1
$\begingroup$

Both Vakil and Hartshorne describe Cohomology and Base Change in the following way:

Suppose $f:X \rightarrow Y$ is a projective (in Vakil, proper) morphism of Noetherian schemes, $F$ a coherent sheaf on $X$ flat over $Y.$ If the natural map $\phi^i(y): R^if_*(F) \rightarrow H^i(X_y,F_y)$ is surjective, then $\phi^{i-1}(y)$ is also surjective if and only if $R^if_*(F)$ is locally free near $y$.

An important special case would be when $i=0$. I'm nervous about the fact that neither author singles out this special case, and my silly question is: does the hypothesis "$\phi^{-1}(y)$ is surjective" hold vacuously when $i=0$? In other words, is surjectivity of $\phi^0(y)$ enough to conclude local freeness of $f_*F$ near $y$? Thanks.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.