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In Serre's GAGA he proves theorem 3: given M on $X^h$ there exists coherent algebraic sheaf $\mathscr{F}$ on X s.t. $\mathscr{F}^h$ is isomorphic to M , unique up to isomorphism.

He proves that it suffices to prove existence of $\mathscr{F}$ on X = $\mathbb{P}_r(\mathbb{C})$, where $\mathscr{F}$ is defined on Y $\subset$ X = $\mathbb{P}_r(\mathbb{C})$ on X.

He states in the proof: given x $\in$ X, let f $\in$ $\mathscr{J}(Y)$, let $\phi$ be multiplication by f on $\mathscr{G}_x$. Thus somehow $\phi^h$ of $\mathscr{G}_x^h$= $\mathscr{M}_x^h$ is the zero map, because $\mathscr{M}$ is a coherent analytic sheaf on $Y^h$??.

where $phi^h$ is defined in definition 2, page 11 in the English translation version of GAGA.

Why is $\phi^h$ the zero map?! Please help.

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    $\mathcal{M}_x$ is a $\mathcal{H}_x$/$\theta(\mathcal{I}_x)$ module in the terminology of the english translation of serre. Thus the multiplication is zero by $\theta(f)$. Hence the multiplication is zero by $f$. – 2017-03-06
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    where $\mathscr{H}(X)_x/\theta(\mathscr{J}(Y)_x)= \mathscr{H}_x(Y)$? – 2017-03-06
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    yes. That seems correct. – 2017-03-07

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