The Stacks project

85.28 The structure sheaf

Let $X$ be a formal algebraic space. A structure sheaf for $X$ is a sheaf of topological rings $\mathcal{O}_ X$ on the étale site $X_{\acute{e}tale}$ (which we defined in Section 85.27) such that

\[ \mathcal{O}_ X(U_{red}) = \mathop{\mathrm{lim}}\nolimits \Gamma (U_\lambda , \mathcal{O}_{U_\lambda }) \]

as topological rings whenever

  1. $\varphi : U \to X$ is a morphism of formal algebraic spaces,

  2. $U$ is an affine formal algebraic space,

  3. $\varphi $ is representable by algebraic spaces and étale,

  4. $U_{red} \to X_{red}$ is the corresponding affine object of $X_{\acute{e}tale}$, see Lemma 85.27.7,

  5. $U = \mathop{\mathrm{colim}}\nolimits U_\lambda $ is a colimit representation for $U$ as in Definition 85.5.1.

Structure sheaves exist but may behave in unexpected manner.

Proof. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. By ( it suffices to construct $\mathcal{O}_ X$ as a sheaf of topological rings on $X_{affine, {\acute{e}tale}}$. Denote $\mathcal{C}$ the category whose objects are morphisms $\varphi : U \to X$ of formal algebraic spaces such that $U$ is an affine formal algebraic space and $\varphi $ is representable by algebraic spaces and étale. By Lemma 85.27.7 the functor $U \mapsto U_{red}$ is an equivalence of categories $\mathcal{C} \to X_{affine, {\acute{e}tale}}$. Hence by the rule given above the lemma, we already have $\mathcal{O}_ X$ as a presheaf of topological rings on $X_{affine, {\acute{e}tale}}$. Thus it suffices to check the sheaf condition.

By definition of $X_{affine, {\acute{e}tale}}$ a covering corresponds to a finite family $\{ g_ i : U_ i \to U\} _{i = 1, \ldots , n}$ of morphisms of $\mathcal{C}$ such that $\{ U_{i, red} \to U_{red}\} $ is an étale covering. The morphisms $g_ i$ are representably by algebraic spaces (Lemma 85.14.3) hence affine (Lemma 85.14.7). Then $g_ i$ is étale (follows formally from Properties of Spaces, Lemma 64.16.6 as $U_ i$ and $U$ are étale over $X$ in the sense of Bootstrap, Section 78.4). Finally, write $U = \mathop{\mathrm{colim}}\nolimits U_\lambda $ as in Definition 85.5.1.

With these preparations out of the way, we can prove the sheaf property as follows. For each $\lambda $ we set $U_{i, \lambda } = U_ i \times _ U U_\lambda $ and $U_{ij, \lambda } = (U_ i \times _ U U_ j) \times _ U U_\lambda $. By the above, these are affine schemes, $\{ U_{i, \lambda } \to U_\lambda \} $ is an étale covering, and $U_{ij, \lambda } = U_{i, \lambda } \times _{U_\lambda } U_{j, \lambda }$. Also we have $U_ i = \mathop{\mathrm{colim}}\nolimits U_{i, \lambda }$ and $U_ i \times _ U U_ j = \mathop{\mathrm{colim}}\nolimits U_{ij, \lambda }$. For each $\lambda $ we have an exact sequence

\[ 0 \to \Gamma (U_\lambda , \mathcal{O}_{U_\lambda }) \to \prod \nolimits _ i \Gamma (U_{i, \lambda }, \mathcal{O}_{U_{i, \lambda }}) \to \prod \nolimits _{i, j} \Gamma (U_{ij, \lambda }, \mathcal{O}_{U_{ij, \lambda }}) \]

as we have the sheaf condition for the structure sheaf on $U_\lambda $ and the étale topology (see Étale Cohomology, Proposition 58.17.1). Since limits commute with limits, the inverse limit of these exact sequences is an exact sequence

\[ 0 \to \mathop{\mathrm{lim}}\nolimits \Gamma (U_\lambda , \mathcal{O}_{U_\lambda }) \to \prod \nolimits _ i \mathop{\mathrm{lim}}\nolimits \Gamma (U_{i, \lambda }, \mathcal{O}_{U_{i, \lambda }}) \to \prod \nolimits _{i, j} \mathop{\mathrm{lim}}\nolimits \Gamma (U_{ij, \lambda }, \mathcal{O}_{U_{ij, \lambda }}) \]

which exactly means that

\[ 0 \to \mathcal{O}_ X(U_{red}) \to \prod \nolimits _ i \mathcal{O}_ X(U_{i, red}) \to \prod \nolimits _{i, j} \mathcal{O}_ X((U_ i \times _ U U_ j)_{red}) \]

is exact and hence the sheaf propery holds as desired. $\square$

Remark 85.28.2. The structure sheaf does not always have “enough sections”. In Examples, Section 108.67 we have seen that there exist affine formal algebraic spaces which aren't McQuillan and there are even examples whose points are not separated by regular functions.

