Almost Witt Vectors

Abstract

Our main goal in this paper is to prove results for Witt vectors in the almost category. To do so, we spend the first section developing almost ring theory, as detailed in Gabber and Ramero's book on Almost Ring Theory [3]. We have tried to keep the exposition linear and digestable in order for the reader to grasp necessities and appreciate their usability. We finish with an application to almost purity, in particular offering a new proof for perfectoid rings with a rank one valuation.

Let us describe the main sections. First, we define all the relevent categorical notions in Sections 1.1 - 1.3 that lead to the interesting algebraic theory pertaining to almost mathematics. We provide some modest generalisations in some places, for example, in Section 1.1 where we discuss derived categories of almost modules. For the most part, chapter 1 follows [3], with the exception of Section 1.4 on flatness criteria, which we belive is entirely new. In some places, we have given new proofs of results found in [3], and moreover, in the authors opinion, Section 1.5 improves on the exposition given in [3]. The main point of the inclusion of this chapter was two-fold; we needed to ensure that all proofs given in [3] did not rely on the condition that m \otimes_R m was flat (a condition present in [3]), and we felt compeled to give a brief introduction to almost mathematics that could serve as a softer and more linear alternative to [3], and therefore more approachable.

Lastly, we believe the results in Chapter 2 are entirely new; in particular, our proofs in Section 2.1 generalise results found in the literature with the added bonus that they work in the almost category. In Section 2.2, we provide a new proof of the almost purity theorem for rank one perfectoid rings, which could be seen as sitting somewhere between the proof provided in [3] and the proofs given by Kedlaya-Liu in their paper on p-adic Hodge theory. What is new, in particular, is the use of finite length witt vectors to exchange between characteristic p and characteristic 0 worlds. This untilting process works for perfectoid rings more generally, which gives some hope that our methods could potentially prove the full statement of the almost purity theorem. Show more