Vector bundles and ideal closure operations

Summary

This is an introduction to the use of vector bundle techniques to ideal closure operations, in particular to tight closure and related closures like solid closure and plus closure. We also briefly introduce the theory of vector bundles in general, with an emphasis on smooth projective curves, and discuss the relationship between forcing algebras and closure operations.

An ideal operation is an assignment which provides for every ideal I in a commutative ring R a further ideal I’ fulfilling certain structural conditions such as I ⊆ I ', I '' = I ', and an inclusion I⊆ J should induce an inclusion It ⊆ J '. The most important examples are the radical of an ideal, whose importance stems from Hilbert’s Nullstellensatz, the integral closure ¯I , which plays a crucial role in the normalization of blow-up algebras, and tight closure I ^∗, which is a closure operation in positive characteristic invented by Hochster and Huneke. In the context of tight closure, many other closure operations were introduced such as plus closure I ⁺, Frobenius closure I ^F , solid closure I^✶, dagger closure I ^†, parasolid closure.

In this survey article we want to describe how the concepts of forcing algebras, vector bundles and their torsors can help to understand closure operations. This approach can be best understood by looking at the fundamental question whether ƒ ∈ ( ƒ1, . . . , ƒn) ‘ = I’ . In his work on solid closure, Hochster considered the forcing algebra