On the existence of embeddings into modules of finite homological dimensions

Let R be a commutative Noetherian local ring. We show that R is Gorenstein if and only if every finitely generated R-module can be embedded in a finitely generated R-module of finite projective dimension. This extends a result of Auslander and Bridger to rings of higher Krull dimension, and it also improves a result due to Foxby where the ring is assumed to be Cohen-Macaulay.


Introduction
Throughout this paper, let R be a commutative Noetherian local ring. All Rmodules in this paper are assumed to be finitely generated.
(2) Every R-module can be embedded in a free R-module.
On the other hand, in [4, Theorem 2] Foxby showed the following.
(2) R is Cohen-Macaulay, and every R-module can be embedded in an R-module of finite projective dimension.
For an R-module C we denote by add R C the class of R-modules which are direct summands of finite direct sums of copies of C. The C-dimension of an R-module X, C-dim R X, is defined as the infimum of nonnegative integers n such that there exists an exact sequence In this paper, we prove the following theorem. This result removes from Theorem 1.2 the assumption that R is Cohen-Macaulay, and it extends Theorem 1.1 to rings of higher Krull dimension. It should be noted that our proof of this result is different from Foxby's proof for the special case C = R. (1) C is dualizing.
(2) Every R-module can be embedded in an R-module of finite C-dimension.
Moreover, if one of these three conditions holds, then R is Cohen-Macaulay.

Proof of Theorem 1.3 and its applications
First of all, we recall the definition of a semidualizing module.
is an isormophism and Ext i R (C, C) = 0 for all i > 0. Note that a dualizing module is nothing but a semidualizing module of finite injective dimension. Another typical example of a semidualizing module is a free module of rank one. Recently a considerable number of authors have studied semidualizing modules and have obtained many results concerning these modules.
We denote by m the maximal ideal of R and by k the residue field of R. To prove our main theorem, we establish two lemmas.
Proof. First of all we prove that M can be embedded in a module C 0 in add R C. For this we set n = C-dim R X. If n = 0, then this is obvious from the assumption, since X ∈ add R C. If n > 0, then there exists an exact sequence f is injective as well. Therefore M has an embedding f into C 0 .
To prove that λ M is injective, we note that λ C0 is an isomorphism, because of C 0 ∈ add R C. Since there is an injective homomorphism f : M → C 0 , the following commutative diagram forces λ M to be injective:

Lemma 2.3. Let C be a semidualizing R-module and let M be an R-module. Assume that M is free on the punctured spectrum of R. Then there is an isomorphism
Proof. Set t = depth R C. Since C is semidualizing, we have a spectral sequence . Note by assumption that the R-module Tor R q (M, C) has finite length for q > 0. By [2, Proposition 1.2.10(e)] we have E p,q 2 = 0 if p < t and q > 0. Hence Let M be an R-module. Take a free resolution (2) ⇒ (3): This implication is obvious.
(3) ⇒ (1): We denote by (−) † the C-dual functor Hom R (−, C). Put t = depth R C and set M = Tr Ω t k. Then we have depth R = t by [5]. Since As M is free on the punctured spectrum of R, By assumption (3), the module M ⊗ R C has an embedding into a module X with C-dim R X < ∞. According to [7,Lemma 4.3], we have C-dim R X ≤ t. Thus we obtain Ext t+1 R (k, C) = 0. By [3, Theorem (1.1)], the R-module C must have finite injective dimension.
As we observed in the proof of the implication (1) ⇒ (2), assertion (1) implies that R is Cohen-Macaulay. Thus the last assertion follows. Now we give applications of our main theorem. Letting C = R in Theorem 1.3, we obtain the following result. This improves Theorem 1.2 and extends Theorem 1.1.
(2) Every R-module can be embedded in an R-module of finite projective dimension.
Corollary 2.5. If every finitely generated R-module can be embedded in a finitely generated R-module of finite projective dimension, then every finitely generated Rmodule can be embedded in a finitely generated R-module of finite injective dimension.