[Paper Review] Some ring-theoretic properties of A_inf

Kiran S. Kedlaya|arXiv (Cornell University)|Feb 29, 2016

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper investigates fundamental ring-theoretic properties of the ring $\mathbf{A}_{\operatorname{inf}}$, defined as the ring of $p$-typical Witt vectors over a perfect valuation ring of characteristic $p$. It shows that $\mathbf{A}_{\operatorname{inf}}$ is typically not coherent when the value group is not isomorphic to $\mathbb{R}$, but vector bundles over the punctured spectrum of $\operatorname{Spec}(\mathbf{A}_{\operatorname{inf}})$ extend uniquely over the closed point, a property shared in Huber's category of adic spaces.

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.

Motivation & Objective

• To investigate coherence and finite generation properties of $\mathbf{A}_{\operatorname{inf}}$, a central ring in $p$-adic Hodge theory.
• To explore analogies between $\mathbf{A}_{\operatorname{inf}}$ and two-dimensional regular local rings, particularly regarding ideal structure and vector bundle extensions.
• To determine whether vector bundles on the punctured spectrum of $\operatorname{Spec}(\mathbf{A}_{\operatorname{inf}})$ extend uniquely to the full spectrum.
• To extend the vector bundle extension result to Huber's category of adic spaces, addressing geometric and cohomological properties in non-noetherian settings.

Proposed method

• Use of Newton polygon theory for elements in $\mathbf{A}_{\operatorname{inf}}$ to analyze the structure of intersections of finitely generated ideals.
• Construction of specific elements $f$ and $g$ in $\mathbf{A}_{\operatorname{inf}}$ whose intersection ideal fails to be finitely generated when the value group is not isomorphic to $\mathbb{R}$.
• Application of the Anderson–Watkins technique to show non-coherence in power series-like rings over nondiscrete valuation rings.
• Adaptation of Serre’s criterion for reflexive modules and use of $\operatorname{Tor}$-vanishing to analyze projective dimension and non-projectivity of certain modules.
• Use of the Zariski and adic spectra to study sheaf-theoretic extension of vector bundles, relying on fully faithful functors between categories of vector bundles on open subsets.
• Translation of classical counterexamples from algebraic geometry (e.g., Example 3.10) into the context of $\mathbf{A}_{\operatorname{inf}}$-modules to demonstrate failure of extension in relative settings.

Experimental results

Research questions

• RQ1 Is $\mathbf{A}_{\operatorname{inf}}$ coherent when the value group of the base field $K$ is not isomorphic to $\mathbb{R}$?
• RQ2 Do vector bundles on the punctured spectrum $\operatorname{Spec}(\mathbf{A}_{\operatorname{inf}}) \setminus \{\mathfrak{m}\}$ extend uniquely to the full spectrum $\operatorname{Spec}(\mathbf{A}_{\operatorname{inf}})$?
• RQ3 Does the unique extension property of vector bundles hold in Huber’s category of adic spaces for $\mathbf{A}_{\operatorname{inf}}$?
• RQ4 Can the extension result be generalized to relative settings where $K$ is replaced by a more general nonarchimedean Banach ring $R$?
• RQ5 What is the projective dimension of the cokernel of a map $W(R^{+})^3 \to I$ for an ideal $I$ generated by a regular sequence in $W(R^{+})$?

Key findings

• When the value group of $K$ is not isomorphic to $\mathbb{R}$, the ring $\mathbf{A}_{\operatorname{inf}}$ is not coherent, as demonstrated by constructing elements $f$ and $g$ such that the ideal $(f) \cap (g)$ is not finitely generated.
• For any perfect field $K$ of characteristic $p$ with nontrivial valuation, the intersection of two finitely generated ideals in $\mathbf{A}_{\operatorname{inf}}$ may fail to be finitely generated, especially when the value group is archimedean or non-archimedean.
• Vector bundles on the punctured spectrum $\operatorname{Spec}(\mathbf{A}_{\operatorname{inf}}) \setminus \{\mathfrak{m}\}$ extend uniquely to the full spectrum, as shown in Theorem 2.7.
• The same unique extension property holds in Huber’s category of adic spaces, as established in Theorem 3.9, where the category of vector bundles on the punctured adic spectrum fully embeds into the category on the full adic spectrum.
• The module $M = \ker(W(R^{+})^3 \to I)$, where $I$ is generated by a regular sequence, has projective dimension at least 1 and is not projective, showing that not all reflexive modules are projective in this context.
• An example is constructed in which a vector bundle on the punctured adic spectrum $Y$ or $Z$ fails to extend to the full spectrum $X$, showing that the extension property does not generalize to relative settings with $R$ replacing $K$.