A Tannakian classification of torsors on the projective line

In this small note we present a Tannakian proof of the theorem of Grothendieck-Harder on the classification of torsors under a reductive group on the projective line over a field.Comment: 13 pages; any comments or hints to existing literature welcom

Reciprocity laws for smooth projective schemes

We prove that two natural isomorphisms between the first mod m Suslin homology and the mod m abelianized etale fundamental group agree for connected smooth projective schemes over algebraically closed fields

Prismatic Dieudonn\'e theory

We define, for each quasi-syntomic ring $R$ (in the sense of Bhatt-Morrow-Scholze), a category $\mathrm{DF}(R)$ of \textit{filtered prismatic Dieudonn\'e crystals over $R$} and a natural functor from $p$-divisible groups over $R$ to $\mathrm{DF}(R)$. We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze.Comment: v2: fixed a gap in Section 3 and removed an unnecessary hypothesis in the statement of the main theore

Hodge-Tate stacks and non-abelian pp-adic Hodge theory of v-perfect complexes on rigid spaces

Let $X$ be a quasi-compact quasi-separated $p$-adic formal scheme that is smooth either over a perfectoid $\mathbb{Z}_p$-algebra or over some ring of integers of a complete discretely valued extension of $\mathbb{Q}_p$ with $p$-finite residue field. We construct a fully faithful functor from perfect complexes on the Hodge-Tate stack of $X$ up to isogeny to perfect complexes on the v-site of the generic fibre of $X$. Moreover, we describe perfect complexes on the Hodge-Tate stack in terms of certain derived categories of Higgs, resp. Higgs-Sen modules. This leads to a derived $p$-adic Simpson functor. We deduce new results about the $p$-adic Simpson correspondence in both cases

Prismatic Dieudonné theory

We define, for each quasi-syntomic ring R (in the sense of Bhatt-Morrow-Scholze), a category DF(R) of filtered prismatic Dieudonné crystals over R and a functor from p-divisible groups over R to DF(R). We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze

The p-completed cyclotomic trace in degree 2

International audienceWe prove that for a quasi-regular semiperfectoid Z cycl p-algebra R (in the sense of Bhatt-Morrow-Scholze), the cyclotomic trace map from the p-completed K-theory spectrum K(R; Zp) of R to the topological cyclic homology TC(R; Zp) of R identifies on π 2 with a q-deformation of the logarithm