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An effective avoidance principle for a class of ideals. (English) Zbl 1388.13028
Let \(S\) be a polynomial ring over a field of characteristic zero and let \(I\subset S\) be a monomial ideal. The ideal \(I\) is said to be of intersection type if it can be written as the intersection of powers of monomial prime ideals. This kind of ideals was introduced by J. Herzog and M. Vladoiu [Electron. J. Comb. 21, No. 1, Research Paper P1.69, 18 p. (2014; Zbl 1307.13014)]. In the paper under review, the author investigates the ideals \(I\) which are of intersection type and moreover, have no embedded primary component. The main goal is to provide an effective sufficient condition for a given monomial prime ideal to avoid the sets of prime divisors of the powers of \(I\), and in particular to avoid the celebrated set of asymptotic prime divisors of \(I\), which will follow from a new and quite surprising double-colon stability property. Further, the author briefly describes some other applications, e.g., on the topology of a suitable blowing-up.
MSC:
13C13 Other special types of modules and ideals in commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
05E40 Combinatorial aspects of commutative algebra
13A99 General commutative ring theory
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