Related papers: A new version of $\mathfrak{a}$-tight closure

We introduce a new variant of tight closure associated to any fixed ideal $\a$, which we call $\a$-tight closure, and study various properties thereof. In our theory, the annihilator ideal $\tau(\a)$ of all $\a$-tight closure relations,…

Closure operations such as tight and integral closure and test ideals have appeared frequently in the study of commutative algebra. This articles serves as a survey of the authors' prior results connecting closure operations, test ideals,…

Using the fact that the structure sheaf of a resolution of singularities, or regular alteration, pushes forward to a Cohen-Macaulay complex in equal characteristic zero with a differential graded algebra structure, we introduce a…

Let $R$ be an excellent Noetherian ring of prime characteristic. Consider an arbitrary nested pair of ideals (or more generally, a nested pair of submodules of a fixed finite module). We do \emph{not} assume that their quotient has finite…

The test ideal $\tau(R)$ of a ring $R$ of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal $\tau(\a^t)$ associated to a given ideal…

We define a duality operation connecting closure operations, interior operations, and test ideals, and describe how the duality acts on common constructions such as trace, torsion, tight and integral closures, and divisible submodules. This…

An analogue of the theory of integral closure and reductions is developed for a more general class of closures, called Nakayama closures. It is shown that tight closure is a Nakayama closure by proving a ``Nakayama lemma for tight…

This paper establishes the fundamental properties of the $s$-closures, a recently introduced family of closure operations on ideals of rings of positive characteristic. The behavior of the $s$-closure of homogeneous ideals in graded rings…

We introduce two closure operations on ideals in commutative rings related to the ring operation of root closure. One closure is the result of iterating a root-like operation on ideals infinitely many times, and the other closure arises as…

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…

Hara and Smith independently proved that in a normal $\mQ$-Gorenstein ring of characteristic $p \gg 0$, the test ideal coincides with the multiplier ideal associated to the trivial divisor. We extend this result for a pair $(R, \Delta)$ of…

This paper is concerned with the tight closure of an ideal $I$ in a commutative Noetherian ring $R$ of prime characteristic $p$. The formal definition requires, on the face of things, an infinite number of checks to determine whether or not…

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…

In this paper, using ultra-Frobenii, we introduce a variant of Schoutens' non-standard tight closure, ultra-tight closure, on ideals of a local domain $R$ essentially of finite type over $\mathbb{C}$. We prove that the ultra-test ideal…

This article is a survey of closure operations on ideals in commutative rings, with an emphasis on structural properties and on using tools from one part of the field to analyze structures in another part. The survey is broad enough to…

Let $(R,\mathfrak m)$ be an analytically unramified local ring of positive prime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure. We introduce the tight Hilbert function $H^*_I(n)=\ell(R/(I^n)^*)$ and the…

We give a survey about some recent work on tight closure and Hilbert-Kunz theory from the viewpoint of vector bundles

The F-threshold $c^J(\a)$ of an ideal $\a$ with respect to the ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\a$ with the Frobenius powers of $J$. We show that under mild assumptions, we can detect…

Given a densely defined and closed operator $A$ acting on a complex Hilbert space $\mathcal{H}$, we establish a one-to-one correspondence between its closed extensions and subspaces $\mathfrak{M}\subset\mathcal{D}(A^*)$, that are closed…

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.