Minimal injective and flat resolutions of modules over Gorenstein rings
In the first part of this paper the author studies the minimal injective resolutions of modules of finite flat dimension over a Gorenstein ring \(R\). The main tools used are the Bass invariants \(\mu_i ({\mathfrak p}, M)\). For instance: Gorenstein rings are characterized in terms of vanishing of \(\mu_i ({\mathfrak p}, F)\), for any flat \(R\)-module \(F\) and any \(i \neq \text{ht} ({\mathfrak p})\); an \(R\)-module \(M\), where \(R\) is Gorenstein, is shown to be of finite flat dimension \(s\) iff \(\mu_i ({\mathfrak p}, M) \neq 0\) implies \(i \leq \text{ht} ({\mathfrak p}) \leq i + s\). In the second part of the paper, the author deals with minimal flat resolutions of modules. He defines new invariants \(\pi_i ({\mathfrak p}, M)\), which are analogous to Bass invariants, when ``injective resolution'' is replaced by ``flat resolution''. Some interesting characterization of the Gorenstein property is given, in terms of vanishing of \(\pi_i ({\mathfrak p}, X)\), when \(X\) is an injective \(R\)- module. Finally, strongly cotorsion modules \(G\) are defined by the condition \(\text{Ext}^1_R (X,C) = 0\), for all \(X\) of finite flat dimension. Strongly cotorsion modules over a Gorenstein ring are exactly those admitting a minimal flat resolution, while a strongly torsion free module \(M\) over a Gorenstein ring \(R\) (which means \(\text{Tor}^R_1 (X,M) = 0\) for all \(X\) of finite flat dimension) is characterized by: \(\mu_i ({\mathfrak p}, M) = 0\) if \(\text{ht} {\mathfrak p} > i\), \(i \geq 0\).
scientific article; zbMATH DE number 783931