Characterizing vector bundles on solid affinoid spaces via finite projective modules

Determine whether, for every solid affinoid animated Z_□-algebra A, an object F in D(AnSpec A) is a vector bundle if and only if F arises from a finite projective module over the underlying condensed ring \underline{A}.

Background

The paper recalls the notion of vector bundles on a solid D-stack X as objects of D(X) that locally (for the D-topology) come from finite projective modules. This generalizes classical coherent sheaf notions to the condensed/solid analytic setting.

For certain well-behaved analytic rings (termed Fredholm), a result of Clausen–Scholze shows that dualizable objects in D(A) are relatively discrete, leading to an identification of vector bundles with finite projective modules over the underlying condensed ring. However, outside the Fredholm context, it is unclear whether this characterization holds.

Clarifying this equivalence for general solid affinoid animated Z_□-algebras would provide a precise algebraic description of vector bundles in the solid D-stack framework.