Remark 99.14.5. Let $B$ be an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Let $B\textit{-Polarized}$ be the category consisting of triples $(X \to S, \mathcal{L}, h : S \to B)$ where $(X \to S, \mathcal{L})$ is an object of $\mathcal{P}\! \mathit{olarized}$ and $h : S \to B$ is a morphism. A morphism $(X' \to S', \mathcal{L}', h') \to (X \to S, \mathcal{L}, h)$ in $B\textit{-Polarized}$ is a morphism $(f, g, \varphi )$ in $\mathcal{P}\! \mathit{olarized}$ such that $h \circ g = h'$. In this situation the diagram
\[ \xymatrix{ B\textit{-Polarized} \ar[r] \ar[d] & \mathcal{P}\! \mathit{olarized}\ar[d] \\ (\mathit{Sch}/B)_{fppf} \ar[r] & \mathit{Sch}_{fppf} } \]
is $2$-fibre product square. This trivial remark will occasionally be useful to deduce results from the absolute case $\mathcal{P}\! \mathit{olarized}$ to the case of families over a given base algebraic space.
Comments (0)