This paper facilitates the computation of tight closure by giving giving upper and lower bounds on the degrees of elements that need to be checked for inclusion in the tight closure of certain homogeneous ideals in a graded ring. Differential operators are introduced to the study of tight closure, and used to prove that the degree of any element in the tight closure of a homogeneous ideal (but ...

Contents Introduction 3 1. Foundations 13 1.1. A survey about the theory of tight closure 13 1.2. Solid closure and forcing algebras 23 1.3. Cohomological dimension 25 1.4. Vector bundles, locally free sheaves and projective bundles 28 2. Geometric interpretation of tight closure via bundles 30 2.1. Relation bundles 30 2.2. Affine-linear bundles arising from forcing algebras 32 2.3. Cohomology ...

In this paper, non-standard tight closure is proposed as an alternative for classical tight closure on finitely generated algebras over C. It has the advantage that it admits a functional definition, similar to the characteristic p definition of tight closure, where instead of the characteristic p Frobenius, its ultraproduct, the non-standard Frobenius, is used. This new closure operation cl(·)...

Closure of the tight junctions of the mammary epithelium has been shown to accompany the onset of copious milk secretion or lactogenesis, stage 2, in both goats and humans. Here we use injection of [(14)C]sucrose and FITC-albumin (fluorescein isothiocyanate-albumin) into the mammary duct to follow the course of tight junction closure during lactogenesis in mice. To examine the hormonal changes ...