On this page on StacksProject about coherent sheaves on projective spaces they define a hypersurface to be the zero scheme $Z(s)$ of a global section $ s \in \Gamma(\mathbb{P}_k^n,\mathcal{O}(d))$, which is defined here. I know that those global sections can be identified with homogeneous polynomials of degree $d$ in $k[X_0, \ldots, X_n]$, so let $f_d$ be the homogeneous polynomial corresponding to $s$. My question is, does this zero scheme coincide with the closed subscheme $V_{+}(f_d)=Proj(k[X_0, \ldots,X_n]/(f_d)) $? This would make sense since this last scheme is sometimes called "zero locus", but I am not able to show this from the abstract definition of $Z(s)$ in stacks project.
I know that $Z(s)$ is, by definition, a closed subscheme of $\mathbb{P}_k^n$, so it must be of the form $Proj(k[X_0, \ldots, X_n]/I)$ for some homogeneous ideal $I$, but I do not know how to show $I=(f_d)$.
EDIT: In fact, the case $d=1$ is showed here.
Best Answer
I think can be seen by just using the standard cover of projective space: if $\text{Spec}\left(k\left[\frac{X_0}{X_i},\ldots,\frac{X_n}{X_i}\right]\right) \cong U_i = D(X_i) \subset \mathbf{P}^n_k$ then $$ U_i \cap V(s) = V(s_{\mid D(X_i)}) \cong \text{Spec}\left(k\left[\frac{X_0}{X_i},\ldots,\frac{X_n}{X_i}\right]/f_d\left(\frac{X_0}{X_i},\ldots,\frac{X_n}{X_i}\right)\right) $$ which agrees with the standard affine cover of $\text{Proj}(k[X_0,\ldots,X_n]/f_d)$, and the gluing isomorphisms on the intersections of course also agrees.
It wish I could say this follows by some abstract principle but, disenchantingly, the Proj construction really fails at functoriality...