39.21 Descent in terms of groupoids
Cartesian morphisms are defined as follows.
Definition 39.21.1. Let $S$ be a scheme. Let $f : (U', R', s', t', c') \to (U, R, s, t, c)$ be a morphism of groupoid schemes over $S$. We say $f$ is cartesian, or that $(U', R', s', t', c')$ is cartesian over $(U, R, s, t, c)$, if the diagram
\[ \xymatrix{ R' \ar[r]_ f \ar[d]_{s'} & R \ar[d]^ s \\ U' \ar[r]^ f & U } \]
is a fibre square in the category of schemes. A morphism of groupoid schemes cartesian over $(U, R, s, t, c)$ is a morphism of groupoid schemes compatible with the structure morphisms towards $(U, R, s, t, c)$.
Cartesian morphisms are related to descent data. First we prove a general lemma describing the category of cartesian groupoid schemes over a fixed groupoid scheme.
Lemma 39.21.2. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. The category of groupoid schemes cartesian over $(U, R, s, t, c)$ is equivalent to the category of pairs $(V, \varphi )$ where $V$ is a scheme over $U$ and
\[ \varphi : V \times _{U, t} R \longrightarrow R \times _{s, U} V \]
is an isomorphism over $R$ such that $e^*\varphi = \text{id}_ V$ and such that
\[ c^*\varphi = \text{pr}_1^*\varphi \circ \text{pr}_0^*\varphi \]
as morphisms of schemes over $R \times _{s, U, t} R$.
Proof.
The pullback notation in the lemma signifies base change. The displayed formula makes sense because
\[ (R \times _{s, U, t} R) \times _{\text{pr}_1, R, \text{pr}_1} (V \times _{U, t} R) = (R \times _{s, U, t} R) \times _{\text{pr}_0, R, \text{pr}_0} (R \times _{s, U} V) \]
as schemes over $R \times _{s, U, t} R$.
Given $(V, \varphi )$ we set $U' = V$ and $R' = V \times _{U, t} R$. We set $t' : R' \to U'$ equal to the projection $V \times _{U, t} R \to V$. We set $s'$ equal to $\varphi $ followed by the projection $R \times _{s, U} V \to V$. We set $c'$ equal to the composition
\begin{align*} R' \times _{s', U', t'} R' & \xrightarrow {\varphi , 1} (R \times _{s, U} V) \times _ V (V \times _{U, t} R) \\ & \xrightarrow {} R \times _{s, U} V \times _{U, t} R \\ & \xrightarrow {\varphi ^{-1}, 1} V \times _{U, t} (R \times _{s, U, t} R) \\ & \xrightarrow {1, c} V \times _{U, t} R = R' \end{align*}
A computation, which we omit shows that we obtain a groupoid scheme over $(U, R, s, t, c)$. It is clear that this groupoid scheme is cartesian over $(U, R, s, t, c)$.
Conversely, given $f : (U', R', s', t', c') \to (U, R, s, t, c)$ cartesian then the morphisms
\[ U' \times _{U, t} R \xleftarrow {t', f} R' \xrightarrow {f, s'} R \times _{s, U} U' \]
are isomorphisms and we can set $V = U'$ and $\varphi $ equal to the composition $(f, s') \circ (t', f)^{-1}$. We omit the proof that $\varphi $ satisfies the conditions in the lemma. We omit the proof that these constructions are mutually inverse.
$\square$
Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Then we obtain a groupoid scheme $(X, X \times _ Y X, \text{pr}_1, \text{pr}_0, c)$ over $S$. Namely, $j : X \times _ Y X \to X \times _ S X$ is an equivalence relation and we can take the associated groupoid, see Lemma 39.13.3.
Lemma 39.21.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. The construction of Lemma 39.21.2 determines an equivalence
\[ \begin{matrix} \text{category of groupoid schemes}
\\ \text{cartesian over } (X, X \times _ Y X, \ldots )
\end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data}
\\ \text{ relative to } X/Y
\end{matrix} \]
Proof.
This is clear from Lemma 39.21.2 and the definition of descent data on schemes in Descent, Definition 35.34.1.
$\square$
Comments (0)