Non-standard tight closure for affine C-algebras

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(aB) ∩ A ⊂ cl(a); (3) if A is local and (x1, . . . , xn) is a system of parameters, then ((x1, . . . , xi) : xi+1) ⊂ cl(x1, . . . , xi) (Colon-Capturing); (4) if a is generated by m elements, then cl(a) contains the integral closure of a and is contained in the integral closure of a (Briançon-Skoda).