Tight Closure in Characteristic Zero. Affine Case

We will develop a tight closure theory in characteristic zero which is different from the Hochster-Huneke approach discussed briefly in §5.6. In this chapter we treat the affine case, that is to say, we develop the theory for algebras of finite type over an uncountable algebraically closed field

K

of characteristic zero; the general local case will be discussed in Chapter 7. Recall that under the Continuum Hypothesis, any uncountable algebraically closed field

K

of characteristic zero is a

Lefschetz field

, that is to say an ultraproduct of fields of positive characteristic, by Theorem 2.4.3 and Remark 2.4.4. In particular, without any set-theoretic assumption, ℂ, the field of complex numbers, is a Lefschetz field. The idea now is to use the ultra-Frobenius, that is to say, the ultraproduct of the Frobenii (see Definition 2.4.21), in the same manner in the definition of tight closure as in positive characteristic. However, the ultra-Frobenius does not act on the affine algebra but rather on its ultra-hull, so that we have to introduce a more general setup. It is instructive to do this first in an axiomatic manner (§6.1) and then specialize to the situation at hand (§6.2). We briefly discuss a variant construction in §6.3, and conclude in §6.4 with another example how ultraproducts can be used to transfer constructions from positive to zero characteristic, to wit, the balanced big Cohen-Macaulay algebras of Hochster and Huneke.