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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