The Stacks project

61.13 Weakly contractible objects

In this section we prove the key fact that our pro-étale sites contain many weakly contractible objects. In fact, the proof of Lemma 61.13.3 is the reason for the shape of the function $Bound$ in Definition 61.12.7 (although for readers who are ignoring set theoretical questions, this information is without content).

We first express the notion of w-contractible rings in terms of pro-étale coverings.

Lemma 61.13.1. Let $T = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. The following are equivalent

  1. $A$ is w-contractible, and

  2. every pro-étale covering of $T$ can be refined by a Zariski covering of the form $T = \coprod _{i = 1, \ldots , n} U_ i$.

Proof. Assume $A$ is w-contractible. By Lemma 61.12.5 it suffices to prove we can refine every standard pro-étale covering $\{ f_ i : T_ i \to T\} _{i = 1, \ldots , n}$ by a Zariski covering of $T$. The morphism $\coprod T_ i \to T$ is a surjective weakly étale morphism of affine schemes. Hence by Definition 61.11.1 there exists a morphism $\sigma : T \to \coprod T_ i$ over $T$. Then the Zariski covering $T = \coprod \sigma ^{-1}(T_ i)$ refines $\{ f_ i : T_ i \to T\} $.

Conversely, assume (2). If $A \to B$ is faithfully flat and weakly étale, then $\{ \mathop{\mathrm{Spec}}(B) \to T\} $ is a pro-étale covering. Hence there exists a Zariski covering $T = \coprod U_ i$ and morphisms $U_ i \to \mathop{\mathrm{Spec}}(B)$ over $T$. Since $T = \coprod U_ i$ we obtain $T \to \mathop{\mathrm{Spec}}(B)$, i.e., an $A$-algebra map $B \to A$. This means $A$ is w-contractible. $\square$

Lemma 61.13.2. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site as in Definition 61.12.7. Let $T = \mathop{\mathrm{Spec}}(A)$ be an affine object of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. The following are equivalent

  1. $A$ is w-contractible,

  2. $T$ is a weakly contractible (Sites, Definition 7.40.2) object of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, and

  3. every pro-étale covering of $T$ can be refined by a Zariski covering of the form $T = \coprod _{i = 1, \ldots , n} U_ i$.

Proof. We have seen the equivalence of (1) and (3) in Lemma 61.13.1.

Assume (3) and let $\mathcal{F} \to \mathcal{G}$ be a surjection of sheaves on $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. Let $s \in \mathcal{G}(T)$. To prove (2) we will show that $s$ is in the image of $\mathcal{F}(T) \to \mathcal{G}(T)$. We can find a covering $\{ T_ i \to T\} $ of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ such that $s$ lifts to a section of $\mathcal{F}$ over $T_ i$ (Sites, Definition 7.11.1). By (3) we may assume we have a finite covering $T = \coprod _{j = 1, \ldots , m} U_ j$ by open and closed subsets and we have $t_ j \in \mathcal{F}(U_ j)$ mapping to $s|_{U_ j}$. Since Zariski coverings are coverings in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ (Lemma 61.12.3) we conclude that $\mathcal{F}(T) = \prod \mathcal{F}(U_ j)$. Thus $t = (t_1, \ldots , t_ m) \in \mathcal{F}(T)$ is a section mapping to $s$.

Assume (2). Let $A \to D$ be as in Proposition 61.11.3. Then $\{ V \to T\} $ is a covering of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. (Note that $V = \mathop{\mathrm{Spec}}(D)$ is an object of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ by Remark 61.11.4 combined with our choice of the function $Bound$ in Definition 61.12.7 and the computation of the size of affine schemes in Sets, Lemma 3.9.5.) Since the topology on $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ is subcanonical (Lemma 61.12.22) we see that $h_ V \to h_ T$ is a surjective map of sheaves (Sites, Lemma 7.12.4). Since $T$ is assumed weakly contractible, we see that there is an element $f \in h_ V(T) = \mathop{\mathrm{Mor}}\nolimits (T, V)$ whose image in $h_ T(T)$ is $\text{id}_ T$. Thus $A \to D$ has a section $\sigma : D \to A$. Now if $A \to B$ is faithfully flat and weakly étale, then $D \to D \otimes _ A B$ has the same properties, hence there is a section $D \otimes _ A B \to D$ and combined with $\sigma $ we get a section $B \to D \otimes _ A B \to D \to A$ of $A \to B$. Thus $A$ is w-contractible and (1) holds. $\square$

Lemma 61.13.3. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site as in Definition 61.12.7. For every object $T$ of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ there exists a covering $\{ T_ i \to T\} $ in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ with each $T_ i$ affine and the spectrum of a w-contractible ring. In particular, $T_ i$ is weakly contractible in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$.

