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

main <strong>the</strong>orem. In Section 5, we will characterize local noe<strong>the</strong>rian algebras <strong>of</strong> finite selfinjective
dimension. Also, we will provide several examples showing what rich properties
local noe<strong>the</strong>rian algebras <strong>of</strong> finite selfinjective dimension enjoy.
2. Full embedding
Let G A /P A be <strong>the</strong> residue category <strong>of</strong> G A over <strong>the</strong> full additive subcategory P A and
D b (mod-A) fGd /D b (mod-A) fpd <strong>the</strong> quotient category <strong>of</strong> D b (mod-A) fGd over <strong>the</strong> épaisse
subcategory D b (mod-A) fpd . Then, as Avramov [3] announced, <strong>the</strong> embedding G A →
D b (mod-A) fGd gives rise to an equivalence
In this section, we will extend this fact.
G A /P A ∼ → D b (mod-A) fGd /D b (mod-A) fpd .
Definition 7. We denote by ĜA <strong>the</strong> full additive subcategory <strong>of</strong> mod-A consisting <strong>of</strong>
modules X ∈ mod-A with Ext i A(X, A) = 0 for i ≠ 0.
We denote by ĜA/P A <strong>the</strong> residue category <strong>of</strong> ĜA over <strong>the</strong> full additive subcategory P A
and by D b (mod-A)/D b (mod-A) fpd <strong>the</strong> quotient category <strong>of</strong> D b (mod-A) over <strong>the</strong> épaisse
subcategory D b (mod-A) fpd . Also, we denote by D b (mod-A) bdh /D b (mod-A) fpd <strong>the</strong> quotient
category <strong>of</strong> D b (mod-A) bdh over <strong>the</strong> épaisse subcategory D b (mod-A) fpd .
Proposition 8. The embedding ĜA → D b (mod-A) bdh gives rise to a full embedding
F : ĜA/P A → D b (mod-A) bdh /D b (mod-A) fpd .
In <strong>the</strong> next section, we will characterize a complex X • ∈ D b (mod-A) bdh which admits
a homomorphism Z[m] → X • in D b ∼
(mod-A) bdh inducing an isomorphism Z[m] −→ X •
in D b (mod-A) bdh /D b (mod-A) fpd for some Z ∈ ĜA and m ∈ Z. Such a complex does not
necessarily belong to D b (mod-A) fGd . Namely, Ĝ A ≠ G A in general (see Proposition 15
below), which has been pointed out by J.-I. Miyachi in oral communication.
Example 9 (Miyachi). Let k be a field and fix a nonzero element c ∈ k which is not
a root <strong>of</strong> unity. Let S = k < x, y > be a non-commutative polynomial ring and I =
(x 2 , y 2 , cxy + yx) a two-sided ideal generated by x 2 , y 2 and cxy + yx. Set R = S/I,
z n = x + c n y + I ∈ R for n ∈ Z and w = xy + I ∈ R. Then R is a selfinjective algebra and
for each n ∈ Z <strong>the</strong>re exist exact sequences R z n+1
−−→ R z n
−→ R in mod-R and R z n
−→ R z n+1
−−→ R
in mod-R op . Since c is not a root <strong>of</strong> unity, z n R ≁ = zm R and Hom R (z n R, z m R) ∼ = k unless
n = m. Thus, since we have a projective resolution · · · → R z 3
−→ R z 2
−→ R z 1
−→ z 1 R → 0 in
mod-R, applying Hom R (−, z 0 R) we have Ext i R(z 1 R, z 0 R) = 0 for all i ≥ 1 (see [14]).
Now, we set
A =
( )
k z0 R
0 R
and e 1 =
( )
1 0
, e
0 0 2 =
( )
0 0
∈ A.
0 1
Then a module X ∈ mod-A is given by a triple (X 1 , X 2 ; φ) <strong>of</strong> X 1 ∈ mod-k, X 2 ∈ mod-R
and φ ∈ Hom R (X 1 ⊗ k z 0 R, X 2 ), and a module M ∈ mod-A op is given by a triple
(M 1 , M 2 ; ψ) <strong>of</strong> M 1 ∈ mod-k, M 2 ∈ mod-R op and ψ ∈ Hom k (z 0 R ⊗ R M 2 , M 1 ) (see [7]).
z
Set X = (0, z 1 R; 0) ∈ mod-A. Since we have a projective resolution · · · −→
3
e2 A z 2
−→
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