Some Ring-Theoretic Properties of $$\mathbf {A}_{{{\mathrm{inf}\,}}}$$

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-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.

Perfectoid Spaces: An Annotated Bibliography

“…is not finitely generated. In fact, A inf (F ) has infinite [37] and even uncountable [13] global dimension, and in general is not even coherent [30].…”

“…On the other hand, it is true that every vector bundle on the punctured spectrum of A inf (F ) extends uniquely over the puncture. See [30].…”

Is Infinite Dimensional

“…As suggested in [5,Remark 1.6], we use Newton polygons to find an infinite chain of prime ideals between p and W (m).…”

“…In this paper, we prove the following theorem. Bhatt [2,Warning 2.24] and Kedlaya [5,Remark 1.6] note that the Krull dimension of A is at least 3. To see this, fix a pseudouniformizer ∈ R and let κ denote the residue field of R. Let W (m) be the kernel of the natural map W (R) → W (κ) and [−] : R → W (R) the Teichmüller map.…”