Derivations, generic formal fibers and bad Noetherian rings

Abstract

We consider a circle of ideas involving differential algebra, local Noetherian rings, and their generic formal fibers. Connecting these ideas gives rise to what we term a "twisted" subring $R$ of a ring $S$. Each such subring $R$ arises as a pullback of a derivation taking values in an $S$-module $K$. The twisting relationship proves to be a kind of inversion of Nagata idealization: whereas idealization extends $S$ to the larger ring $S \star K$, twisting produces a subring of $S$ which behaves much like the ring $S \star K$. The rings produced in this manner exhibit pathological features, such as failing to have finite normalization, but in spite of this they are quite tractable and conceptually (if not practically) easy to locate, and in this way provide a rich but manageable source of non-standard Noetherian rings. The theory developed to support this construction involves extensive use of both Noetherian and non-Noetherian commutative ring theory, as well as differential algebra.