The rank-$r\ge 2$ case of the non-archimedean analytic Bass–Quillen question of Kerz–Saito–Tamme (Question 3)

For a smooth affinoid algebra $A$ over a complete non-archimedean field $K$, Kerz–Saito–Tamme proposed a non-archimedean analytic analog of the Bass–Quillen conjecture: for the pro-system of Tate algebras whose $n$-th stage is $A\langle\pi^{-n}t\rangle$ with transition maps induced by $t\mapsto\pi t$, approximating the analytic affine line $(\mathbb{A}^1_A)^{\mathrm{an}}$, is the natural map from rank-$r$ vector bundles on $A$ to vector bundles on this pro-system a pro-isomorphism? The authors proved the rank-$1$ case (Picard groups) and, for $K$ discretely valued and $A$ admitting a regular model, the corresponding statement for the Grothendieck group $K_0$ of vector bundles. As of the 2016–2026 follow-up literature, the rank-$r\ge 2$ case, and the $K_0$ statement without the discreteness and regular-model hypotheses, remain open.

Background

In commutative algebra, Serre's problem asked whether every finitely generated projective module over a polynomial ring $k[t_1,\dots,t_n]$ over a field $k$ is free; Quillen and Suslin proved this in 1976. Bass then asked the relative version now known as the Bass–Quillen conjecture: for a regular ring $R$, is every finitely generated projective $R[T]$-module extended from $R$, i.e. of the form $N\otimes_R R[T]$? Equivalently, vector bundles on the affine space $\mathbb{A}^1_R$ are pullbacks of vector bundles on $\operatorname{Spec} R$. This is known for regular rings essentially of finite type over a field (Lindel's theorem) and in several further cases, but not in full generality.

The non-archimedean analog takes place in rigid analytic geometry. A complete non-archimedean field $K$ is a field complete with respect to a nontrivial absolute value $|\cdot|$; fix an element $\pi\in K\setminus\{0\}$ with $|\pi|<1$. The one-variable Tate algebra $K\langle t\rangle$ consists of the power series $\sum_k c_k t^k$ with coefficients tending to $0$, i.e. the functions holomorphic on the closed unit disc, and an affinoid $K$-algebra $A$ is a quotient of such a convergent power series ring; “smooth” means geometrically regular, i.e. the associated rigid analytic space $\operatorname{Sp}(A)$ has no singularities. Vector bundles over $A$ are the same as finitely generated projective $A$-modules; $\operatorname{Vec}_r(A)$ denotes the set of isomorphism classes of rank-$r$ vector bundles, and the Picard group $\operatorname{Pic}(A)=\operatorname{Vec}_1(A)$ (with tensor product) classifies line bundles.

The naive analytic analog of Bass–Quillen fails: Lütkebohmert proved that finitely generated projective modules over $K\langle t_1,\dots,t_n\rangle$ are free (the analytic Quillen–Suslin theorem), but Gerritzen constructed a smooth affinoid algebra $A=K\langle t_1,t_2\rangle/(t_1^2-t_2^3-\pi)$ for which the map $\operatorname{Pic}(A)\to\operatorname{Pic}(A\langle t\rangle)$ is not bijective, where $A\langle t\rangle=A\widehat\otimes_K K\langle t\rangle$ is the ring of functions convergent on the closed unit disc over $A$. The culprit is that the closed unit disc is too small an analytic replacement for the polynomial variable. Kerz–Saito–Tamme therefore replaced $A\langle t\rangle$ by the pro-system with stages $A\langle\pi^{-n}t\rangle$: its $n$-th stage consists of the functions convergent on the closed disc of radius $|\pi|^{-n}$, with transition maps induced by $t\mapsto\pi t$. These growing closed discs exhaust the analytic affine line $(\mathbb{A}^1_A)^{\mathrm{an}}$, a rigid analytic space that is not quasi-compact (not affinoid), and indeed the limit of the pro-system is the ring $H^0((\mathbb{A}^1_A)^{\mathrm{an}},\mathcal{O})$ of global analytic functions.

A map from a fixed set (or group) $S$ to such a pro-system is a pro-isomorphism when two conditions hold: first, for every stage $n$ and every vector bundle $V$ on $A\langle\pi^{-n}t\rangle$ there is a later stage $m\ge n$ such that the pullback of $V$ to stage $m$ is the pullback of a vector bundle on $A$; second, two vector bundles on $A$ whose pullbacks become isomorphic on some stage are already isomorphic over $A$. Kerz–Saito–Tamme posed Question 3, asked for every rank $r\ge 1$: for a smooth affinoid algebra $A/K$, is the natural map

$$\operatorname{Vec}_r(A)\longrightarrow\lim_{t\mapsto\pi t}\operatorname{Vec}_r(A\langle t\rangle)$$

