Recall that if \(R\) is a commutative noetherian ring, \(M\) is an \(R\)-module and \(\mathfrak{p}\) is a prime ideal in \(R\), then the \(i\)-th Bass number \(\mu_i(\mathfrak{p},M)\) is defined to be the number of the copies of the injective hull \(E(R/\mathfrak{p})\) of \(R/\mathfrak{p}\) which occurs in the \(i\)-th term \(E^i(M)\) of the minimal injective resolution of \(M\). A result going back toH. Bass [Math. Z. 82, 8–28 (1963;Zbl 0112.26604)] says that \(\mu_i(\mathfrak{p},M)=\dim_{k(\mathfrak{p})}\mathrm{Ext}_R^i(R/\mathfrak{p},M)_{\mathfrak{p}}\), where \(k(\mathfrak{p})\) is the residue field of \(\mathfrak{p}\). In the paper under review it is generalized this classical result, the place of the module category over \(R\) being taken by a locally noetherian Grothendieck category \(\mathcal A\). In this new setting the rôle of quotients \(R/\mathfrak{p}\) with \(\mathfrak{p}\) a prime ideal is taken by atoms. An atom is an equivalence classes of so called monoform objects of \(\mathcal A\) (that is objects \(H\in\mathcal{A}\), for which \(H\) and \(H/N\) have no nonzero isomorphic subobjects, for every \(0\neq N\leq H\)) modulo an appropriate equivalence relation.

Reviewer: George Ciprian Modoi (Cluj-Napoca)

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