There are several definitions of injective dimension of a complex \(J\) consisting of modules over a commutative ring \(R\). For instance, for any class \(\mathfrak{I}\) of complexes of \(R\)-modules, consider the number \[ \inf\{n \in \mathbb{Z} | J \simeq I \text{ with }I\in\mathfrak{I},\text{ and }I_j = 0\text{ for }j <-n\}, \] where \(J \simeq I\) means that \(J\) and \(I\) are quasi-isomorphic. The paper deals with the following classes of complexes. A complex \(I\) is called graded-injective if each \(R\)-module \(I_n\) is injective. Also, \(I\) is called semi-injective if it is graded-injective and \(\operatorname{Hom}_R(-, I)\) sends quasi-isomorphism to quasi-isomorphism. Taking \(\mathfrak{I}\) the class of semi-injective complexes gives an invariant \(\text{id}_R(J)\). Taking \(\mathfrak{I}\) the class of graded-injective complexes gives an invariant \(\text{gr-id}_R(J)\). When \(M\) is a module, viewed as a complex with \(M\) in degree zero and zero otherwise. Clearly, \(\text{gr-id}_R(M)\leq \text{id}_R(M)\).
By [L. L. Avramov and H.-B. Foxby, J. Pure Appl. Algebra 71, No. 2–3, 129–155 (1991; Zbl 0737.16002)], the equality holds whenever the ring \(R\) has finite global dimension, and they asked is the converse true? Also recall that \(R\) is called regular if every ideal has a finite resolution by finitely generated projective modules. This yields that \(R\) is noetherian. The main result of the paper under review gives the affirmative answer to the question while \(R\) is regular (i.e., in the noetherian case). This uses a beautiful result of Krause in the context of triangulated subcategories of the derived category and the calculus of fractions in the sense of Gabriel and Verdier.
In their next results the authors of the paper show the similar results for flat dimension and projective dimension.