17 research outputs found
Specialization of Integral Closure of Ideals by General Elements
In this paper, we prove a result similar to results of Itoh and Hong-Ulrich,
proving that integral closure of an ideal is compatible with specialization by
a general element of that ideal for ideals of height at least two in a large
class of rings. Moreover, we show integral closure of sufficiently large powers
of the ideal is compatible with specialization by a general element of the
original ideal. In a polynomial ring over an infinite field, we give a class of
squarefree monomial ideals for which the integral closure is compatible with
specialization by a general linear form
Integral closure and generic elements
AbstractLet (R,m) be a formally equidimensional local ring with depthR⩾2 and I=(a1,…,an) an m-primary ideal in R. The main result of this paper shows that if I is integrally closed, then so is its image modulo a generic element, that is, if T=R[X1,…,Xn]/(a1X1+⋯+anXn), then IT¯=I¯T
Limit Behavior of the Rational Powers of Monomial Ideals
We investigate the rational powers of ideals. We find that in the case of
monomial ideals, the canonical indexing leads to a characterization of the
rational powers yielding that symbolic powers of squarefree monomial ideals are
indeed rational powers themselves. Using the connection with symbolic powers
techniques, we use splittings to show the convergence of depths and normalized
Castelnuovo-Mumford regularities. We show the convergence of Stanley depths for
rational powers, and as a consequence of this we show the before-now unknown
convergence of Stanley depths of integral closure powers. In addition, we show
the finiteness of asymptotic associated primes, and we find that the normalized
lengths of local cohomology modules converge for rational powers, and hence for
symbolic powers of squarefree monomial ideals