Dieudonné theory via classifying stacks and prismatic $F$-gauges
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915624312111104 |
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| author | Mondal, Shubhodip |
| author_facet | Mondal, Shubhodip |
| contents | In this paper, we apply stack theoretic ideas to the classification problem in Dieudonné theory. First, we use crystalline cohomology of classifying stacks to directly reconstruct the classical Dieudonné module of a finite, $p$-power rank, commutative group scheme $G$ over a perfect field $k$ of characteristic $p>0$. As a consequence, we give a new, much shorter proof of the isomorphism $σ^* M(G) \simeq \mathrm{Ext}^1 (G, \mathcal{O}^{\mathrm{crys}})$ due to Berthelot--Breen--Messing using stacky methods combined with the theory of de Rham--Witt complexes. Additionally, we show that finite locally free commutative group schemes of $p$-power rank over a quasisyntomic base can be classified in terms of ``prismatic Dieudonné $F$-gauges", which we introduce by making constructions using (higher) classifying stacks. The latter generalizes the result of Anschütz and Le Bras on classification of $p$-divisible groups, which we also reprove using our approach. Along the way, we prove a description of cohomology with coefficients in group schemes, compatibility with Cartier duality, and reconstruction of Galois representations in terms of our prismatic Dieudonné $F$-gauges. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12967 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dieudonné theory via classifying stacks and prismatic $F$-gauges Mondal, Shubhodip Algebraic Geometry Number Theory In this paper, we apply stack theoretic ideas to the classification problem in Dieudonné theory. First, we use crystalline cohomology of classifying stacks to directly reconstruct the classical Dieudonné module of a finite, $p$-power rank, commutative group scheme $G$ over a perfect field $k$ of characteristic $p>0$. As a consequence, we give a new, much shorter proof of the isomorphism $σ^* M(G) \simeq \mathrm{Ext}^1 (G, \mathcal{O}^{\mathrm{crys}})$ due to Berthelot--Breen--Messing using stacky methods combined with the theory of de Rham--Witt complexes. Additionally, we show that finite locally free commutative group schemes of $p$-power rank over a quasisyntomic base can be classified in terms of ``prismatic Dieudonné $F$-gauges", which we introduce by making constructions using (higher) classifying stacks. The latter generalizes the result of Anschütz and Le Bras on classification of $p$-divisible groups, which we also reprove using our approach. Along the way, we prove a description of cohomology with coefficients in group schemes, compatibility with Cartier duality, and reconstruction of Galois representations in terms of our prismatic Dieudonné $F$-gauges. |
| title | Dieudonné theory via classifying stacks and prismatic $F$-gauges |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2405.12967 |