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

e 2 A z 1
−→ X → 0 in mod-A, it follows that Ext i A(X, e 1 A) ∼ = Ext i R(z 1 R, z 0 R) = 0 and
Ext i A(X, e 2 A) ∼ = Ext i R(z 1 R, R) = 0 for i > 0. Thus X ∈ ĜA. On <strong>the</strong> o<strong>the</strong>r hand, we have
Hom A (X, A) ∼ z
= Ker(Ae
2
2 −→ Ae2 )
∼ = (wR, Rz1 ; 0)
∼ = (wR, 0; 0) ⊕ (0, Rz1 ; 0)
in mod-A op and hence Hom A op(Hom A (X, A), A) is decomposable, so that we have X ≇
Hom A op(Hom A (X, A), A) and X /∈ G A .
3. Weak Gorenstein dimension
In this section, we will introduce <strong>the</strong> notion <strong>of</strong> weak Gorenstein dimension for finitely
presented modules and study finitely presented modules <strong>of</strong> finite weak Gorenstein dimension.
Definition 10. A complex X • ∈ D b (mod-A) with sup{ i | H i (X • ) ≠ 0} = d < ∞
is said to have finite weak Gorenstein dimension if X • ∈ D b (mod-A) bdh , H i (η X •) is an
isomorphism for i < d and H d (η X •) is a monomorphism.
For a module X ∈ mod-A <strong>of</strong> finite weak Gorenstein dimension we set
Ĝ-dim X = sup{ i | Ext i A(X, A) ≠ 0}
if X ≠ 0 and Ĝ-dim X = 0 if X = 0. Also, we set Ĝ-dim X = ∞ if X ∈ mod-A does
not have finite weak Gorenstein dimension. Then Ĝ-dim X is called <strong>the</strong> weak Gorenstein
dimension <strong>of</strong> X ∈ mod-A.
Remark 11. For any X ∈ mod-A <strong>the</strong> following hold.
(1) Ĝ-dim X = 0 if and only if X is embedded in some P ∈ P A, i.e., <strong>the</strong> canonical
homomorphism
X → Hom A op(Hom A (X, A), A), x ↦→ (f ↦→ f(x))
is a monomorphism and X ∈ ĜA.
(2) If G-dim X = d < ∞ <strong>the</strong>n Ĝ-dim X = d.
(3) If Ĝ-dim X = d < ∞ <strong>the</strong>n Ĝ-dim X′ ≤ d for all X ′ ∈ add(X), <strong>the</strong> full additive
subcategory <strong>of</strong> mod-A consisting <strong>of</strong> direct summands <strong>of</strong> finite direct sums <strong>of</strong> copies
<strong>of</strong> X.
Lemma 12. A complex X • ∈ D b (mod-A) with sup{ i | H i (X • ) ≠ 0} = d < ∞ has
finite weak Gorenstein dimension if and only if <strong>the</strong>re exists a distinguished triangle in
D b (mod-A)
X • → Y • → Z[−d] →
with Y • ∈ K b (P A ), Y i = 0 for i > d, and Z ∈ ĜA.
Corollary 13 (cf. [6, Lemma 2.17]). For any X ∈ mod-A with Ĝ-dim X < ∞ <strong>the</strong>re
exists an exact sequence 0 → X → Y → Z → 0 in mod-A with Ĝ-dim X = proj dim Y
and Z ∈ ĜA.
Lemma 14. For any exact sequence 0 → X → Y → Z → 0 in mod-A <strong>the</strong> following hold.
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