Cartesian morphisms are defined as follows.
Definition 85.27.1. Let $a : Y \to X$ be a morphism of simplicial schemes. We say $a$ is cartesian, or that $Y$ is cartesian over $X$, if for every morphism $\varphi : [n] \to [m]$ of $\Delta $ the corresponding diagram is a fibre square in the category of schemes.
\[ \xymatrix{ Y_ m \ar[r]_ a \ar[d]_{Y(\varphi )} & X_ m \ar[d]^{X(\varphi )}\\ Y_ n \ar[r]^{a} & X_ n } \]
Cartesian morphisms are related to descent data. First we prove a general lemma describing the category of cartesian simplicial schemes over a fixed simplicial scheme. In this lemma we denote $f^* : \mathit{Sch}/X \to \mathit{Sch}/Y$ the base change functor associated to a morphism of schemes $f :Y \to X$.
Lemma 85.27.2. Let $X$ be a simplicial scheme. The category of simplicial schemes cartesian over $X$ is equivalent to the category of pairs $(V, \varphi )$ where $V$ is a scheme over $X_0$ and is an isomorphism over $X_1$ such that $(s_0^0)^*\varphi = \text{id}_ V$ and such that as morphisms of schemes over $X_2$.
\[ \varphi : V \times _{X_0, d^1_1} X_1 \longrightarrow X_1 \times _{d^1_0, X_0} V \]
\[ (d^2_1)^*\varphi = (d^2_0)^*\varphi \circ (d^2_2)^*\varphi \]
Proof. The statement of the displayed equality makes sense because $d^1_1 \circ d^2_2 = d^1_1 \circ d^2_1$, $d^1_1 \circ d^2_0 = d^1_0 \circ d^2_2$, and $d^1_0 \circ d^2_0 = d^1_0 \circ d^2_1$ as morphisms $X_2 \to X_0$, see Simplicial, Remark 14.3.3 hence we can picture these maps as follows
and the condition signifies the diagram is commutative. It is clear that given a simplicial scheme $Y$ cartesian over $X$ we can set $V = Y_0$ and $\varphi $ equal to the composition
of identifications given by the cartesian structure. To prove this functor is an equivalence we construct a quasi-inverse. The construction of the quasi-inverse is analogous to the construction discussed in Descent, Section 35.3 from which we borrow the notation $\tau ^ n_ i : [0] \to [n]$, $0 \mapsto i$ and $\tau ^ n_{ij} : [1] \to [n]$, $0 \mapsto i$, $1 \mapsto j$. Namely, given a pair $(V, \varphi )$ as in the lemma we set $Y_ n = X_ n \times _{X(\tau ^ n_ n), X_0} V$. Then given $\beta : [n] \to [m]$ we define $V(\beta ) : Y_ m \to Y_ n$ as the pullback by $X(\tau ^ m_{\beta (n)m})$ of the map $\varphi $ postcomposed by the projection $X_ m \times _{X(\beta ), X_ n} Y_ n \to Y_ n$. This makes sense because
and
We omit the verification that the commutativity of the displayed diagram above implies the maps compose correctly. We also omit the verification that the two functors are quasi-inverse to each other. $\square$
Definition 85.27.3. Let $f : X \to S$ be a morphism of schemes. The simplicial scheme associated to $f$, denoted $(X/S)_\bullet $, is the functor $\Delta ^{opp} \to \mathit{Sch}$, $[n] \mapsto X \times _ S \ldots \times _ S X$ described in Simplicial, Example 14.3.5.
Thus $(X/S)_ n$ is the $(n + 1)$-fold fibre product of $X$ over $S$. The morphism $d^1_0 : X \times _ S X \to X$ is the map $(x_0, x_1) \mapsto x_1$ and the morphism $d^1_1$ is the other projection. The morphism $s^0_0$ is the diagonal morphism $X \to X \times _ S X$.
Lemma 85.27.4. Let $f : X \to S$ be a morphism of schemes. Let $\pi : Y \to (X/S)_\bullet $ be a cartesian morphism of simplicial schemes. Set $V = Y_0$ considered as a scheme over $X$. The morphisms $d^1_0, d^1_1 : Y_1 \to Y_0$ and the morphism $\pi _1 : Y_1 \to X \times _ S X$ induce isomorphisms Denote $\varphi : V \times _ S X \to X \times _ S V$ the resulting isomorphism. Then the pair $(V, \varphi )$ is a descent datum relative to $X \to S$.
\[ \xymatrix{ V \times _ S X & & Y_1 \ar[ll]_-{(d^1_1, \text{pr}_1 \circ \pi _1)} \ar[rr]^-{(\text{pr}_0 \circ \pi _1, d^1_0)} & & X \times _ S V. } \]
Proof. This is a special case of (part of) Lemma 85.27.2 as the displayed equation of that lemma is equivalent to the cocycle condition of Descent, Definition 35.34.1. $\square$
Lemma 85.27.5. Let $f : X \to S$ be a morphism of schemes. The construction of Lemma 85.27.4 is an equivalence of categories.
\[ \begin{matrix} \text{category of cartesian }
\\ \text{schemes over } (X/S)_\bullet
\end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data}
\\ \text{ relative to } X/S
\end{matrix} \]
Proof. The functor from left to right is given in Lemma 85.27.4. Hence this is a special case of Lemma 85.27.2. $\square$
We may reinterpret the pullback of Descent, Lemma 35.34.6 as follows. Suppose given a morphism of simplicial schemes $f : X' \to X$ and a cartesian morphism of simplicial schemes $Y \to X$. Then the fibre product (viewed as a “pullback”)
of simplicial schemes is a simplicial scheme cartesian over $X'$. Suppose given a commutative diagram of morphisms of schemes
This gives rise to a morphism of simplicial schemes
We claim that the “pullback” $f_\bullet ^*$ along the morphism $f_\bullet : (X'/S')_\bullet \to (X/S)_\bullet $ corresponds via Lemma 85.27.5 with the pullback defined in terms of descent data in the aforementioned Descent, Lemma 35.34.6.
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