Suppose that we have a commutative ring $R$ which i) is local ii) is the quotient of a regular ring and iii) it is a $k$-algebra, where $k$ is a field.
I am trying to prove that in that case we can choose a presentation $R = S/I$, where $S$ is a local regular $k$-algebra.
Begin by taking a presentation $R=S/I$ with $S$ regular ring. If $m$ is the maximal ideal of $R$ then $m = P/I$, where $P$ is a prime ideal of $S$ that contains $I$. Then $R = R_m = (S/I)_{P/I} = S_P / I S_P$. Since $S_P$ is again regular, we can replace $S$ with $S_P$ and we obtain a presentation $R=S/I$ with $S$ regular local.
Question: How can we show that we can also take $S$ to be a $k$-algebra?
Remark: I can see that $S/I \otimes_k k = S/I = R$ and ideally i would like to write $S\otimes k / I \otimes k = R$, but i don't see why i can do that.
Reference: Bruns and Herzog, CMR, top of page 78.
