# Fargues's Theorem for $Spa(C,C^+)$ (rather than $Spa(C,O_C)$

$$\DeclareMathOperator\Spa{Spa}$$Fargues's Theorem for $$\Spa(C,O_C)$$ states that the category of (mixed characteristic) shtukas with one paw at $$x_C$$ is equivalent to the category of Breuil-Kisin-Fargues modules over $$A_{inf}=W(O_C)$$. (Here, $$C$$ is an algebraically closed perfectoid field, and $$O_C$$ is the ring of integers of $$C$$.)

My question is concerned with extending this theorem to shtukas over $$\Spa(C,C^+)$$, where $$C^+$$ is an open and bounded valuation subring of $$C$$ and BKF-modules over $$W(C^+)=A_{inf}(C,C^+)$$ (and $$C$$ is still an algebraically closed perfectoid field).

First question: Where in the proof of Fargues's theorem does this slighly bigger generality fail?

A sketch of the proof of Fargues's theorem goes as follows:

Step 1: Turn a shtuka over $$\mathcal{Y}_{[0,\infty)}$$ to a shtuka over $$\mathcal{Y}_{[0,\infty]}$$. This step is done by glueing the shtuka with a "spread" vector bundle over the Fargues–Fontaine (that is the corresponding $$\phi$$-module over $$\mathcal{Y}_{[r,\infty]}$$ for some $$r$$). Here we use that we understand vector bundles over $$\mathrm{FF}_{(C,O_C)}$$ to be able to extend over $$\infty$$. Now theorem 8.7.7 in Kedlaya-Liu ("Relative $$p$$-adic Hodge Theory: Foundations" (MSN,arxiv)), tell us that vector bundles on $$\mathrm{FF}_{(C,C^+)}$$ coincide with those of $$\mathrm{FF}_{(C,O_C)}$$. That's why I think this step can be done as well.

Step 2: Turning an "analytic" shtuka over $$\mathcal{Y}_{[0,\infty]}$$ into an "algebraic" shtuka over $$Y_{[0,\infty]}$$. This is the content of Kedlaya's theorem 4.5.10 a) in his AWS (revised) notes, and that step is stated in full generality for any perfectoid Huber pair $$(R,R^+)$$.

Step 3: Turning the "algebraic" shtuka over $$\operatorname{Spec}(A_{inf})\setminus \{\varpi=0=p\}$$ into a BKF-module over $$\operatorname{Spec}(A_{inf})$$. This is now Kedlaya's theorem 4.5.10 b) of his AWS notes again. It works for any perfectoid field $$K$$ regardless of the chosen $$K^+$$.

All steps seem to work to me for $$\Spa(C,C^+)$$, but I might be missing a subtlety and I would like to know what that is. Step 1 is the one I would have most doubts about.

Step 1 indeed fails. The functor from BKF-modules over $$C^+$$ to shtukas over $$C^+$$ is not going to be full. You can still glue so you still get an essentially surjective functor but there could be more interesting $$\varphi$$-modules over $$\mathcal{Y}_{(0,\infty]}$$ by which you could attempt to glue. If $$k=O_C/\mathfrak{m}$$ and $$k^+=C^+/\mathfrak{m}$$ then the information lost is similar to the information that you lose by passing from a $$\varphi$$-module over $$W(k^+)$$ to a $$\varphi$$-module over $$W(k)$$, certain things that weren't isomorphic become isomorphic.

• Sorry I didn't post this earlier. Commented Apr 30, 2021 at 5:32