Blow-Up Algebra definition Suppose $R$ is a ring and $I$ is an ideal of that ring, then the blow-up algebra of $I$ in $R$, as defined in Eisenbud, is: $B_IR := R \oplus I \oplus I^2 \oplus \dots \cong R[tI]$ Why is this direct sum isomorphic to $R[tI]$?

I'm not sure what Eisenbud's definition of "$R[tI]$" is but it seems like he's trying to suggest this picture:

$\\{p(t)\in R[t]\mid p_0\in R, p_i\in I^i\text{ for } i> 0\\}$

That is, the natural grading on the algebra with the (external) direct sums of powers of $I$ reflects that the multiplication is convolution of the entries. (After distribution, an entry from $I^j$ multiplied from one in $I^k$ contributes to the $I^{j+k}$ entry.)

This is the same as if you had used powers of $t$ to track the positions of the coefficients rather than the tuples in the external direct sum.

The map would be, then, $(p_0, p_1, \ldots, p_k,\ldots)\mapsto\sum p_it^i$.

There are only finitely many nonzero entries in the tuple on the left, of course, so the right hand side is only a finite sum.