Blowup algebra

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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_IR := R \oplus I \oplus I^2 \oplus \ldots \cong R[tI] \subseteq R[t]

Note that B_IR/IB_IR = gr_IR, the associated graded ring to I in R.

Particular cases

When I is the zero ideal, B_IR = R. In other words, R does not get blown up anywhere.

When I = R, B_IR is the whole polynomial ring over R.