Extensions of primes, flatness, and intersection flatness

Abstract

We study when R → S has the property that prime ideals of R extend to prime ideals or the unit ideal of S, and the situation where this property continues to hold after adjoining the same indeterminates to both rings. We prove that if R is reduced, every maximal ideal of R contains only finitely many minimal primes of R, and prime ideals of R[X 1 , . . . , X n ] extend to prime ideals of S[X 1 , . . . , X n ] for all n, then S is flat over R. We give a counterexample to flatness over a reduced quasilocal ring R with infinitely many minimal primes by constructing a non-flat R-module M such that M = P M for every minimal prime P of R. We study the notion of intersection flatness and use it to prove that in certain graded cases it suffices to examine just one closed fiber to prove the stable prime extension property. Dedicated to Roger and Sylvia Wiegand onthe occasion of their 150th birthday

Test elements, excellent rings, and content functions

“…(See Corollary 5.3.) As a corollary to these, combined with the fact [HJ20] that algebras over complete local rings tend to be intersection-flat, we get a handy criterion that is all about Noetherianity.…”

Regularity and intersections of bracket powers

“…Remark 7 (Ohm-Rush content and Frobenius roots). Recall that for a commutative ring R, a module M is Ohm-Rush [OR72,ES16] or weakly intersection flat for ideals [HJ20], if for all collections {I α } of ideals of R, we have α (I α M ) = ( α I α )M . The condition on a Noetherian local ring of positive prime characteristic that the Frobenius itself is a flat Ohm-Rush module is an important condition in tight closure theory (cf.…”