Theorem of Formal Functions Hartshorne

Question

I have following question in the proof of thm in Hartshoren's AG (II, page 277):

Since $X$ is projective then of cource we have an embedding $i:X \to P^N _Y$. But what about the coherent sheaf $\mathcal{F}$ on $X$? How do we consider $\mathcal{F}$ as a sheaf on $P^N _Y$?

Via $i_*\mathcal{F}$ or $i_!\mathcal{F}$?

Answer 1

Via $i_* \mathcal{F}$.

The terminology is explained at page 68, see Exercise 1.19 (a).

Answer 2

Hartshorne means $i_* \mathcal F$, also because $j_!$ is defined for the inclusion $j\colon U \rightarrow X$ of an open subset $U$ (see Ex. II.1.19).

Indeed, you want to reduce to the case $X = \mathbf P^N_Y$. Suppose the theorem holds in this case. Then, for a general projective morphism $X \rightarrow Y$, consider - as you say - a closed immersion $i\colon X\rightarrow \mathbf P^N_Y$. A closed immersion is a finite morphism (Ex. II.5.5), so in particular it is an affine morphism. Now use Ex. III.8.2 to conclude that $H^i(X,\mathcal F) \cong H^i(\mathbf P^N_Y, i_*\mathcal F)$.