The derived $\infty$-category of Cartier Modules

Abstract

For an endofunctor $F:\mathcal{C}\to \mathcal{C}$ on an ($\infty$-)category $\mathcal{C}$ we define the $\infty$-category $\mathrm{Cart}(\mathcal{C},F)$ of generalized Cartier modules as the lax equalizer of $F$ and the identity. This generalizes the notion of Cartier modules on ${\mathbb{F}}_p$-schemes considered in the literature. We show that in favorable cases $\mathrm{Cart}(\mathcal{C},F)$ is monadic over $\mathcal{C}$. If $\mathcal{A}$ is a Grothendieck abelian category and $F:\mathcal{A}\to \mathcal{A}$ is an exact and colimit-preserving endofunctor, we use this fact to construct an equivalence $\mathcal{D}(\mathrm{Cart}(\mathcal{A},F))\simeq \mathrm{Cart}(\mathcal{D}(\mathcal{A}),\mathcal{D}(F))$ of stable $\infty$-categories. We use this equivalence to construct a perverse t-structure on $\mathcal{D}(\mathrm{Cart}(\mathrm{Mod}(X),F_{\ast }))$ for any Noetherian ${\mathbb{F}}_p$-scheme $X$ with absolute Frobenius $F$. If $F$ is finite, this coincides with the perverse t-structure constructed by Baudin.

37 pages, final published version