Lemma 87.35.3 (0DEL)—The Stacks project

Lemma 87.35.3. 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 59.22.1). 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 87.34.4 and 87.34.5 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 87.19.14. 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 $n > 0$. 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 87.4.5. This proves the lemma. $\square$

The Stacks project

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