Chapter 5 - Crystals with coefficients

Summary

Crystals with coefficients

τ-sheaves and crystals with coefficients

In this chapter C will always denote the spectrum of a commutative F_q-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 F_q. A τ-sheaf on X with coefficients in A is a pair ℱ = (ℱ, τ) consisting of a quasi-coherent O_C×X-module ℱ and an O_C×X-linear map

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

or an additive map

τ_s : ℱ → ℱ

satisfying τ_s((a⊗r)s) = (a⊗r^q)τ_s(s) for all a ∈ A and all local sections r and s of O_X 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 F_q. We say that a τ-sheaf on X with coefficients in A is coherent if the underlying O_C×X-module is coherent. The category of such objects is denoted Coh_τ (X,A).

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

is the zero map for some n > 0, or equivalently, if τⁿ_s = 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 inCoh_τ (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 Coh_τ (X,A) is a thick subcategory. We define the category of A-crystals on X as the quotient category of Coh_τ (X,A) by the thick subcategory of nilpotent objects. It is denoted Crys(X,A).