Definition 10.69.1. 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\[ \text{Bl}_ I(R) = \bigoplus \nolimits _{n \geq 0} I^ n = R \oplus I \oplus I^2 \oplus \ldots \]where 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 10.56.

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