Absolutely integral homomorphisms (Q2370139)

This paper deals with the notion of ``absolutely integral homomorphisms''. Recall that if \(f: A\rightarrow B\) is a homomorphism of rings and \(P\) is a property of rings, then we say that \(f\) is: (1) absolutely \(P\) (or \(B\) is absolutely \(P\) algebra over \(A\)) if \(C\bigotimes_{A}B\) verifies \(P\) for any base change \(A\rightarrow C\) where \(C\) satisfies \(P\). When \(P\) is the integral (respectively reduced) property, we say that \(A\rightarrow B\) is absolutely integral (respectively reduced). (2) geometrically \(P\) if \(f\) is flat and \(K\bigotimes_{A}B\) verifies \(P\) for every base change \(A\rightarrow K\), (3) almost absolutely integral if \(C\bigotimes_{A}B\) is either an integral domain or zero for any domain base change \(A\rightarrow C\), (4) almost geometrically integral if \(f\) is flat and \(K\bigotimes_{A}B\) is either a domain or zero for any field base change \(A\rightarrow K\), (5) integrally closed if \(f\) is injective and \(A\) is integrally closed in \(B\). The paper is divided into five sections. In the second section, among others, the following characterizations of the notion of (almost) absolutely integral homomorphisms are provided: A homomorphism \(f: A\rightarrow B\) is almost absolutely integral iff \(A/I\rightarrow B/IB\) is almost absolutely integral for each ideal \(I\) of \(A\) iff \(A/Q\rightarrow B/QB\) is almost absolutely integral for each prime ideal \(Q\) of \(A\) iff \(f\) is integrally flat with almost geometrically integral fibers iff \(f\) is integrally flat and \({\overline{k(Q)}}\bigotimes_{A}B\) is an integral domain for each \(Q\in \chi(f)\). The main theorem of the third section asserts that if \(f: A\rightarrow B\) is absolutely integral, then it is absolutely integrally closed. The fourth section shows that absolutely integral homomorphisms have good transfer properties, for instance, it is proved that if \(M\) is an \(A\)-module, then \(A\rightarrow S_{A}(M)\) is absolutely integral iff \(M\) is integrally flat, where \(S_{A}(M)\) is its symetric algebra. Other results concerning homomorphism ring \(A\rightarrow B\) between noetherian rings and the above properties are given. The paper closes with a short section collecting a few results on the above notions and polynomial rings.