 (η) equals <strong>the</strong> length <strong>of</strong> <strong>the</strong> linear code defined by η after any all-zero positions have been removed. (In terms <strong>of</strong> a generator matrix, one removes all <strong>the</strong> zero columns.) Assume <strong>the</strong> extension property holds with respect to <strong>the</strong> Hamming weight. This means that if η, η ′ ∈ F (O ♯ , N) satisfy l(η) = l(η ′ ) and W η = W η ′, <strong>the</strong>n η = η ′ . That is, W is injective along <strong>the</strong> level sets <strong>of</strong> <strong>the</strong> length function l. If l(η ′ ) < l(η) and W η = W η ′, <strong>the</strong>n we can append zeros to η ′ until its length is <strong>the</strong> same as l(η) without changing W η ′. More precisely, define η ′′ by η ′′ (λ) = η ′ (λ) for λ ≠ 0 and set η ′′ (0) = η ′ (0) + l(η) − l(η ′ ). Then l(η ′′ ) = l(η) and W η ′′ = W η . Then η ′′ = η, by <strong>the</strong> extension property. In particular, <strong>the</strong> reduced lengths are equal: l 0 (η) = l 0 (η ′ ) = l 0 (η ′′ ). There is a projection pr : F (O ♯ , N) → F 0 (O ♯ , N) which sets (pr η)(0) = 0 and leaves <strong>the</strong> o<strong>the</strong>r values unchanged, (pr η)(λ) = η(λ), λ ≠ 0. This projection splits <strong>the</strong> monoid as F (O ♯ , N) = F 0 (O ♯ , N) ⊕ N. The argument <strong>of</strong> <strong>the</strong> previous paragraph shows that if W η = W η ′, <strong>the</strong>n pr η = pr η ′ as elements <strong>of</strong> F 0 (O ♯ , N). Conversely, suppose W : F 0 (O ♯ , N) → F 0 (O, N) is injective. Let η, η ′ ∈ F (O ♯ , N) satisfy l(η) = l(η ′ ) and W η = W η ′. Because <strong>the</strong> value <strong>of</strong> η(0) does not affect W η , we see that W pr η = W pr η ′. By assumption, W is injective on F 0 (O ♯ , N), so that pr η = pr η ′ . In particular, l 0 (η) = l 0 (η ′ ). Since l(η) = l(η ′ ), we must also have η(0) = η ′ (0), and thus η = η ′ . 6. Self-Dual Codes I want to finish this article by touching on a very active research topic: self-dual codes. As we saw in subsection 3.3, if C ⊂ F n is a linear code <strong>of</strong> length n over a finite field F, <strong>the</strong>n its dual code C ⊥ is defined by C ⊥ = {y ∈ F n : x · y = 0, x ∈ C}. A linear code C is self-orthogonal if C ⊂ C ⊥ and is self-dual if C = C ⊥ . Because dim C ⊥ = n − dim C, a necessary condition for <strong>the</strong> existence <strong>of</strong> a self-dual code C over a finite field is that <strong>the</strong> length n must be even; <strong>the</strong>n dim C = n/2. The Hamming weight enumerator <strong>of</strong> a self-dual code appears on both sides <strong>of</strong> <strong>the</strong> MacWilliams identities: W C (X, Y ) = 1 |C| W C(X + (q − 1)Y, X − Y ), where C is self-dual over F q . As |C| = q n/2 and <strong>the</strong> total degree <strong>of</strong> <strong>the</strong> polynomial W C (X, Y ) is n, <strong>the</strong> MacWilliams identities for a self-dual code can be written in <strong>the</strong> form W C (X, Y ) = W C ( X + (q − 1)Y √ q , X √ − Y ) . q Every element x <strong>of</strong> a self-dual code satisfies x · x = 0. In <strong>the</strong> binary case, q = 2, notice that x · x ≡ wt(x) mod 2. Thus, every element <strong>of</strong> a binary self-dual code C has even length. This implies that W C (X, −Y ) = W C (X, Y ). Restrict to <strong>the</strong> binary case, q = 2. Define two complex 2 × 2 matrices P, Q by P = ( 1/ √ 2 1/ √ 2 1/ √ 2 −1/ √ 2 –242– ) , Q = ( 1 0 0 −1 ) .
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 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
