Related papers: Tight closure and continuous closure
We prove that in normal rings the tight closure of an ideal can be computed as the sum of the ideal and a piece of the tight closure, called the special tight closure.
We characterize the tight closure of graded primary ideals in a homogeneous coordinate ring over an elliptic curve by numerical conditions and we show that it is in positive characteristic the same as the plus closure.
We prove that the tight closure and the graded plus closure of a homogeneous ideal coincide for a two-dimensional N-graded domain of finite type over the algebraic closure of a finite field. This answers in this case a ``tantalizing…
We formulate problems of tight closure theory in terms of projective bundles and subbundles. This provides a geometric interpretation of such problems and allows us to apply intersection theory to them. This yields new results concerning…
We define a closure operation for ideals in a commutative ring which has all the good properties of solid closure (at least in the case of equal characteristic) but such that also every ideal in a regular ring is closed. This gives in…
There has arisen in recent years a substantial theory of "multiplier ideals'' in commutative rings. These are integrally closed ideals with properties that lend themselves to highly interesting applications. But how special are they among…
In this paper, we introduce the notions of tight closure of ideals on Witt rings and quasi-tightly closedness of system of parameters. By using the notions, we obtain a characterization of quasi-$F$-rationality. Furthermore, we study the…
Let $R$ be a (commutative Noetherian) local ring of prime characteristic that is $F$-pure. This paper studies a certain finite set ${\mathcal I}$ of radical ideals of $R$ that is naturally defined by the injective envelope of the simple…
Tight closure test ideals have been central to the classification of singularities in rings of characteristic $p>0$, and via reduction to characteristic $p$, in equal characteristic zero as well. A summary of their properties and…
We prove tight closure analogues of results of Watanabe about chains and families of integrally closed ideals.
Let R be an excellent local ring, m its maximal ideal and I an ideal. Then there exists a positive integer c such that for all integers n, the integral closure of (I + m^n) is contained in m^(n/c) + the integral closure of I. In the proof,…
We look at how the equivalence of tight closure and plus closure (or Frobenius closure) in the homogeneous m-coprimary case implies the same closure equivalence in the non-homogeneous m-coprimary case in standard graded rings. Although our…
It is proved that tight closure commutes with localization in any domain which has a module finite extension in which tight closure is known to commute with localization. It follows that tight closure commutes with localization in binomial…
We introduce a graded version of dagger closure and prove that it coincides with solid closure for homogeneous ideals in two dimensional $\mathbb{N}$-graded domains of finite type over a field.
We characterize the rings in which the equality $(\tau I:\tau)= I^*$ holds for every ideal $I \subset R$. Under certain assumptions, these rings must be either weakly F-regular or one-dimensional.
We study the relationship between the tight closure of an ideal and the sum of all ideals in its linkage class.
We ask for a given system of polynomials f_1,...,f_n and f over the complex numbers when there exist continuous functions q_1,...,q_n such that q_1 f_1+...+q_n f_n = f. This condition defines the continuous closure of an ideal. We give…
In this paper, we study some properties of the closure operator in the Mac\'ias topology on infinite integral domains. Moreover, under certain conditions, we present topological proofs of the infiniteness of maximal ideals and…
We prove a characterization of F-rationality in terms of tight closure of products of parameter ideals. Our results are inspired by the theory of complete ideals for surfaces and, in particular, the fundamental results of Lipman-Teissier…
We study computational aspects of the tight closure of a homogeneous primary ideal in a two-dimensional normal standard-graded domain. We show how to use slope criteria for the sheaf of syzygies for generators of the ideal to compute the…