In the next lemma we prove that the structure sheaf on a countably indexed affine formal scheme has vanishing higher cohomology. For non-countably indexed ones, presumably this generally doesn't hold.

Lemma 85.28.3. If $X$ is a countably indexed affine formal algebraic space, then we have $H^ n(X_{\acute{e}tale}, \mathcal{O}_ X) = 0$ for $n > 0$.

Proof. We may work with $X_{affine, {\acute{e}tale}}$ as this gives the same topos. We will apply Cohomology on Sites, Lemma 21.10.9 to show we have vanishing. Since $X_{affine, {\acute{e}tale}}$ has finite disjoint unions, this reduces us to the Čech complex of a covering given by a single arrow $\{ U_{red} \to V_{red}\} $ in $X_{affine, {\acute{e}tale}} = X_{red, affine, {\acute{e}tale}}$ (see Étale Cohomology, Lemma 58.22.1). Thus we have to show that

\[ 0 \to \mathcal{O}_ X(V_{red}) \to \mathcal{O}_ X(U_{red}) \to \mathcal{O}_ X(U_{red} \times _{V_{red}} U_{red}) \to \ldots \]

is exact. We will do this below in the case $V_{red} = X_{red}$. The general case is proven in exactly the same way.

Recall that $X = \text{Spf}(A)$ where $A$ is a weakly admissible topological ring having a countable fundamental system of weak ideals of definition. We have seen in Lemmas 85.27.4 and 85.27.5 that the object $U_{red}$ in $X_{affine, {\acute{e}tale}}$ corresponds to a morphism $U \to X$ of affine formal algebraic spaces which is representable by algebraic space and étale and $U = \text{Spf}(B^\wedge )$ where $B$ is an étale $A$-algebra. By our rule for the structure sheaf we see

\[ \mathcal{O}_ X(U_{red}) = B^\wedge \]

We recall that $B^\wedge = \mathop{\mathrm{lim}}\nolimits B/JB$ where the limit is over weak ideals of definition $J \subset A$. Working through the definitions we obtain

\[ \mathcal{O}_ X(U_{red} \times _{X_{red}} U_{red}) = (B \otimes _ A B)^\wedge \]

and so on. Since $U \to X$ is a covering the map $A \to B$ is faithfully flat, see Lemma 85.14.14. Hence the complex

\[ 0 \to A \to B \to B \otimes _ A B \to B \otimes _ A B \otimes _ A B \to \ldots \]

is universally exact, see Descent, Lemma 35.3.6. Our goal is to show that

\[ H^ n(0 \to A^\wedge \to B^\wedge \to (B \otimes _ A B)^\wedge \to (B \otimes _ A B \otimes _ A B)^\wedge \to \ldots ) \]

is zero for $p > 0$. To see what is going on, let's split our exact complex (before completion) into short exact sequences

\[ 0 \to A \to B \to M_1 \to 0,\quad 0 \to M_ i \to B^{\otimes _ A i + 1} \to M_{i + 1} \to 0 \]

By what we said above, these are universally exact short exact sequences. Hence $JM_ i = M_ i \cap J(B^{\otimes _ A i + 1})$ for every ideal $J$ of $A$. In particular, the topology on $M_ i$ as a submodule of $B^{\otimes _ A i + 1}$ is the same as the topology on $M_ i$ as a quotient module of $B^{\otimes _ A i}$. Therefore, since there exists a countable fundamental system of weak ideals of definition in $A$, the sequences

\[ 0 \to A^\wedge \to B^\wedge \to M_1^\wedge \to 0,\quad 0 \to M_ i^\wedge \to (B^{\otimes _ A i + 1})^\wedge \to M_{i + 1}^\wedge \to 0 \]

remain exact by Lemma 85.4.5. This proves the lemma. $\square$

Remark 85.28.4. Even if the structure sheaf has good properties, this does not mean there is a good theory of quasi-coherent modules. For example, in Examples, Section 108.12 we have seen that for almost any Noetherian affine formal algebraic spaces the most natural notion of a quasi-coherent module leads to a category of modules which is not abelian.

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