I am reading section III.12 in Hartshorne, the one about the Semicontinuity Theorem. For $f:X \rightarrow Y$, where $Y=\mathrm{Spec}A$ and $\mathcal{F}$ a coherent sheaf on $X$, he writes $H^i(X,\mathcal{F}\otimes_A M)$, where $M$ is an $A$-module.
This notation confuses me. Does he mean $H^i(X,\mathcal{F}\otimes_{\mathcal{O}_X}f^*\tilde{M}$)? It is the only way I can make sense of what we have there, since we want some sheaf on $X$ (we take cohomology there, and in general the cohomology of the pushforward on $Y$ is different). Also, $\mathcal{F}$ does not have a structure of $\mathcal{O}_Y$ module.
Is my interpretation correct? Otherwise, what is the right one?
Thank you!