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

$\xymatrix{ X_2 \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & X_1 \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & X_0 \ar@<0ex>[l] }$

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

$f_\varphi = X(\varphi )_{spaces, {\acute{e}tale}} : X_{n, spaces, {\acute{e}tale}} \to X_{m, spaces, {\acute{e}tale}},$

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

$\xymatrix{ U \ar[r]_ f \ar[d] & V \ar[d] \\ X_ n \ar[r]^{f_\varphi } & X_ m }$

is commutative. Consider the full subcategories

$X_{affine, {\acute{e}tale}} \subset X_{\acute{e}tale}\subset X_{spaces, {\acute{e}tale}}$

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

$\mathop{\mathit{Sh}}\nolimits (X_{affine, {\acute{e}tale}}) = \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) = \mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}})$

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

$h_{\acute{e}tale}: (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y)$

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

$a : (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (X_{-1}), \mathcal{O}_{X_{-1}})$

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

$\xymatrix{ X_0 \ar[r]_{h_0} \ar[d]_{a_0} & Y_0 \ar[d]^{b_0} \\ X_{-1} \ar[r]^{h_{-1}} & Y_{-1} }$

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

$\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \ar[r]_{h_{\acute{e}tale}} \ar[d]_ a & (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y) \ar[d]^ b \\ (\mathop{\mathit{Sh}}\nolimits (X_{-1}), \mathcal{O}_{X_{-1}}) \ar[r]^{h_{-1}} & (\mathop{\mathit{Sh}}\nolimits (Y_{-1}), \mathcal{O}_{Y_{-1}}) }$

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