injective hull of $k[x_1,...,x_n]/m$ where $m=(x_1,...,x_n)$

Question

Is there any source in which I can find the exact relation between the injective hulls of $k[x_1,...,x_n]/m$ and $k[|x_1,...,x_n|]/m$ where $m$ is the maximal ideal $m=(x_1,...,x_n)$?

Answer 1

The injective hulls are the same, though the rings are different. Let $A$ denote either of the rings and let $A_p=A/(x_1^p,\ldots,x_n^p)$. Then you have a natural inclusion $A_p\to A_{p+1}$ given by $1\mapsto \prod x_i$. Take the direct limit to get the injective hull.