Rees algebra sheaf of a finite type ideal

Assume the Axiom of Choice, inherited from the quasi-coherence suppliers used below (The Axiom of Choice).

Let $X$ be a scheme and let $\mathcal{I}\subseteq {\mathcal{O}}_X$ be a quasi-coherent ideal sheaf of finite type (Quasi-coherent ideal sheaves). Put ${\mathcal{I}}^0:={\mathcal{O}}_X$, and for $n\ge 1$ let ${\mathcal{I}}^n$ be the $n$-fold product of the ideal sheaf $\mathcal{I}$ inside ${\mathcal{O}}_X$, that is, the image of the multiplication map ${\mathcal{I}}^{\otimes n}\to {\mathcal{O}}_X$ (equivalently, the ideal sheaf generated by all local products of $n$ sections of $\mathcal{I}$). The Rees algebra sheaf of $\mathcal{I}$ is the sheaf of graded ${\mathcal{O}}_X$-algebras

$\mathcal{R}(\mathcal{I}):={⨁}_{n\ge 0}{\mathcal{I}}^n,$

whose degree-$n$ piece is the ideal sheaf ${\mathcal{I}}^n$, whose multiplication ${\mathcal{I}}^m{\otimes }_{{\mathcal{O}}_X}{\mathcal{I}}^n\to {\mathcal{I}}^{m+n}$ is induced by multiplication in ${\mathcal{O}}_X$, and whose unit is the identification ${\mathcal{O}}_X={\mathcal{I}}^0$. Since multiplication of ideals is associative and commutative and ${\mathcal{I}}^m{\mathcal{I}}^n\subseteq {\mathcal{I}}^{m+n}$ with equality for the product ideal, $\mathcal{R}(\mathcal{I})$ is a commutative graded ${\mathcal{O}}_X$-algebra with $\mathcal{R}(\mathcal{I}{)}_0={\mathcal{O}}_X$ and $\mathcal{R}(\mathcal{I}{)}_1=\mathcal{I}$.

The construction is local on $X$ and agrees with the affine Rees algebra: if $U=\mathrm{Spec}A\subseteq X$ is affine and $\mathcal{I}{\mathrm{\mid }}_U=\tilde{I}$ for an ideal $I\subseteq A$ (Quasi-coherent ideal sheaves), then ${\mathcal{I}}^n{\mathrm{\mid }}_U=\widetilde{I^n}$ for every $n\ge 0$, and taking sections on $U$ gives the affine Rees algebra ${⨁}_{n\ge 0}{I}^n{t}^n$ of (The Rees algebra of an ideal and the Rees module of a filtered module) with its degree-$n$ piece $I^n$. Two affine covers therefore glue to canonically isomorphic sheaves, and the graded pieces are the ideal powers defined above.

The following properties are part of the definition and are used by the consumers of this item.

1. The powers are quasi-coherent. Each ${\mathcal{I}}^n$ is a quasi-coherent ${\mathcal{O}}_X$-module (Quasi-coherent module on a scheme). Indeed, cover $X$ by affine opens $U=\mathrm{Spec}A$ with $\mathcal{I}{\mathrm{\mid }}_U=\tilde{I}$; then ${\mathcal{I}}^n{\mathrm{\mid }}_U=\widetilde{I^n}$ is the associated sheaf of an $A$-module, and quasi-coherence is local on $X$. Alternatively, for $n\ge 1$ the multiplication map ${\mathcal{I}}^{n-1}{\otimes }_{{\mathcal{O}}_X}\mathcal{I}\to {\mathcal{I}}^n$ is surjective from a quasi-coherent source by Tensor product preserves quasi-coherence.
2. The Rees algebra sheaf is quasi-coherent. On each affine chart $U=\mathrm{Spec}A$ one has $\mathcal{R}(\mathcal{I}){\mathrm{\mid }}_U=\widetilde{{⨁}_{n\ge 0}{I}^n}$, the associated sheaf of the graded $A$-module ${⨁}_{n\ge 0}{I}^n$; since quasi-coherence is local on $X$, the direct sum of the quasi-coherent sheaves ${\mathcal{I}}^n$ is quasi-coherent.
3. Degree-one generation. $\mathcal{R}(\mathcal{I}{)}_0={\mathcal{O}}_X$ and $\mathcal{R}(\mathcal{I}{)}_1=\mathcal{I}$ generate $\mathcal{R}(\mathcal{I})$ as an ${\mathcal{O}}_X$-algebra: for every $n\ge 1$ the product map ${\mathcal{I}}^{\otimes n}\to {\mathcal{I}}^n$ is surjective, because ${\mathcal{I}}^n$ is by construction generated by products of $n$ local sections of $\mathcal{I}$. Equivalently, the canonical graded ${\mathcal{O}}_X$-algebra homomorphism ${⨁}_{n\ge 0}{\mathrm{Sym}}_{{\mathcal{O}}_X}^n(\mathcal{I})\to \mathcal{R}(\mathcal{I})$ is surjective. The finite type hypothesis is used in subsequent finite-type structural results, not in the relative Proj construction; the Rees algebra sheaf and its relative Proj are defined for any quasi-coherent ideal.