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85.26 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 ith coordinate” and to think of s^ n_ j : X_ n \to X_{n + 1} as the “diagonal map repeating the jth 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 85.2 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 85.3.1 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 85.2.2 or from Lemma 85.3.4. 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 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 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 85.12.10.

Let h : X \to Y be a morphism of simplicial schemes. Either by Lemma 85.2.3 or by (the proof of) Lemma 85.5.2 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 85.2.4 or Lemma 85.5.2. 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 85.2.8 or by Lemma 85.4.2 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 85.11. 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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