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, −)
Page 73 and 74:
where r v ij := a i a j −a j a i
Page 75 and 76:
ON A DEGENERATION PROBLEM FOR COHEN
Page 77 and 78:
We make several othe</stron
Page 79 and 80:
Let A be a commutative Gorenstein r
Page 81 and 82:
3. Extended orders In the</
Page 83 and 84:
WEAK GORENSTEIN DIMENSION FOR MODUL
Page 85 and 86:
1.2. Introduction. The notion <stro
Page 87 and 88:
e 2 A z 1 −→ X → 0 in mod-A,
Page 89 and 90:
5. Gorenstein algebras In this sect
Page 91 and 92:
ON Ω-PERFECT MODULES AND SEQUENCES
Page 93 and 94:
when R is a Nakayama algebra <stron
Page 95 and 96:
says that components of</st
Page 97 and 98:
then we obtain a c
Page 99 and 100:
Proposition 16. Let C s be a stable
Page 101 and 102:
3.1. ZD ∞ -components. We assume
Page 103 and 104:
Theorem 26. Let R be a local artini
Page 105 and 106:
where each f i is an irreducible ep
Page 107 and 108:
QUANTUM UNIPOTENT SUBGROUP AND DUAL
Page 109 and 110:
{F i } i∈I and q-Serre relations
Page 111 and 112:
Theorem 8. (1)Let U − q (w) → U
Page 113 and 114:
E-mail address: ykimura@kurims.kyot
Page 115 and 116:
Indeed, (1, 2, 3, 4) {1,2} −−
Page 117 and 118:
By comparing this the</stro
Page 119 and 120:
(a) (b) Proposition 17. Let G be a
Page 121 and 122:
POLYCYCLIC CODES AND SEQUENTIAL COD
Page 123 and 124:
⎛ ⎞ 0 0 c 0 1 c t D c = ⎜ 1
Page 125 and 126:
References [1] D. Boucher and P. So
Page 127 and 128:
2. Dimension of tr
Page 129 and 130:
APR TILTING MODULES AND QUIVER MUTA
Page 131 and 132:
(1) Q ′ is a quiver obtained from
Page 133 and 134:
1 arrows of ˜µ L
Page 135 and 136:
[11] S. Fomin, A. Zelevinsky, Clust
Page 137 and 138:
Theorem. Assume r is a common prime
Page 139 and 140:
References [1] Apostol, T. M., The
Page 141 and 142:
e(n, s) n = 6 7 8 s = 1 36 63 120 2
Page 143 and 144:
Now we will show that the</
Page 145 and 146:
is equal to ( n 2s−1 ); hence we
Page 147 and 148:
THE NOETHERIAN PROPERTIES OF THE RI
Page 149 and 150:
Proposition 4 (Paper III, Propositi
Page 151 and 152:
HOCHSCHILD COHOMOLOGY OF QUIVER ALG
Page 153 and 154:
(4) In [10], Green, Snashall and So
Page 155 and 156:
4. Quiver algebras defined by two c
Page 157 and 158:
[6] L. Evens, The cohomology ring <
Page 159 and 160:
Then there is an i
Page 161 and 162:
• ( ˜F , ˜m) = ({ (1, 3), (2, 3
Page 163 and 164:
Corollary 7 ([16, Theorem 3.4]). Th
Page 165 and 166:
We have the direct
Page 167 and 168:
We say a CW complex is regular, if
Page 169 and 170:
SHARP BOUNDS FOR HILBERT COEFFICIEN
Page 171 and 172:
Let us briefly note how this paper
Page 173 and 174:
whence the set Λ
Page 175 and 176:
Proof of</
Page 177 and 178:
We have Q·H j m(A/(a 1 , a 2 , ·
Page 179 and 180:
Let B = A/U(a d−1 ). We t
Page 181 and 182:
• quiver Γ = (I, Ω) I Ω
Page 183 and 184:
B = (B τ ) relations ρ 1 , · ·
Page 185 and 186:
Definition 1. double quiver ˜Γ p
Page 187 and 188:
◦ side Γ = (I, Ω) cycle (ac
Page 189 and 190:
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 and 242: (2.3) soc( R R) ∼ = k⊕ s i T i
Page 243 and 244: principal ideal Rr ⊂ ker ϖ. Thus
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: is injective for every finite left
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
Page 265 and 266: By the above resul
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
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
Page 287 and 288: field of V . We de
Page 289 and 290: (Proof) ( ) p 0 0
Page 291 and 292: we have a sequence of</stro
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 <
show all

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

Create successful ePaper yourself

Delete template?

Save as template?