The Stacks project

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

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

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 09XT. Beware of the difference between the letter 'O' and the digit '0'.