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

More documents

Recommendations

Info

(4) for every object M <strong>of</strong> D(A) <strong>the</strong>re are two triangles and i ! i ! M M j ∗ j ∗ M Σi ! i ! M j ! j ! M M i ∗ i ∗ M Σj ! j ! M , where <strong>the</strong> four morphisms starting from and ending at M are <strong>the</strong> units and counits. Necessary and sufficient conditions under which such a recollement exists were discussed in [13, 16]. Example 1. Let A be <strong>the</strong> path algebra <strong>of</strong> <strong>the</strong> Kronecker quiver 1 The trivial path e 1 at 1 is an idempotent <strong>of</strong> A and e 1 A is a projective A-module. The following diagram is a recollement ? L ⊗ A A/Ae 1 A 2 . ? L ⊗ e1 Ae 1 e 1 A D(A/Ae 1 A) ? ⊗ L A/Ae1 AA/Ae 1 A D(A) ? ⊗ L A Ae 1 D(e 1 Ae 1 ). RHom A (A/Ae 1 A,?) RHom e1 Ae 1 (Ae 1 ,?) Note that both e 1 Ae 1 and A/Ae 1 A are isomorphic to k. 2. Algebraic stratifications <strong>of</strong> derived module categories Let A be an algebra. An algebraic stratification <strong>of</strong> D(A) is a sequence <strong>of</strong> iterated nontrivial recollements <strong>of</strong> derived module categories. It can be depicted as a binary tree as below, where each edge represents an adjoint triple <strong>of</strong> triangle functors and each hook represents a recollement D(A) ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ D(B) D(C) ✈ ✈✈✈✈✈✈✈✈ D(B ′ ) ✽ ✽✽✽✽ ✽ ❍ ❍❍❍❍❍❍ ❍ ❍ D(B ′′ ) ✽ ✽✽✽✽ ✽ ✈ ✈✈✈✈✈✈✈✈ D(C ′ ) ❍ ❍❍❍❍❍❍ ❍ ❍ D(C ′′ ) ✞ ✞✞✞✞✞✞ ✽ ✝ ✝✝✝✝✝✝ ✽ ✞ ✞✞✞✞✞✞ ✼ ✝ ✝✝✝✝✝✝ ✽ · · · · · · · · ✼ ✼✼✼✼ ✼ . . . ✽ ✽✽✽✽ ✽ –257–
The leaves <strong>of</strong> <strong>the</strong> tree are <strong>the</strong> simple factors <strong>of</strong> <strong>the</strong> stratification. The following questions are basic: (a) Does every derived module category admit a finite algebraic stratification? (b) Do two finite algebraic stratifications <strong>of</strong> a derived module category have <strong>the</strong> same number <strong>of</strong> simple factors? Do <strong>the</strong>y have <strong>the</strong> same simple factors (up to triangle equivalence and up to reordering)? (c) Which derived module categories occur as simple factors <strong>of</strong> some algebraic stratifications? The question (c) will be discussed in <strong>the</strong> next section. The questions (a) and (b) ask for a Jordan–Hölder type result for derived module categories. For general (possibly infinitedimensional) algebras <strong>the</strong> answers are negative. Below we give some (counter-)examples. Example 2. ([2]) Let A = ∏ N k. Then D(A) does not admit a finite algebraic stratification. Example 3. ([6]) Let A be as in Example 1. Let V be a regular simple A-module, namely, V corresponds to one <strong>of</strong> <strong>the</strong> following representations <strong>of</strong> <strong>the</strong> Kronecker quiver k 1 λ k (λ ∈ k), k 0 1 k . Let ϕ : A → A V be <strong>the</strong> corresponding universal localisation. Then T = A ⊕ A V /ϕ(A) is an (infinitely generated) tilting A-module. We refer to [6] for <strong>the</strong> unexplained notions. Let B = End A (T ). Then <strong>the</strong>re are two algebraic stratifications <strong>of</strong> D(B) <strong>of</strong> length 3 and 2 respectively : D(B) D(B) D(k (t )) ❍ ❍❍❍❍❍❍ ❍ D(A) ● ●●●●● ● ✉ ✉✉✉✉✉✉✉✉ D(k[t]) ❏ ❏❏❏❏❏❏ ❏ ❏ D(k [t]) ✈ ✈✈✈✈✈✈✈✈ D(k) ● D(k) Examples <strong>of</strong> this type are systematically studied in [7]. Notice that <strong>the</strong> algebra B in <strong>the</strong> preceding example is infinite-dimensional. For finitedimensional algebras, <strong>the</strong> questions (a) and (b) are open. For piecewise hereditary algebras <strong>the</strong> answers to <strong>the</strong>m are positive. Recall that a finite-dimensional algebra is piecewise hereditary if it is derived equivalent to a hereditary abelian category. Theorem 4. ([1, 3]) Let A be a piecewise hereditary algebra. Then any algebraic stratification <strong>of</strong> D(A) has <strong>the</strong> same set (with multiplicities) <strong>of</strong> simple factors: <strong>the</strong>y are precisely <strong>the</strong> derived categories <strong>of</strong> <strong>the</strong> endomorphism algebras <strong>of</strong> <strong>the</strong> simple A-modules. –258–
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 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
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: ALGEBRAIC STRATIFICATIONS OF DERIVE
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 ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?