Example 10.70.4. Let $R$ be a ring. Let $P = R[t_1, \ldots , t_ n]$ be the polynomial algebra. Let $I = (t_1, \ldots , t_ n) \subset P$. With notation as in Definition 10.70.1 there is an isomorphism

$P[T_1, \ldots , T_ n]/(t_ iT_ j - t_ jT_ i) \longrightarrow \text{Bl}_ I(P)$

sending $T_ i$ to $t_ i^{(1)}$. We leave it to the reader to show that this map is well defined. Since $I$ is generated by $t_1, \ldots , t_ n$ we see that our map is surjective. To see that our map is injective one has to show: for each $e \geq 1$ the $P$-module $I^ e$ is generated by the monomials $t^ E = t_1^{e_1} \ldots x_ n^{e_ n}$ for multiindices $E = (e_1, \ldots , e_ n)$ of degree $|E| = e$ subject only to the relations $t_ i t^ E = t_ j t^{E'}$ when $|E| = |E'| = e$ and $e_ a + \delta _{a i} = e'_ a + \delta _{a j},\ a = 1, \ldots , n$ (Kronecker delta). We omit the details.

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