a pro-isomorphism? They proved it for $r=1$: the map $\operatorname{Pic}(A)\to\lim_{t\mapsto\pi t}\operatorname{Pic}(A\langle t\rangle)$ is an isomorphism of pro-abelian groups (Theorem 4), which implies $\operatorname{Pic}(A)\cong\operatorname{Pic}((\mathbb{A}^1_A)^{\mathrm{an}})$ (Corollary 5). The rank-$1$ proof reduces, via the exponential sequence in residue characteristic $0$ and a direct argument in positive characteristic, to vanishing results for additive rigid cohomology going back to Bartenwerfer. They also proved a “stable” version: if $K$ is discretely valued and $A$ is a regular affinoid algebra admitting a regular model — a proper morphism of schemes $X\to\operatorname{Spec}(A^\circ)$, with $A^\circ$ the subring of power-bounded elements of $A$, that is an isomorphism over $\operatorname{Spec}(A)$ and has $X$ regular — then the induced map on the Grothendieck group $K_0(A)$ of vector bundles (the group completion of the monoid of finitely generated projective $A$-modules under direct sum) is a pro-isomorphism (Theorem 6). The existence of a regular model is automatic when the residue field of $K$ (the quotient of the valuation ring by its maximal ideal) has characteristic $0$, by resolution of singularities; the proof of Theorem 6 uses pro-cdh descent for algebraic $K$-theory, a descent property for blow-up covers, and is therefore unavailable in positive residue characteristic without a regular model.

In two subsequent papers the same authors developed “analytic $K$-theory” $K^{\mathrm{an}}$ of Tate rings, an $\mathbb{A}^1$-local (homotopy-invariant) variant of algebraic $K$-theory in the spirit of Weibel's homotopy $K$-theory, and showed that $K^{\mathrm{an}}$ satisfies pro-homotopy invariance unconditionally for every complete Tate ring admitting a noetherian finite-dimensional ring of definition. This, however, concerns the $\mathbb{A}^1$-local replacement functor: it sidesteps rather than answers the question about honest vector bundles or their Grothendieck group, and the comparison of $K^{\mathrm{an}}$ with the $K$-theory of vector bundles still requires regularity and weak-resolution hypotheses on a ring of definition. Thus the vector-bundle question beyond rank $1$, and the stable statement without the discreteness and regular-model hypotheses, survive as open problems.

Problem Statement

Let $K$ be a complete non-archimedean non-trivially valued field, let $\pi\in K\setminus\{0\}$ with $|\pi|<1$, and let $A$ be a smooth affinoid $K$-algebra. Consider the pro-system of sets of isomorphism classes of vector bundles over the stages $A\langle\pi^{-n}t\rangle$, $n\ge 0$, with transition maps induced by $t\mapsto\pi t$, denoted $\lim_{t\mapsto\pi t}\operatorname{Vec}_r(A\langle t\rangle)$.

For every rank $r\ge 2$, every such $K$, and every smooth affinoid $K$-algebra $A$, is the natural pullback map

$$\operatorname{Vec}_r(A)\longrightarrow\lim_{t\mapsto\pi t}\operatorname{Vec}_r(A\langle t\rangle)$$

a pro-isomorphism? That is: for every $n\ge 0$ and every vector bundle $V$ on $A\langle\pi^{-n}t\rangle$, does there exist $m\ge n$ such that the pullback of $V$ to $A\langle\pi^{-m}t\rangle$ is the pullback of a vector bundle on $A$ (pro-surjectivity); and if two vector bundles on $A$ become isomorphic after pullback to some stage $A\langle\pi^{-n}t\rangle$, are they already isomorphic over $A$ (pro-injectivity)?

The rank-$1$ case is Theorem 4 of the source and is settled; the open problem is every rank $r\ge 2$.

References

1. Moritz Kerz, Shuji Saito, Georg Tamme, Towards a non-archimedean analytic analog of the Bass–Quillen conjecture, Journal of the Institute of Mathematics of Jussieu 19 (2020), no. 6, 1931–1946, doi:10.1017/S147474801900001X, https://arxiv.org/abs/1608.00703
2. Moritz Kerz, Shuji Saito, Georg Tamme, K-theory of non-archimedean rings. I, Documenta Mathematica 24 (2019), 1365–1411, doi:10.4171/dm/707, arXiv:1802.09819, https://arxiv.org/abs/1802.09819
3. Moritz Kerz, Shuji Saito, Georg Tamme, K-theory of non-archimedean rings II, Nagoya Mathematical Journal 251 (2023), 669–685, doi:10.1017/nmj.2023.4, arXiv:2103.06711, https://arxiv.org/abs/2103.06711
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8. Anastasia Stavrova, On the generalized Bass–Quillen conjecture in dimension 2, arXiv:2512.18868, https://arxiv.org/abs/2512.18868
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