I.6: Sheafy rings

In the definition of adic spaces, one demands {\mathcal{O}_X} to be a sheaf. Hence it is neccessary to know for which {X} the presheaf {\mathcal{O}_X} is a sheaf. The first criterion is due to Huber (from “`A generalization of formal schemes and rigid-anaytic varieties”‘). Recall that a Huber ring is Tate if it contains a topologically nilpotent unit.

Proposition 23 Let {(A, A^+)} be a Huber pair. Then {\mathcal{O} _{\mathrm{Spa}(A,A^+)}} is a sheaf, and its higher cohomology vanishes on rational subdomains, if either of the following conditions holds:

  1. {\hat{A}} has a noetherian ring of definition.
  2. {A} is a strongly noetherian Tate ring (i.e. for all {n} the ring {A\langle t_1,...,t_n \rangle} are noetherian).

Huber uses this theorem to show that locally noetherian formal schemes give rise to adic spaces (point (1) ensures that) as well as rigid-analytic varieties (by point (2)). On the other hand Scholze proved that {\mathcal{O} _{\mathrm{Spa}(A,A^+)}} is a sheaf when {A} is perfectoid, in particular highly non-noetherian (we will discuss perfectoid spaces later on). Thus we would like to have a criterion for being sheafy which would apply to both these contexts: noetherian and non-noetherian. Before stating it, let us introduce two definitions. For what follows, we let {(A,A^+)} be a Huber pair with {A} Tate.

Definition 24 The pair {(A,A^+)} is uniform if the open subring {A^{\circ}} of power-bounded elements is bounded.

An example of non-uniform Tate ring is {\mathbb{Q}_p[\varepsilon]/(\varepsilon ^2)} with the {p}-adic topology. We have in this case {A^{\circ} = \mathbb{Z}_p \oplus \mathbb{Q}_p \cdot \varepsilon}, which is not bounded as it contains a {\mathbb{Q}_p}-line.

Definition 25 The pair {(A,A^+)} is stably uniform if for all rational subsets {U \subset \mathrm{Spa}(A,A^+)}, the ring {\mathcal{O}_{\mathrm{Spa}(A,A^+)}(U)} is unifom.

There are examples of pairs which are stably uniform, but not uniform. The main result of this section is the theorem of Buzzard and Verberkmoes:

Theorem 26 Let {(A,A^+)} be a stably uniform Huber pair with {A} Tate. Then {\mathcal{O}_{\mathrm{Spa}(A,A^+)}} is a sheaf and its higher sheaf cohomology vanishes on rational domains.

We have followed in this statement the seminar of Conrad (see L13). Observe that there is no finiteness assumption in the theorem which permits to apply it to perfectoid algebras. Let us now give a sketch of the proof of the sheaf property.

There are two steps. Let {X = \mathrm{Spa} (A, A^+)}. First, one verifies the sheaf property for the cover of the form {U = \{ v \in X \ : \ v(t) \geq 1\}} and {V = \{v \in X \ : \ v(t) \leq 1 \}}, where {t} is any element of {A}, only assuming that {(A,A^+)} is uniform. Second, one reduces the general case to this case, adding a stably uniform hypothesis.

In the first step we want to prove that

\displaystyle 0 \rightarrow \mathcal{O} _X(X) \rightarrow \mathcal{O} _X(U) \oplus \mathcal{O}_X(V) \rightarrow \mathcal{O}_X(U \cap V) \rightarrow 0

is exact. But before completion, this sequence is

\displaystyle 0 \rightarrow A \rightarrow ^f A(t/1) \oplus A(1/t) \rightarrow ^g A(t/1,1/t) \rightarrow 0

where {f} is the localisation map to both factors and {g} is the substraciton of the second factor from the first. It is not hard to establish the exactness of this sequence. In order to conclude, we want to know that when we complete, we still end up with an exact sequence. This is not automatic! In order to do so, we need to prove that both {f} and {g} are strict, i.e. a map of topological abelian groups {h : B \rightarrow C} is strict if the quotient topology on {h(B)} from {B} coincides with the subspace topology from {C}. It is equivalent to {h} being continuous and the induced map {B \rightarrow h(B)} being open. In our setting {g} is always strict, but in order to prove that {f} is strict we have to use the uniformity. We leave it to the reader.

The second step, deducing the full theorem, follows the reasoning of Tate. By induction one checks the sheaf property for Laurent covers i.e. {U_I}, where {I} is a subset of {\{1,...,n\}}, {t_1,...,t_n \in A} and

\displaystyle U_I = \{v \in X \ : \ v(t_i) \leq 1 \textrm{ for } i \in I, \ v(t_i) \geq 1 \textrm{ for } i \notin I \}

Then by using a refinement one passes to any rational domains and then to arbitrary covers by rational domains. See § 8.2 of [BGR].

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