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.

## 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 38.35.2. 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.

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 $\mathcal{V}$.

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 $f$.

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, 32.8.15, 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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