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Convex bodies and multiplicities of ideals. (English) Zbl 1315.13013
Proc. Steklov Inst. Math. 286, 268-284 (2014) and Tr. Mat. Inst. Steklova 286, 291-307 (2014).
The connection between multiplicities of monomial ideals in a polynomial ring and convex geometry is known by the work of Bernstein and Kouchnirenko. The aim of this article is to extend this connection to a more general setting.
Let \((R, {\mathfrak m})\) be an analytically irreducible local domain of dimension \(n\), and \({\mathbf k}:=R/{\mathfrak m}\) the residue field. In this article it is proved that \(R\) has a “good” valuation, that is, a valuation \(v\) with values in \({\mathbb Z}^n\), satisfying some technical properties. On the other hand, let \(({\mathfrak a}_\bullet )\) be a \({\mathfrak m}-\) primary sequence of subspaces of \({\mathfrak m}\), such that \({\mathfrak a}_k {\mathfrak a}_m\subset {\mathfrak a}_{k+m}\) for any natural integers \(k,m\), and \({\mathfrak a}_1\) contains a power of \({\mathfrak m}\). The authors prove that the limit: \[ e({\mathfrak a}_\bullet ):=n!\lim_{k\rightarrow +\infty } \frac{\text{dim}_{\mathbf k}(R/{\mathfrak a}_k)}{k^n} \] exists. The number \(e({\mathfrak a}_\bullet )\) is called the multiplicity of \(({\mathfrak a}_\bullet) \).
Let \(C\subset {\mathbb R}^n\) be the closure of the convex hull of the value semigroup \(S\) of the valuation \(v\). The authors define a convex region \(\Gamma ({\mathfrak a}_\bullet)\subset C\) such that \(C\setminus \Gamma ({\mathfrak a}_\bullet)\) is bounded.
The main result in this paper is that \(e({\mathfrak a}_\bullet )=n! \text{vol}(C\setminus \Gamma ({\mathfrak a}_\bullet))\). As a consequence they get the Brunn-Minkowski inequality: \[ e({\mathfrak a}_\bullet )^{1/n}+ e({\mathfrak b}_\bullet )^{1/n}\geq e({\mathfrak a}_\bullet {\mathfrak b}_\bullet )^{1/n} \]

MSC:
13A18 Valuations and their generalizations for commutative rings
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
13D10 Deformations and infinitesimal methods in commutative ring theory
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