Definition 10.70.1. Let R be a ring. Let I \subset R be an ideal.
The blowup algebra, or the Rees algebra, associated to the pair (R, I) is the graded R-algebra
\text{Bl}_ I(R) = \bigoplus \nolimits _{n \geq 0} I^ n = R \oplus I \oplus I^2 \oplus \ldotswhere the summand I^ n is placed in degree n.
Let a \in I be an element. Denote a^{(1)} the element a seen as an element of degree 1 in the Rees algebra. Then the affine blowup algebra R[\frac{I}{a}] is the algebra (\text{Bl}_ I(R))_{(a^{(1)})} constructed in Section 10.57.
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