Blowup algebra

Definition

Let $R$ be a commutative unital ring and $I$ be an ideal in $R$ . The blowup algebra of $I$ in $R$ is defined as:

$B_{I} R : = R \oplus I \oplus I^{2} \oplus \dots ≅ R [t I] \subseteq R [t]$

Note that $B_{I} R / I B_{I} R = g r_{I} R$ , the associated graded ring to $I$ in $R$ .

When $I$ is the zero ideal, $B_{I} R = R$ . In other words, $R$ does not get blown up anywhere.

When $I = R$ , $B_{I} R$ is the whole polynomial ring over $R$ .