A Conjecture of Bhatt--Lurie and weakly $p$-nilpotent Hodge--Tate stacks

Let $k$ be a perfect field of characteristic $p$, and let $X\mathrm{/}k$ 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 ${\text{W}}_2(k)$, 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 $T_{X\mathrm{/}k}\otimes {\alpha }_p$, 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.