Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A numerical characterization of reduction for arbitrary modules   Let $(R, \mathfrak m)$ be a $d$-dimensional Noetherian local ring and $E$ a
finitely generated $R$-submodule of a free module $R^p.$ In this work we
introduce a multiplicity sequence $c_k(E), k=0,..., d+p-1$ for $E$ that
generalize the Buchsbaum-Rim multiplicity defined when $E$ has finite colength
in $R^p$ as well as the Achilles-Manaresi multiplicity sequence that applies
when $E\subseteq R$ is an ideal. Our main result is that the new multiplicity
sequence can indeed be used to detect integral dependence of modules. Our proof
is self-contained and implies known numerical criteria for integral dependence
of ideals and modules.

13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
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The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
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The categories of this paper are: eess
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys
1
The categories of this paper are: cs
17
The categories of this paper are: q-fin