In this paper, we find the tight closure of powers parameter ideals certain diagonal hypersurface rings. many cases, associated graded ring with respect to filtration turns out be Cohen–Macaulay. This helps us Hilbert polynomial in these hypersurfaces. We determine following cases: (1) F-pure hypersurfaces where number variables is equal degree defining equation, (2) rings characteristic one le...

This paper establishes the fundamental properties of s-closures, a recently introduced family closure operations on ideals rings positive characteristic. The behavior s-closure homogeneous in graded is studied, and criteria are given for when an ideal can be described exactly terms its tight rational powers. Sufficient conditions established weak to equal s-closure. A generalization Briançon-Sk...

We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from below for the tight closure of a homogeneous R+-primary ideal in a two-dimensional normal standard-graded algebra R in terms of the minimal and the maximal slope...

We introduce a new variant of tight closure and give an interpretation of adjoint ideals via this tight closure. As a corollary, we prove that a log pair (X,∆) is plt if and only if the modulo p reduction of (X,∆) is divisorially F-regular for all large p ≫ 0. Here, divisorially F-regular pairs are a class of singularities in positive characteristic introduced by Hara and Watanabe [HW] in terms...

Finitely generated modules with finite F -representation type (or FFRT for short) over Noetherian (local) rings of prime characteristic p are studied. If a ring R has FFRT or, more generally, if a faithful R-module has FFRT, then tight closure commutes with localizations over R. We also define F -contributors and use them to give an effective way to characterize tight closure. Then we show lime...

We associate to every equicharacteristic zero Noetherian local ring R a faithfully flat ring extension which is an ultraproduct of rings of various prime characteristics, in a weakly functorial way. Since such ultraproducts carry naturally a non-standard Frobenius, we can define a new tight closure operation on R by mimicking the positive characteristic functional definition of tight closure. T...

During neural tube closure, regulated changes at the level of individual cells are translated into large-scale morphogenetic movements to facilitate conversion of the flat neural plate into a closed tube. Throughout this process, the integrity of the neural epithelium is maintained via cell interactions through intercellular junctions, including apical tight junctions. Members of the claudin fa...