The Stacks project

Lemma 69.17.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme ├ętale over $X$ and let $I \subset A$ be the ideal corresponding to $\mathcal{I}|_ U$. If $X' \to X$ is the blowup of $X$ in $\mathcal{I}$, then there is a canonical isomorphism

\[ U \times _ X X' = \text{Proj}(\bigoplus \nolimits _{d \geq 0} I^ d) \]

of schemes over $U$, where the right hand side is the homogeneous spectrum of the Rees algebra of $I$ in $A$. Moreover, $U \times _ X X'$ has an affine open covering by spectra of the affine blowup algebras $A[\frac{I}{a}]$.

Proof. Note that the restriction $\mathcal{I}|_ U$ is equal to the pullback of $\mathcal{I}$ via the morphism $U \to X$, see Properties of Spaces, Section 64.26. Thus the lemma follows on combining Lemma 69.11.2 with Divisors, Lemma 31.32.2. $\square$


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