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84.26 Simplicial schemes

A simplicial scheme is a simplicial object in the category of schemes, see Simplicial, Definition 14.3.1. Recall that a simplicial scheme 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 schemes $h : X \to Y$ is the same thing as a morphism of simplicial objects in the category of schemes, see Simplicial, Definition 14.3.1. Thus $h$ consists of morphisms of schemes $h_ n : X_ n \to Y_ n$ 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 of a simplicial scheme $X$ is a morphism of schemes $a_0 : X_0 \to S$ such that $a_0 \circ d^1_0 = a_0 \circ d^1_1$. See Simplicial, Section 14.20.

Let $X$ be a simplicial scheme. The construction of Section 84.2 applied to the underlying simplicial topological space gives a site $X_{Zar}$. On the other hand, for every $n$ we have the small Zariski site $X_{n, Zar}$ (Topologies, Definition 34.3.7) and for every morphism $\varphi : [m] \to [n]$ we have a morphism of sites $f_\varphi = X(\varphi )_{small} : X_{n, Zar} \to X_{m, Zar}$, associated to the morphism of schemes $X(\varphi ) : X_ n \to X_ m$ (Topologies, Lemma 34.3.17). This gives a simplicial object $\mathcal{C}$ in the category of sites. In Lemma 84.3.1 we constructed an associated site $\mathcal{C}_{total}$. Assigning to an open immersion its image defines an equivalence $\mathcal{C}_{total} \to X_{Zar}$ which identifies sheaves, i.e., $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}) = \mathop{\mathit{Sh}}\nolimits (X_{Zar})$. The difference between $\mathcal{C}_{total}$ and $X_{Zar}$ is similar to the difference between the small Zariski site $S_{Zar}$ and the underlying topological space of $S$. We will silently identify these sites in what follows.

Let $X_{Zar}$ be the site associated to a simplicial scheme $X$. There is a sheaf of rings $\mathcal{O}$ on $X_{Zar}$ whose restriction to $X_ n$ is the structure sheaf $\mathcal{O}_{X_ n}$. This follows from Lemma 84.2.2 or from Lemma 84.3.4. We will say $\mathcal{O}$ is the structure sheaf of the simplicial scheme $X$. At this point all the material developed for simplicial (ringed) sites applies, see Sections 84.3, 84.4, 84.5, 84.6, 84.8, 84.9, 84.10, 84.11, 84.12, 84.13, and 84.14.

Let $X$ be a simplicial scheme with structure sheaf $\mathcal{O}$. As on any ringed topos, there is a notion of a quasi-coherent $\mathcal{O}$-module on $X_{Zar}$, see Modules on Sites, Definition 18.23.1. However, a quasi-coherent $\mathcal{O}$-module on $X_{Zar}$ is just a cartesian $\mathcal{O}$-module $\mathcal{F}$ whose restrictions $\mathcal{F}_ n$ are quasi-coherent on $X_ n$, see Lemma 84.12.10.

Let $h : X \to Y$ be a morphism of simplicial schemes. Either by Lemma 84.2.3 or by (the proof of) Lemma 84.5.2 we obtain a morphism of sites $h_{Zar} : X_{Zar} \to Y_{Zar}$. Recall that $h_{Zar}^{-1}$ and $h_{Zar, *}$ have a simple description in terms of the components, see Lemma 84.2.4 or Lemma 84.5.2. Let $\mathcal{O}_ X$, resp. $\mathcal{O}_ Y$ denote the structure sheaf of $X$, resp. $Y$. We define $h_{Zar}^\sharp : h_{Zar, *}\mathcal{O}_ X \to \mathcal{O}_ Y$ to be the map of sheaves of rings on $Y_{Zar}$ 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 sites

\[ h_{Zar} : (X_{Zar}, \mathcal{O}_ X) \longrightarrow (Y_{Zar}, \mathcal{O}_ Y) \]

Let $X$ be a simplicial scheme with structure sheaf $\mathcal{O}$. Let $S$ be a scheme and let $a_0 : X_0 \to S$ be an augmentation of $X$. Either by Lemma 84.2.8 or by Lemma 84.4.2 we obtain a corresponding morphism of topoi $a : \mathop{\mathit{Sh}}\nolimits (X_{Zar}) \to \mathop{\mathit{Sh}}\nolimits (S)$. Observe that $a^{-1}\mathcal{G}$ is the sheaf on $X_{Zar}$ with components $a_ n^{-1}\mathcal{G}$. Hence we can use the maps $a_ n^\sharp : a_ n^{-1}\mathcal{O}_ S \to \mathcal{O}_{X_ n}$ to define a map $a^\sharp : a^{-1}\mathcal{O}_ S \to \mathcal{O}$, or equivalently by adjunction a map $a^\sharp : \mathcal{O}_ S \to a_*\mathcal{O}$ (which as usual has the same name). This puts us in the situation discussed in Section 84.11. Therefore we obtain a morphism of ringed topoi

\[ a : (\mathop{\mathit{Sh}}\nolimits (X_{Zar}), \mathcal{O}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (S), \mathcal{O}_ S) \]

A final observation is the following. Suppose we are given a morphism $h : X \to Y$ of simplicial schemes $X$ 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

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

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