The Stacks project

38.35 More on the h topology

In this section we prove a few more results on the h topology. First, some non-examples.

Example 38.35.1. The “structure sheaf” $\mathcal{O}$ is not a sheaf in the h topology. For example, consider a surjective closed immersion of finite presentation $X \to Y$. Then $\{ X \to Y\} $ is an h covering for example by Lemma 38.34.6. Moreover, note that $X \times _ Y X = X$. Thus if $\mathcal{O}$ where a sheaf in the h topology, then $\mathcal{O}_ Y(Y) \to \mathcal{O}_ X(X)$ would be bijective. This is not the case as soon as $X$, $Y$ are affine and the morphism $X \to Y$ is not an isomorphism.

Example 38.35.2. On any of the sites $(\mathit{Sch}/S)_ h$ the topology is not subcanonical, in other words, representable sheaves are not sheaves. Namely, the “structure sheaf” $\mathcal{O}$ is representable because $\mathcal{O}(X) = \mathop{\mathrm{Mor}}\nolimits _ S(X, \mathbf{A}^1_ S)$ in $(\mathit{Sch}/S)_ h$ and we saw in Example 38.35.1 that $\mathcal{O}$ is not a sheaf.

Lemma 38.35.3. Let $T$ be an affine scheme which is written as a limit $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ of a directed inverse system of affine schemes.

  1. Let $\mathcal{V} = \{ V_ j \to T\} _{j = 1, \ldots , m}$ be a standard h covering of $T$, see Definition 38.34.10. Then there exists an index $i$ and a standard h covering $\mathcal{V}_ i = \{ V_{i, j} \to T_ i\} _{j = 1, \ldots , m}$ whose base change $T \times _{T_ i} \mathcal{V}_ i$ to $T$ is isomorphic to $\mathcal{V}$.

  2. Let $\mathcal{V}_ i$, $\mathcal{V}'_ i$ be a pair of standard h coverings of $T_ i$. If $f : T \times _{T_ i} \mathcal{V}_ i \to T \times _{T_ i} \mathcal{V}'_ i$ is a morphism of coverings of $T$, then there exists an index $i' \geq i$ and a morphism $f_{i'} : T_{i'} \times _{T_ i} \mathcal{V} \to T_{i'} \times _{T_ i} \mathcal{V}'_ i$ whose base change to $T$ is $f$.

  3. If $f, g : \mathcal{V} \to \mathcal{V}'_ i$ are morphisms of standard h coverings of $T_ i$ whose base changes $f_ T, g_ T$ to $T$ are equal then there exists an index $i' \geq i$ such that $f_{T_{i'}} = g_{T_{i'}}$.

In other words, the category of standard h coverings of $T$ is the colimit over $I$ of the categories of standard h coverings of $T_ i$.

Proof. By Limits, Lemma 32.10.1 the category of schemes of finite presentation over $T$ is the colimit over $I$ of the categories of finite presentation over $T_ i$. By Limits, Lemma 32.8.2 the same is true for category of schemes which are affine and of finite presentation over $T$. To finish the proof of the lemma it suffices to show that if $\{ V_{j, i} \to T_ i\} _{j = 1, \ldots , m}$ is a finite family of finitely presented morphisms with $V_{j, i}$ affine, and the base change family $\{ T \times _{T_ i} V_{j, i} \to T\} $ is an h covering, then for some $i' \geq i$ the family $\{ T_{i'} \times _{T_ i} V_{j, i} \to T_{i'}\} $ is an h covering. To see this we use Lemma 38.34.4 to choose a finitely presented, proper, surjective morphism $Y \to T$ and a finite affine open covering $Y = \bigcup _{k = 1, \ldots , n} Y_ k$ such that $\{ Y_ k \to T\} _{k = 1, \ldots , n}$ refines $\{ T \times _{T_ i} V_{j, i} \to T\} $. Using the arguments above and Limits, Lemmas 32.13.1, 32.8.14, and 32.4.11 we can find an $i' \geq i$ and a finitely presented, surjective, proper morphism $Y_{i'} \to T_{i'}$ and an affine open covering $Y_{i'} = \bigcup _{k = 1, \ldots , n} Y_{i', k}$ such that moreover $\{ Y_{i', k} \to Y_{i'}\} $ refines $\{ T_{i'} \times _{T_ i} V_{j, i} \to T_{i'}\} $. It follows that this last mentioned family is a h covering and the proof is complete. $\square$

Lemma 38.35.4. Let $S$ be a scheme contained in a big site $\mathit{Sch}_ h$. Let $F : (\mathit{Sch}/S)_ h^{opp} \to \textit{Sets}$ be an h sheaf satisfying property (b) of Topologies, Lemma 34.13.1 with $\mathcal{C} = (\mathit{Sch}/S)_ h$. Then the extension $F'$ of $F$ to the category of all schemes over $S$ satisfies the sheaf condition for all h coverings and is limit preserving (Limits, Remark 32.6.2).

Proof. This is proven by the arguments given in the proofs of Topologies, Lemmas 34.13.3 and 34.13.4 using Lemmas 38.35.3 and 38.34.11. Details omitted. $\square$


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