Formal smoothness of the Artin-Mazur formal groups
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908574572085248 |
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| author | Grammatica, Livia |
| author_facet | Grammatica, Livia |
| contents | Let $X$ be a smooth proper variety over an algebraically closed field of positive characteristic $p$. We find cohomological conditions for the Artin-Mazur formal group functors $Φ^{i}(X,\mathbb{G}_m)$ to be formally smooth. We show that if all crystalline cohomology groups of $X$ are torsion-free (e.g. if $X$ is an abelian variety) then all of the $Φ^{i}(X,\mathbb{G}_m)$ are representable and formally smooth. We then identify a necessary condition for formal smoothness, which we use to give examples, for any $d\ge2$, of varieties $X$ for which $Φ^{i}(X,\mathbb{G}_m)$ is formally smooth when $i<d$, whereas $Φ^{d}(X,\mathbb{G}_m)$ is not. The constructions are inspired by Igusa's surface with non-smooth Picard scheme. Finally, we give a condition equivalent to formal smoothness in terms of Serre's Witt vector cohomology. The strategy relies on the notion of $C$-smoothness - where $C$ is the group algebra of $\mathbb{Q}_p/\mathbb{Z}_p$ - which is a condition that detects when a formal group is formally smooth, and on the use of the Nygaard filtration to relate fppf cohomology to crystalline cohomology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_03001 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Formal smoothness of the Artin-Mazur formal groups Grammatica, Livia Algebraic Geometry 14F30, 14L05 Let $X$ be a smooth proper variety over an algebraically closed field of positive characteristic $p$. We find cohomological conditions for the Artin-Mazur formal group functors $Φ^{i}(X,\mathbb{G}_m)$ to be formally smooth. We show that if all crystalline cohomology groups of $X$ are torsion-free (e.g. if $X$ is an abelian variety) then all of the $Φ^{i}(X,\mathbb{G}_m)$ are representable and formally smooth. We then identify a necessary condition for formal smoothness, which we use to give examples, for any $d\ge2$, of varieties $X$ for which $Φ^{i}(X,\mathbb{G}_m)$ is formally smooth when $i<d$, whereas $Φ^{d}(X,\mathbb{G}_m)$ is not. The constructions are inspired by Igusa's surface with non-smooth Picard scheme. Finally, we give a condition equivalent to formal smoothness in terms of Serre's Witt vector cohomology. The strategy relies on the notion of $C$-smoothness - where $C$ is the group algebra of $\mathbb{Q}_p/\mathbb{Z}_p$ - which is a condition that detects when a formal group is formally smooth, and on the use of the Nygaard filtration to relate fppf cohomology to crystalline cohomology. |
| title | Formal smoothness of the Artin-Mazur formal groups |
| topic | Algebraic Geometry 14F30, 14L05 |
| url | https://arxiv.org/abs/2510.03001 |