Morphisms of schemes

 Conference topic

This is from discussions at the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Zijian Yao for explaining the material.

Consider a morphism of schemes $\varphi:S'\to S$ and coherent sheaves $\mathcal F,\mathcal G$ over $S$. Consider also a map of sheaves $f:\mathcal F\to \mathcal G$ and a map $f'$ between the pullbacks of $\mathcal F$ and $\mathcal G$, as described by the diagram below.

There are two natural questions to ask.

1. When is $f' = \varphi^*f$?
2. If we start with $\mathcal G'$ over $S'$, when is $\mathcal G' = \varphi^*\mathcal G$?

To answer these questions, consider fiber products of schemes and projections from them, as given below.

Remark: If 1. is true, then $p_1^*(f') = p_2^*(f')$. If the previous statement is an equivalence, then $\varphi$ is a morphism of descent.

Remark: If 2. is true, then there exists $\alpha:p_1^*(\mathcal G') \to p_2^*(\mathcal G')$ such that $\pi_{32}^*(\alpha)\pi_{21}^*(\alpha) = \pi_{31}^*(\alpha)$ and $\pi^*(\Delta) = \alpha$. If the previous statement is an equivalence, then $\varphi$ is effective.