Lemma
Take \(X\) and \(Y\) to be smooth projective schemes over \(\Spec k\). All structure maps to a field are flat and flatness is preserved under base change, so everything in the following diagram is flat:
[diagram failed to compile]
latex failed:
e named `tikz@f@1-2-2' is known.
See the pgf package documentation for explanation.
Type H <return> for immediate help
...
l.5 I think the culprit is a tikzcd arrow in cell 1-2.
\errmessage ...currentrow -\tikzcd@currentcolumn }
l.7 \end{tikzcd}
! Package pgf Error: No shape named `tikz@f@1-2-2' is known.
See th
Then for an \(\mathcal O_Y\)-module \(\mathcal F\) we get a cohomological flat base change formula \[R^if_*g^*\mathcal F\simeq \mathcal O_X\otimes H^i(Y, \mathcal F)\] and a derived version: \[Rf_*Lg^*\mathcal F^\bullet \simeq \mathcal O_X \otimes^L_k H^\bullet (Y, \mathcal F^\bullet).\]