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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2410.06630 |
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| _version_ | 1866911135204114432 |
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| author | Yu, Jiahong |
| author_facet | Yu, Jiahong |
| contents | Let $k$ be a perfect field of characteristic $p$, and let $X/k$ be a smooth variety. It is known that given a Frobenius lifting of $X$, we can identify prismatic crystals and nilpotent Higgs bundles, known as a positive characteristic version of the Simpson correspondence of $X$. However, Ogus--Vologodsky point out in their original paper of non-abelian Hodge theory in characteristic $p$ that, if we are just given a smooth lifting over $\W_2(k)$, there is a non-abelian Hodge theory on $p$-nilpotent Higgs bundles. Hence, it is natural to ask that whether there exists a subcategory of Hodge--Tate crystals on $X$, which can be described as $p$-nilpotent Higgs bundles. In this paper, we construct an analogue of the Hodge--Tate stack, so called the weakly $p$-nilpotent Hodge--Tate stack, on which the vector bundles are identified with certain Hodge--Tate crystals on $X$ that can be locally described by weakly $p$-nilpotent Higgs bundles. Furthermore, we prove that the weakly $p$-nilpotent Hodge--Tate stack is indeed a gerbe banded by $T_{X/k}\otimes{α_p}$, and the obstruction class coincides with the obstruction of the existence of a Frobenius lifting of $X$, which is a conjecture of Bhatt and Lurie. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06630 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Conjecture of Bhatt--Lurie and weakly $p$-nilpotent Hodge--Tate stacks Yu, Jiahong Algebraic Geometry Let $k$ be a perfect field of characteristic $p$, and let $X/k$ be a smooth variety. It is known that given a Frobenius lifting of $X$, we can identify prismatic crystals and nilpotent Higgs bundles, known as a positive characteristic version of the Simpson correspondence of $X$. However, Ogus--Vologodsky point out in their original paper of non-abelian Hodge theory in characteristic $p$ that, if we are just given a smooth lifting over $\W_2(k)$, there is a non-abelian Hodge theory on $p$-nilpotent Higgs bundles. Hence, it is natural to ask that whether there exists a subcategory of Hodge--Tate crystals on $X$, which can be described as $p$-nilpotent Higgs bundles. In this paper, we construct an analogue of the Hodge--Tate stack, so called the weakly $p$-nilpotent Hodge--Tate stack, on which the vector bundles are identified with certain Hodge--Tate crystals on $X$ that can be locally described by weakly $p$-nilpotent Higgs bundles. Furthermore, we prove that the weakly $p$-nilpotent Hodge--Tate stack is indeed a gerbe banded by $T_{X/k}\otimes{α_p}$, and the obstruction class coincides with the obstruction of the existence of a Frobenius lifting of $X$, which is a conjecture of Bhatt and Lurie. |
| title | A Conjecture of Bhatt--Lurie and weakly $p$-nilpotent Hodge--Tate stacks |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2410.06630 |