Let $X$ be a normal projective variety over the field of complex numbers. Let $Y$ be a subvariety of $X$, and let $I_Y$ be the ideal sheaf of $Y$. From what I know, I can define the blow-up of $X$ along $Y$ as the projective spectrum
$$Bl_YX=\operatorname{Proj} (\mathcal{O}_X\oplus I_Y \oplus I_Y^2 \oplus \ldots),$$
which comes with a surjective morphism $\phi:Bl_YX\to X$ (isomorphism everywhere except on $Y$).
The question is the following: the surjective morphism $\phi$ comes as an arrow-reversing map at the level of finitely generated, $\mathbb{Z}$-graded algebras? Since $X$ is projective, I can write $X=\operatorname{Proj}(\bigoplus_{m\geq 0} H^0(X,\mathcal{O}_X(m)))$, but
$$\bigoplus_{m\geq 0} H^0(X,\mathcal{O}_X(m)) \hookrightarrow \bigoplus_{m\geq 0} I_Y^m$$
doesn't look as an inclusion of algebras. I apologize for the probably dumb question but I'm having an hard time understanding the Proj.
Best Answer
The map you want isn't quite what you've written down: for instance, if $X=\Bbb P^2$ and $Y$ is a point, the LHS is $k[x,y,z]$ while the RHS is just $k$. Instead, you want to apply $\Gamma_*(-)=\bigoplus_d \Gamma(-(d))$ to the inclusion of $\mathcal{O}_X$ in to $\bigoplus_m I_Y^m$. The resulting map is injective because $\mathcal{O}_X$ is a direct summand of $\bigoplus_m I_Y^m$.