Proof. For those readers who do not care about set-theoretical issues this lemma is a trivial consequence of Lemma 61.13.2 and Proposition 61.11.3. Here are the details. Choose an affine open covering $T = \bigcup U_ i$. Write $U_ i = \mathop{\mathrm{Spec}}(A_ i)$. Choose faithfully flat, ind-étale ring maps $A_ i \to D_ i$ such that $D_ i$ is w-contractible as in Proposition 61.11.3. The family of morphisms $\{ \mathop{\mathrm{Spec}}(D_ i) \to T\} $ is a pro-étale covering. If we can show that $\mathop{\mathrm{Spec}}(D_ i)$ is isomorphic to an object, say $T_ i$, of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, then $\{ T_ i \to T\} $ will be combinatorially equivalent to a covering of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ by the construction of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ in Definition 61.12.7 and more precisely the application of Sets, Lemma 3.11.1 in the last step. To prove $\mathop{\mathrm{Spec}}(D_ i)$ is isomorphic to an object of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, it suffices to prove that $|D_ i| \leq Bound(\text{size}(T))$ by the construction of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ in Definition 61.12.7 and more precisely the application of Sets, Lemma 3.9.2 in step (3). Since $|A_ i| \leq \text{size}(U_ i) \leq \text{size}(T)$ by Sets, Lemmas 3.9.4 and 3.9.7 we get $|D_ i| \leq \kappa ^{2^{2^{2^\kappa }}}$ where $\kappa = \text{size}(T)$ by Remark 61.11.4. Thus by our choice of the function $Bound$ in Definition 61.12.7 we win. $\square$

Lemma 61.13.4. Let $S$ be a scheme. The pro-étale sites $S_{pro\text{-}\acute{e}tale}$, $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{affine, {pro\text{-}\acute{e}tale}}$, and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ and if $S$ is affine $S_{app}$ have enough (affine) quasi-compact, weakly contractible objects, see Sites, Definition 7.40.2.

Proof. Follows immediately from Lemma 61.13.3. $\square$

Lemma 61.13.5. Let $S$ be a scheme. The pro-étale sites $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, $S_{pro\text{-}\acute{e}tale}$, $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ have the following property: for any object $U$ there exists a covering $\{ V \to U\} $ with $V$ a weakly contractible object. If $U$ is quasi-compact, then we may choose $V$ affine and weakly contractible.

Proof. Suppose that $V = \coprod _{j \in J} V_ j$ is an object of $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ which is the disjoint union of weakly contractible objects $V_ j$. Since a disjoint union decomposition is a pro-étale covering we see that $\mathcal{F}(V) = \prod _{j \in J} \mathcal{F}(V_ j)$ for any pro-étale sheaf $\mathcal{F}$. Let $\mathcal{F} \to \mathcal{G}$ be a surjective map of sheaves of sets. Since $V_ j$ is weakly contractible, the map $\mathcal{F}(V_ j) \to \mathcal{G}(V_ j)$ is surjective, see Sites, Definition 7.40.2. Thus $\mathcal{F}(V) \to \mathcal{G}(V)$ is surjective as a product of surjective maps of sets and we conclude that $V$ is weakly contractible.

Choose a covering $\{ U_ i \to U\} _{i \in I}$ with $U_ i$ affine and weakly contractible as in Lemma 61.13.3. Take $V = \coprod _{i \in I} U_ i$ (there is a set theoretic issue here which we will address below). Then $\{ V \to U\} $ is the desired pro-étale covering by a weakly contractible object (to check it is a covering use Lemma 61.12.2). If $U$ is quasi-compact, then it follows immediately from Lemma 61.12.2 that we can choose a finite subset $I' \subset I$ such that $\{ U_ i \to U\} _{i \in I'}$ is still a covering and then $\{ \coprod _{i \in I'} U_ i \to U\} $ is the desired covering by an affine and weakly contractible object.

In this paragraph, which we urge the reader to skip, we address set theoretic problems. In order to know that the disjoint union lies in our partial universe, we need to bound the cardinality of the index set $I$. It is seen immediately from the construction of the covering $\{ U_ i \to U\} _{i \in I}$ in the proof of Lemma 61.13.3 that $|I| \leq \text{size}(U)$ where the size of a scheme is as defined in Sets, Section 3.9. Moreover, for each $i$ we have $\text{size}(U_ i) \leq Bound(\text{size}(U))$; this follows for the bound of the cardinality of $\Gamma (U_ i, \mathcal{O}_{U_ i})$ in the proof of Lemma 61.13.3 and Sets, Lemma 3.9.4. Thus $\text{size}(\coprod _{i \in I} U_ i)) \leq Bound(\text{size}(U))$ by Sets, Lemma 3.9.5. Hence by construction of the big pro-étale site through Sets, Lemma 3.9.2 we see that $\coprod _{i \in I} U_ i$ is isomorphic to an object of our site and the proof is complete. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F4N. Beware of the difference between the letter 'O' and the digit '0'.