Lemma 31.32.2 (0804)—The Stacks project

Lemma 31.32.2. Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine open subscheme of $X$ and let $I \subset A$ be the ideal corresponding to $\mathcal{I}|_ U$. If $b : X' \to X$ is the blowup of $X$ in $\mathcal{I}$, then there is a canonical isomorphism

\[ b^{-1}(U) = \text{Proj}(\bigoplus \nolimits _{d \geq 0} I^ d) \]

of $b^{-1}(U)$ with the homogeneous spectrum of the Rees algebra of $I$ in $A$. Moreover, $b^{-1}(U)$ has an affine open covering by spectra of the affine blowup algebras $A[\frac{I}{a}]$.

Proof. The first statement is clear from the construction of the relative Proj via glueing, see Constructions, Section 27.15. For $a \in I$ denote $a^{(1)}$ the element $a$ seen as an element of degree $1$ in the Rees algebra $\bigoplus _{n \geq 0} I^ n$. Since these elements generate the Rees algebra over $A$ we see that $\text{Proj}(\bigoplus _{d \geq 0} I^ d)$ is covered by the affine opens $D_{+}(a^{(1)})$. The affine scheme $D_{+}(a^{(1)})$ is the spectrum of the affine blowup algebra $A' = A[\frac{I}{a}]$, see Algebra, Definition 10.70.1. This finishes the proof. $\square$

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