Minimal injective resolutions and Bass invariants of modules over left stable left Noetherian rings
Let R be a left Noetherian ring which is left stable (i.e. every torsion theory in R-Mod is stable). In a certain sense, rings of this type are closer to commutative Noetherian rings than left Noetherian rings in general. The author illustrates this point by considering two constructions from homological algebra. We quote some sample results. First, it is shown, that an injective resolution \(0\to M\to E_ 0\to E_ 1\to...\) in R-Mod is minimal if and only if the natural map \(Hom_ R(W,Q_{\pi}(E_ i))\to Hom_ R(W,Q_{\pi}(E_{i+1}))\) is zero for each i, for each prime torsion theory \(\pi\) and for each \(\pi\)-cocritical module W. If this holds and if \(\tau\) is a perfect torsion theory, then \(0\to Q_{\tau}(M)\to Q_{\tau}(E_ 0)\to...\) is a minimal injective resolution of \(Q_{\tau}(M)\) in \(R_{\tau}\)-Mod. (Q denotes the localization functor.) For any R-module M write \(E(M)=\oplus E(\pi)^{\mu(\pi,M)},\) where \(\pi\) runs through all prime torsion theories and \(E(\pi)\) is the unique (up to isomorphism) injective hull of an \(\pi\)-cocritical module. The cardinal numbers \(\mu\) (\(\pi\),M) are called the Bass invariants. Among other results on these invariants, it is shown, that the \(\tau\)-dominant dimension of M (in the sense of the author and \textit{J. Raynaud} [J. Aust. Math. Soc., Ser. A 35, 162-177 (1983; Zbl 0516.16020)]) is the smallest i such that \(\mu(\pi,E_ i)\neq 0\) for some prime \(\pi\) not greater or equal to \(\tau\), \(0\to M\to E_ 0\to...\) being a minimal injective resolution. If \(\pi\) is a minimal prime and if \(\mu = \mu(\pi,E_ i)\) is finite, then \(\text{Ext}^ i_ R(C(\pi),M)\) is a \(\mu\)-dimensional vector space over the division ring \(D(\pi)\), the endomorphism ring of the core \(C(\pi)\) of \(E(\pi)\).
scientific article; zbMATH DE number 3845786
scientific article