Erratum and Addenda

“Gorenstein test modules” by J. Asadollahi, Sh. Salarian, Communication in Algebra, 33, pp. 3439–3446, 2005 and “Gorenstein test modules” by A. Tehranian, M. Tousi, S. Yassemi, Communication in Algebra, 34, pp. 879–882, 2006.

We are grateful to Sean Sather-Wagstaff for pointing out that Asadollahi and Salarian (2005, Theorem 3.2) as well as the results in Asadollahi and Salarian (2005) and Tehranian et al. (2006) that used this theorem in their proof are wrong.

The problem is that we assume that in a minimal strict G-resolution G, one has . This is not true in general. Consider the following observation from Avramov and Martsinkovsky (2002, Example 9.2).

Example 1

Let (R, m, k) be a nonregular local Gorenstein ring of dimension 1. Then Gdim _R (k) = 1 and for all i > 0. In particular, k is not a G-test module for R.

This contradicts Asadollahi and Salarian (2005, Theorem 3.2) (and so Theorem 3.4 and Corollary 3.5) as well as the key lemma in Tehranian et al. (2006, Lemma 2), (and so Theorem 3, Corollary 4, and Theorem 7).

We point out that the key lemma in Tehranian et al. (2006, Lemma 2), is valid if we put the extra condition Gdim _R (M) ≠ 1. In this way, Lemma 2 and Corollary 4 in Tehranian et al. (2006) will read as follows.

Lemma 2

Let (R, 𝔪) be a local ring, and let M be a nonzero finitely generated R-module. If 1 < Gdim _R (M) = t < ∞, then for any nonzero finite R-module N we have .

Proof

By using Avramov and Martsinkovsky (2002, Theorem 8.5(1)–(2)), there exists an exact sequence

where F _i is free R-module of finite rank for i ≥ 1 and G ₀ is totally reflexive R-module. In addition, for i ≥ 2, ∂ _i (F _i ) ⊆ 𝔪 F _i−1 and also ∂₁(F ₁) has no nonzero free summand of G ₀.

Let t ≥ 2. The module is the cokernel of ∂* _t :Hom _R (F _t−1, N) → Hom _R (F _t , N). On the other hand, Im ∂* _t  ⊆ 𝔪Hom _R (F _t , N). Thus by NAK .

Corollary 3

Let M be a finitely generated R-module with 1 < Gdim _R (M) < ∞. Then for any nonzero finitely generated R-module N,

Proof

Let Gdim _R M = t. Then by Lemma 2, . Thus “≤” holds. Now the assertion holds by Avramov and Martsinkovsky (2002, Theorem 4.2(2.a)).

Similarly, the following definition for a Gorenstein test module looks like the best one can hope for, considering the limitations provided by the example.

Definition 4

We say that a nonzero R-module A is a Gorenstein test module if for any R-module M with finite Gorenstein dimension and any integer t (t > 1), the following holds:

The following shows that Theorem 3 in Tehranian et al. (2006) is valid using the above definition.

Theorem 5

Any nonzero finitely generated R-module N is a Gorenstein test module.

Proof

Let M be an R-module with finite Gorenstein dimension. Let t > 1 be an integer and assume that for any j ≥ t. We have to show that Gdim _R M < t. If Gdim _R M = n ≥ t, then by Lemma 2, . This is a contradiction.

Notes

Communicated by I. Swanson.