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

More documents

Recommendations

Info

A NOTE ON DIMENSION OF TRIANGULATED CATEGORIES. HIROYUKI MINAMOTO Abstract. In this note we study <strong>the</strong> behavior <strong>of</strong> <strong>the</strong> dimension <strong>of</strong> <strong>the</strong> perfect derived category Perf(A) <strong>of</strong> a dg-algebra A over a field k under a base field extension K/k. In particular we show that <strong>the</strong> dimension <strong>of</strong> a perfect derived category is invariant under a separable algebraic extension K/k. As an application we prove <strong>the</strong> following statement: Let A be a self-injective algebra over a perfect field k. If <strong>the</strong> dimension <strong>of</strong> <strong>the</strong> stable category modA is 0, <strong>the</strong>n A is <strong>of</strong> finite representation type. This <strong>the</strong>orem is proved by M. Yoshiwaki in <strong>the</strong> case when k is an algebraically closed field. Our pro<strong>of</strong> depends on his result. 1. Introduction In [3] R. Rouquier introduced <strong>the</strong> dimension <strong>of</strong> triangulated categories and showed that it gives an upper bound or a lower bound <strong>of</strong> o<strong>the</strong>r dimensions in algebraic geometry or in representation <strong>the</strong>ory(see also [4]). The dimension <strong>of</strong> triangulated categories is studied by many researchers. In this note we study <strong>the</strong> behavior <strong>of</strong> <strong>the</strong> dimension <strong>of</strong> <strong>the</strong> perfect derived category Perf(A) <strong>of</strong> a dg-algebra A over a field k under a base field extension K/k. For a field extension K/k, we denote A ⊗ k K by A K . Theorem 1. (1) For an algebraic extension K/k, we have tridim Perf(A) ≤ tridim Perf(A K ). (2) If moreover K/k is separable, <strong>the</strong>n equality holds. As an application we prove <strong>the</strong> following <strong>the</strong>orem, which gives evidence that dimension <strong>of</strong> triangulated categories captures some representation <strong>the</strong>oretic properties. The stable category modA plays an important role in <strong>the</strong> study <strong>of</strong> self-injective algebra A (cf. [2, 4]). If a self-injective algebra A is <strong>of</strong> finite representation type <strong>the</strong>n <strong>the</strong> dimension <strong>of</strong> <strong>the</strong> stable category modA is zero. Then a natural question arises as to whe<strong>the</strong>r <strong>the</strong> converse should also hold. Theorem 2. Let A be a self-injective finite dimensional algebra over a perfect field k. If tridim modA = 0, <strong>the</strong>n A is <strong>of</strong> finite representation type. In <strong>the</strong> case when k is an algebraically closed field, this <strong>the</strong>orem is proved by M. Yoshiwaki in [5]. Our pro<strong>of</strong> depends on his result. The final version <strong>of</strong> this paper has been submitted for publication elsewhere. –111–
2. Dimension <strong>of</strong> triangulated categories. We review <strong>the</strong> definition <strong>of</strong> dimension <strong>of</strong> triangulated categories due to R. Rouquier We need to prepare a bit <strong>of</strong> notations. Let T be a triangulated category. For a full subcategory I <strong>of</strong> T we denote by 〈I〉 <strong>the</strong> smallest full subcategory <strong>of</strong> T containing I which is closed under taking shifts, finite direct sums, direct summands and isomorphisms. For full subcategories I and J <strong>of</strong> T we denote by I ∗ J <strong>the</strong> full subcategory <strong>of</strong> T consisting <strong>of</strong> those object M ∈ T such that <strong>the</strong>re exists an exact triangle I → M → J −→ [1] with I ∈ I and J ∈ J . Set I ⋄J := 〈I ∗J 〉. For n ≥ 1 we define inductively { 〈I〉 for n = 1; 〈I〉 n := 〈I〉 ⋄ 〈I〉 n−1 for n ≥ 2. Now we define <strong>the</strong> dimension <strong>of</strong> a triangulated category T to be tridim T := min{n | 〈E〉 n+1 = T for some E ∈ T }. 3. Sketch <strong>of</strong> pro<strong>of</strong> <strong>of</strong> Theorem 1 and 2 First we consider <strong>the</strong> case when K/k is a finite extension. Let E be an object <strong>of</strong> Perf(A K ) such that 〈E〉 n = Perf(A K ) for some n ∈ N. Then we see that 〈UE〉 n = Perf(A) where U : Perf(A K ) → Perf(A) is <strong>the</strong> forgetful functor. In <strong>the</strong> case K/k is an infinite algebraic extension, <strong>the</strong> key <strong>of</strong> <strong>the</strong> pro<strong>of</strong> is <strong>the</strong> following lemma. Lemma 3. Let K/k be an algebraic extension and E an object <strong>of</strong> D(A). If an object G <strong>of</strong> D(A K ) belongs to 〈E ⊗ k K〉 n , <strong>the</strong>n <strong>the</strong>re exists an intermediate field k ⊂ K 0 ⊂ K which is finite dimensional over k such that <strong>the</strong>re exists an object G ′ <strong>of</strong> 〈E ⊗ k K 0 〉 n , such that G ′ ⊗ K0 K ∼ = G in D(A K ). Let E be an object <strong>of</strong> Perf(A K ) such that 〈E〉 n = Perf(A K ) for some n ∈ N. Since Perf(A K ) = ∪ i∈N 〈A K 〉 i , by <strong>the</strong> above lemma <strong>the</strong>re exists an intermediate field k ⊂ K 0 ⊂ K which is finite dimensional over k such that <strong>the</strong>re exists an object E ′ <strong>of</strong> Perf(A K0 ) such that E ′ ⊗ K0 K ≃ E. Then we see that 〈U 0 (E ′ )〉 n = Perf(A) where U 0 : Perf(A K0 ) → Perf(A) is <strong>the</strong> forgetful functor. To prove <strong>the</strong> second statement, we use <strong>the</strong> fact that when K/k is a finite separable field extension, <strong>the</strong> canonical morphism K ⊗ k K → K splits as K − K bimodules. In <strong>the</strong> case when K/k is an infinite separable field extension, we reduce to <strong>the</strong> finite separable extension case by <strong>the</strong> above lemma. Theorem 2 is reduced to <strong>the</strong> case when <strong>the</strong> base field k is an algebraically closed field by Theorem 1 and <strong>the</strong> following lemma. Lemma 4. Let A be a finite dimensional k-algebra. If A k is <strong>of</strong> finite representation type, <strong>the</strong>n A is <strong>of</strong> finite representation type. –112–
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: References [1] D. Boucher and P. So
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 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?