There is a satisfactory theory of B1-homotopy theory for rigid analytic spaces defined by Ayoub in the style of Voevodsky, and I'm aware of some work about the corresponding theory of motives, e.g. the very interesting papers of Vezzani (1, 2). However, I'm unable to find any reference analogous to Lecture notes on motivic cohomology for the resulting cohomological theory. In particular, are there similar descriptions for some low-dimensional motivic cohomology groups? Are there known/predicted vanishing results? Coniveau or Bloch-style complexes? What about computations of motivic cohomology groups for simple spaces like $\mathbb{G}_m$, projective space, maybe even Drinfeld's $p$-adic upper half space?

I guess probably a number of results carry over verbatim just due to the general formalism, but I'm not well-versed enough in the details of the constructions to understand which ones.

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    $\begingroup$ One interesting question is what kind of transfers should one expect (i.e. what's the replacement for Voevosky's category of correspondences). This is the fundamental geometric input you need to get the machinery going... $\endgroup$ Jul 12, 2021 at 7:46
  • $\begingroup$ According to page vii in Ayoub's article, he defines correspondances using the Suslin-Voevodsky approach of maps between between $fh$-sheaves; this is done in section 2.2. $\endgroup$
    – xir
    Jul 12, 2021 at 14:54

1 Answer 1


I think a very powerful tool for computations in the category of (rational, étale) rigid motives is the equivalence between "unipotent" motives and rigid motives "of good reduction" (see for example Theorem 2.5.57 in Ayoub's book, later expanded here (Section 5) or here (Section 3)) which reduces many rigid problems into algebraic statements for motives over the residue field.

In particular, over a discretely valued field $K$ with residue $k$ one has $$ \mathrm{Map}_{\mathrm{RigDA}(K)}(1(n),1)\cong\mathrm{Map}_{\mathrm{DA}(k)}(1(n),1\oplus 1(-1)[-1]). $$

Generalizations can be obtained using the most recent reference above. It is well-known (Remark 2.9.12) that using the (Nisnevich or étale) version with transfers doesn't alter the category of motives, under suitable hypotheses.


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