Non-standard tight closure for affine \(\mathbb C\)-algebras
The paper under review proposes another approach to tight closure in characteristic zero based on methods from model theory. Tight closure is a closure operation for ideals in a Noetherian ring \(R\) containing a field of positive characteristic, and the containment \(z \in I^*\) is given by the condition that there exists a \(c \in R\), not in any minimal prime, such that \(c z^q \in I^{[q]}\) holds for all prime powers. By reduction to positive characteristic, one also obtains a notion of tight closure in characteristic zero, which means in the easiest case that for a finitely generated \(\mathbb{Z}\)-algebra \(S\) the containment \(z \in I^*\) in \(S_\mathbb{Q}\) is by definition the containment \(z \in I^*_p\) in \(S_{k(p)}\) for almost all prime numbers. The approach of the paper is that there is an isomorphism \(\mathbb{C} = \lim \overline{\mathbb F}_p\), where the limit runs over an ultrafilter of sets of prime numbers. In this sense a complex number is represented by a sequence of numbers in \(\overline{\mathbb F}_p\), and in the same manner polynomials \(f_i\) over \(\mathbb{C}\) may be considered as a sequence of polynomials over \(\overline{\mathbb F}_p\). The corresponding factor rings \(R= \mathbb{C}[X_j]/(f_i)\) and \(R_p= \overline{\mathbb F}_p[X_j]/(f_i)\) are related in such a way that the limit of the \(R_p\), denoted by \(R_\infty\), is a (non-Noetherian) faithfully flat extension of \(R\). Moreover, the characteristic\,\(p\) Frobenius endomorphisms of \(R_p\) yield a non-standard Frobenius endomorphism \(F_\infty\) of \(R_\infty\), and with this characteristic zero Frobenius the author defines the non-standard tight closure containment \(z \in I^*\) (\(I \subseteq R\)) by the requirement that \(cF^m_\infty(z) \in F^m _\infty(I) R_\infty\) holds for some \(c\) not in any minimal prime of \(R\) and all \(m\). The first half of the paper (sections 2--5) gives the logical foundations for this closure operation and several variants, while the second half (sections 6--10) shows that it fulfills the standard properties of tight closure.
scientific article; zbMATH DE number 1965511