Let $X$ be a preperfectoid space over $\mathrm{Spa}(\mathbb{Q}_p,\mathbb{Z}_p)$. It has several associated sites, with successively finer topologies: $$X_{an} \subset X_{et} \subset X_{proet} \subset X_v.$$

I was wondering: what is the relationship between vector bundles on these different sites? Here, by a vector bundle I mean a locally finite free $\mathcal{O}_X$-sheaf for $X_{an}, X_{et}$ and $X_v$, and a locally finite free $\widehat{\mathcal{O}}_X$-sheaf for $X_{proet}$.

If $X$ is perfectoid, it is known that the categories of vector bundles on all of these sites are equivalent, by Theorem 3.5.8 of Kedlaya-Liu's paper "Relative p-adic Hodge Theory II".

Is this also true for preperfectoid spaces? What about sousperfectoid spaces? If this statement is not true, is it still true that some of these categories are equivalent? I expect that at the very least the pullback functor from $X_{an}$ to $X_v$ is fully faithful. My apologies if there is some silly counterexample.



1 Answer 1


There is some relevant work of Ben Heuer on this.

In short, analytic and etale vector bundles agree on all sousperfectoid adic spaces (and much more generally, for all "etale sheafy" adic spaces), while proetale and v-vector bundles also agree (because proetale locally, the space is perfectoid, and so this reduces to the assertion for perfectoids). The pullback functor from etale vector bundles to proetale vector bundles is fully faithful. So in summary, there are two distinct categories (etale vector bundles and proetale vector bundles), one being a full subcategory of the other.

However, etale and proetale vector bundles are very different. This is already a well-known phenomenon for $p$-adic fields $K$ (complete discretely valued with perfectly residue field), where letting $C=\widehat{\overline{K}}$ it is the difference between $K$-vector spaces and $G_K$-equivariant $C$-vector spaces, which forms the subject of Sen--Tate theory.

By the way, there is some recent line of work (by Heuer, Mann, Werner, ...) that takes a fresh look at the $p$-adic Simpson correspondence from the perspective of proetale vector bundles. Roughly speaking, proetale vector bundles are what Faltings called "generalized representations" in his first paper on the $p$-adic Simpson correspondence, and should be more-or-less equivalent to Higgs bundles in case $X$ is smooth over an algebraically closed nonarchimedean $C/\mathbb Q_p$. Noth that Higgs bundles also contain usual vector bundles on $X$ fully faithfully, as those with vanishing Higgs field.

(On the other hand, if $X$ is proper, then proetale vector bundles contain $C$-local systems fully faithfully, by the primitive comparison isomorphisms.)


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