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Definition 10.70.1. Let $R$ be a ring. Let $I \subset R$ be an ideal.

  1. 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 $n$.

  2. 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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Repeated word: "the the"

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