Let $X \to S$ is be a morphism of schemes, and $F$ an $\mathcal O_X$-modules.
Let $P=\mathcal O_X\otimes_{f^{-1}(\mathcal O_S)}\mathcal O_X$
I don't understand the following two structures of $\mathcal O_X$-Algebras, namely how we get $P \otimes F$ from $d_1$ and the $\mathcal O_X$-module structure with $d_0$
I can see the usage of $d_0$ in two ways:
- $d_1$ would map $\mathcal O_X$ to $P$ but in this case $P$ remains only partially an $\mathscr O_X$-module, it is exactly an $f^{-1}(\mathcal O_S)$-module, so we didn't form the tensor product over $\mathcal O_X$
- $d_1: a\to 1\otimes a$ would send $a\in \mathcal O_X$ to $F$ so it doesn't make sense.
The first way makes more sense to see $P$ as an $\mathcal O_X$- module and hence being able to for the tensor product over $\mathcal O_X$ but I don't see how
Thank you for your lights.