The derived ∞-category of Cartier modules

Abstract

For an endofunctor F : 𝒞 → 𝒞 on an (∞-)category 𝒞 we define the ∞-category Cart(𝒞, F ) of generalized Cartier modules as the lax equalizer of F and the identity. This generalizes the notion of Cartier modules on Fp-schemes considered in [4]. We show that in favorable cases Cart(𝒞, F ) is monadic over 𝒞. If 𝒜 is a Grothendieck abelian category and F : 𝒜 → 𝒜 is an exact and colimit-preserving endofunctor, we use this fact to construct an equivalence 𝒟(Cart(𝒜, F )) ≃ Cart(𝒟(𝒜), 𝒟(F )) of stable ∞-categories. We use this equivalence to construct a perverse t-structure on 𝒟(Cart(Mod(X), F∗)) for any Noetherian Fp-scheme X with absolute Frobenius F . If F is finite, this coincides with the perverse t-structure constructed in [3].