The Stacks project

85.32 Simplicial algebraic spaces

Let $S$ be a scheme. A simplicial algebraic space is a simplicial object in the category of algebraic spaces over $S$, see Simplicial, Definition 14.3.1. Recall that a simplicial algebraic space looks like

Here there are two morphisms $d^1_0, d^1_1 : X_1 \to X_0$ and a single morphism $s^0_0 : X_0 \to X_1$, etc. These morphisms satisfy some required relations such as $d^1_0 \circ s^0_0 = \text{id}_{X_0} = d^1_1 \circ s^0_0$, see Simplicial, Lemma 14.3.2. It is useful to think of $d^ n_ i : X_ n \to X_{n - 1}$ as the “projection forgetting the $i$th coordinate” and to think of $s^ n_ j : X_ n \to X_{n + 1}$ as the “diagonal map repeating the $j$th coordinate”.

A morphism of simplicial algebraic spaces $h : X \to Y$ is the same thing as a morphism of simplicial objects in the category of algebraic spaces over $S$, see Simplicial, Definition 14.3.1. Thus $h$ consists of morphisms of algebraic spaces $h_ n : X_ n \to Y_ n$ such that $h_{n - 1} \circ d^ n_ j = d^ n_ j \circ h_ n$ and $h_{n + 1} \circ s^ n_ j = s^ n_ j \circ h_ n$ whenever this makes sense.

An augmentation $a : X \to X_{-1}$ of a simplicial algebraic space $X$ is given by a morphism of algebraic spaces $a_0 : X_0 \to X_{-1}$ such that $a_0 \circ d^1_0 = a_0 \circ d^1_1$. See Simplicial, Section 14.20. In this situation we always indicate $a_ n : X_ n \to X_{-1}$ the induced morphisms for $n \geq 0$.

Let $X$ be a simplicial algebraic space. For every $n$ we have the site $X_{n, spaces, {\acute{e}tale}}$ (Properties of Spaces, Definition 66.18.2) and for every morphism $\varphi : [m] \to [n]$ we have a morphism of sites

associated to the morphism of algebraic spaces $X(\varphi ) : X_ n \to X_ m$ (Properties of Spaces, Lemma 66.18.8). This gives a simplicial object in the category of sites. In Lemma 85.3.1 we constructed an associated site which we denote $X_{spaces, {\acute{e}tale}}$. An object of the site $X_{spaces, {\acute{e}tale}}$ is a an algebraic space $U$ étale over $X_ n$ for some $n$ and a morphism $(\varphi , f) : U/X_ n \to V/X_ m$ is given by a morphism $\varphi : [m] \to [n]$ in $\Delta $ and a morphism $f : U \to V$ of algebraic spaces such that the diagram

is commutative. Consider the full subcategories

whose objects are $U/X_ n$ with $U$ affine, respectively a scheme. Endowing these categories with their natural topologies (see Properties of Spaces, Lemma 66.18.6, Definition 66.18.1, and Lemma 66.18.3) these inclusion functors define equivalences of topoi

In the following we will silently identify these topoi. We will say that $X_{\acute{e}tale}$ is the small étale site of $X$ and its topos is the small étale topos of $X$.

Let $X_{\acute{e}tale}$ be the small étale site of a simplicial algebraic space $X$. There is a sheaf of rings $\mathcal{O}$ on $X_{\acute{e}tale}$ whose restriction to $X_ n$ is the structure sheaf $\mathcal{O}_{X_ n}$. This follows from Lemma 85.3.4. We will say $\mathcal{O}$ is the structure sheaf of the simplicial algebraic space $X$. At this point all the material developed for simplicial (ringed) sites applies, see Sections 85.3, 85.4, 85.5, 85.6, 85.8, 85.9, 85.10, 85.11, 85.12, 85.13, and 85.14.

Let $X$ be a simplicial algebraic space with structure sheaf $\mathcal{O}$. As on any ringed topos, there is a notion of a quasi-coherent $\mathcal{O}$-module on $X_{\acute{e}tale}$, see Modules on Sites, Definition 18.23.1. However, a quasi-coherent $\mathcal{O}$-module on $X_{\acute{e}tale}$ is just a cartesian $\mathcal{O}$-module $\mathcal{F}$ whose restrictions $\mathcal{F}_ n$ are quasi-coherent on $X_ n$, see Lemma 85.12.10.

Let $h : X \to Y$ be a morphism of simplicial algebraic spaces over $S$. By Lemma 85.5.2 applied to the morphisms of sites $(h_ n)_{spaces, {\acute{e}tale}} : X_{spaces, {\acute{e}tale}} \to Y_{spaces, {\acute{e}tale}}$ (Properties of Spaces, Lemma 66.18.8) we obtain a morphism of small étale topoi $h_{\acute{e}tale}: \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$. Recall that $h_{\acute{e}tale}^{-1}$ and $h_{{\acute{e}tale}, *}$ have a simple description in terms of the components, see Lemma 85.5.2. Let $\mathcal{O}_ X$, resp. $\mathcal{O}_ Y$ denote the structure sheaf of $X$, resp. $Y$. We define $h_{\acute{e}tale}^\sharp : h_{{\acute{e}tale}, *}\mathcal{O}_ X \to \mathcal{O}_ Y$ to be the map of sheaves of rings on $Y_{\acute{e}tale}$ given by $h_ n^\sharp : h_{n, *}\mathcal{O}_{X_ n} \to \mathcal{O}_{Y_ n}$ on $Y_ n$. We obtain a morphism of ringed topoi

Let $X$ be a simplicial algebraic space with structure sheaf $\mathcal{O}$. Let $X_{-1}$ be an algebraic space over $S$ and let $a_0 : X_0 \to X_{-1}$ be an augmentation of $X$. By Lemma 85.4.2 applied to the morphism of sites $(a_0)_{spaces, {\acute{e}tale}} : X_{0, spaces, {\acute{e}tale}} \to X_{-1, spaces, {\acute{e}tale}}$ we obtain a corresponding morphism of topoi $a : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{-1, {\acute{e}tale}})$. Observe that $a^{-1}\mathcal{G}$ is the sheaf on $X_{\acute{e}tale}$ with components $a_ n^{-1}\mathcal{G}$. Hence we can use the maps $a_ n^\sharp : a_ n^{-1}\mathcal{O}_{X_{-1}} \to \mathcal{O}_{X_ n}$ to define a map $a^\sharp : a^{-1}\mathcal{O}_{X_{-1}} \to \mathcal{O}$, or equivalently by adjunction a map $a^\sharp : \mathcal{O}_{X_{-1}} \to a_*\mathcal{O}$ (which as usual has the same name). This puts us in the situation discussed in Section 85.11. Therefore we obtain a morphism of ringed topoi

A final observation is the following. Suppose we are given a morphism $h : X \to Y$ of simplicial algebraic spaces $X$ and $Y$ with structure sheaves $\mathcal{O}_ X$, $\mathcal{O}_ Y$, augmentations $a_0 : X_0 \to X_{-1}$, $b_0 : Y_0 \to Y_{-1}$ and a morphism $h_{-1} : X_{-1} \to Y_{-1}$ such that

commutes. Then from the constructions elucidated above we obtain a commutative diagram of morphisms of ringed topoi as follows