Some ring-theoretic properties of A_inf

Abstract

The ring of Witt vectors over a perfect valuation ring of characteristic p, often denoted A_inf, plays a pivotal role in p-adic Hodge theory; for instance, Bhatt, Morrow, and Scholze have recently reinterpreted and refined the crystalline comparison isomorphism by relating it to a certain A_inf-valued cohomology theory. We address some basic ring-theoretic questions about A_inf motivated by analogies with two-dimensional regular local rings. For example, we show that in most cases A_inf, which is manifestly not noetherian, is also not coherent. On the other hand, it does have the property that vector bundles over the complement of the closed point in Spec A_inf do extend uniquely over the puncture; moreover, a similar statement holds in Huber's category of adic spaces.