Danilov, V. I.; Koshevoj, G. A. Nilpotent operators and discretely concave functions. (English. Russian original) Zbl 1072.13007 Izv. Math. 67, No. 1, 1-15 (2003); translation from Izv. Ross. Akad. Nauk Ser. Mat. 67, No. 1, 3-20 (2003). Considered are constructions related to modules over a discrete valuation ring (such as the power series ring \(A=K[[T]]\)). If \(M\) is such an \(A\)-module of finite length and \(T\) is an operator (such as a uniformization) on it, the capacity function of \(M\) relative to its submodules \(N_1,\dots,N_k\) is \[ c(M; N_1,\dots,N_k; a)=\ell(M/N(a)), \] where \(a=(a_1,\dots,a_k)\in\mathbb Z^k_+\) and \(N(a)=T^{a_1}N_1+\dots+T^{a_k}N_k\). The authors then establish that (1) The capacity function \(c(M;N_1,N_2):\mathbb Z^2_+\longrightarrow\mathbb Z\) of the module \(M\), relative to a pair of its submodules is discretely concave. A converse is also valid as follows: (2) Let \(f:\mathbb Z^2_+\longrightarrow\mathbb Z\) be a discretely concave function such that \(f(0)=0\) and \(f\) stabilizes for large values of the argument. Then there are a module \(M\) of finite length and two submodules \(N_1\) and \(N_2\) such that \(f(\cdot)=c(M;N_1,N_2; \cdot)\). A number of other notions, examples and auxiliary results ensures that this paper will induce further contributions in the field. Reviewer: Radoslav M. Dimitrić (Uniontown) MSC: 13C05 Structure, classification theorems for modules and ideals in commutative rings 13F30 Valuation rings Keywords:discretely concave function; capacity function of a module; spider web of a function; modules over a discrete valuation ring; power series ring PDFBibTeX XMLCite \textit{V. I. Danilov} and \textit{G. A. Koshevoj}, Izv. Math. 67, No. 1, 1--15 (2003; Zbl 1072.13007); translation from Izv. Ross. Akad. Nauk Ser. Mat. 67, No. 1, 3--20 (2003) Full Text: DOI