Non-standard tight closure for affine ℂ-algebras

Abstract.

 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(⋅) has the same properties as classical tight closure, to wit, (1) if A is regular, then 𝔞=cl(𝔞); (2) if A⊂B is an integral extension of domains, then cl(𝔞 B)∩A⊂cl(𝔞); (3) if A is local and is a system of parameters, then (Colon-Capturing); (4) if 𝔞 is generated by m elements, then cl𝔞 contains the integral closure of 𝔞^m and is contained in the integral closure of 𝔞 (Briançon-Skoda).