Crystals with coefficients

τ-sheaves and crystals with coefficients

In this chapter C will always denote the spectrum of a commutative Fq-algebra A (on which we will impose various conditions). The C stands for coefficients. We will be considering objects on which the algebra A acts linearly.

algebra A acts linearly. Let X be a scheme over Fq. A τ-sheaf on X with coefficients in A is a pair ℱ = (ℱ, τ) consisting of a quasi-coherent OC×X-module ℱ and an OC×X-linear map

Morphisms are defined in the obvious way. We denote the category of such objects by QCohτ (X,A). By adjunction, specifying τ is equivalent to either giving an OC×X-linear map

or an additive map

τs : ℱ → ℱ

satisfying τs((a⊗r)s) = (a⊗rq)τs(s) for all a ∈ A and all local sections r and s of OX and ℱ respectively.

Now assume that C×X is noetherian. This is the case, for example, if X is noetherian and A of finite type over Fq. We say that a τ-sheaf on X with coefficients in A is coherent if the underlying OC×X-module is coherent. The category of such objects is denoted Cohτ (X,A).

An object ℱ = (ℱ, τ) in Cohτ (X,A) is said to be nilpotent if

is the zero map for some n > 0, or equivalently, if τns = 0 for some n > 0.

Proposition 5.1. Assume that C × X is noetherian. Let

0→ ℱ1→ ℱ2→F3→ 0

be a short exact sequence inCohτ (X,A). Then ℱ2is nilpotent if and only if both ℱ1and ℱ3are nilpotent.

Proof. The proof is identical to that of Proposition 1.17.

In other words, the full subcategory of nilpotent objects of Cohτ (X,A) is a thick subcategory. We define the category of A-crystals on X as the quotient category of Cohτ (X,A) by the thick subcategory of nilpotent objects. It is denoted Crys(X,A).