10 - Tight closure

Summary

The final chapter extends the characteristic p methods by introducing the tight closure of an ideal, a concept that, via the comparison to a regular subring or overring, conveys the flatness of the Frobenius to non-regular rings. It was invented by Hochster and Huneke about ten years ago and is still in rapid development.

The principal classes of rings whose definition is suggested by tight closure theory consist of the F-regular and F-rational rings; they are characterized by the condition that all ideals or, in the case of F-rationality, the ideals of the principal class are tightly closed. Under a mild extra hypothesis F-rationality implies the Cohen–Macaulay property. More is true: F-rational rings are the characteristic p counterparts of rings with rational singularities; we will at least indicate this connection – a full treatment would require methods of algebraic geometry beyond our scope.

Tight closure theory has many powerful applications. Among them we have selected the Briançon–Skoda theorem, whose proof is based on the relationship of tight closure and integral closure, and the theorem of Hochster and Huneke that equicharacteristic direct summands of regular rings are Cohen–Macaulay.

The tight closure of an ideal

Throughout this section we suppose that all rings are Noetherian and of prime characteristic p, unless stated otherwise. Recall from Section 8.2 that I^[q], q = p^e, denotes the q-th Frobenius power of an ideal I, that is, I^[q] is the ideal generated by the q-th powers of the elements of I; equivalently, I^[q] is the ideal generated by the image of I under the e-fold iteration F^e of the Frobenius homomorphism F : R → R, F(a) = a^p.