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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.

MSC:

13C05 Structure, classification theorems for modules and ideals in commutative rings
13F30 Valuation rings
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