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

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–