Noether's normalization lemma over a ring A

The geometric interpretation of Noether's normalization lemma is that any affine algebraic variety has a finite surjective morphism to the affine space $\mathbb A^d_k$ of dimension $d=\dim X$. When $k$ is an integral domain, and $X$ dominates $\mathrm{Spec}(k)$, the finite surjective morphism of the generic fiber of $X$ to $\mathbb A^d_{K}$, where $K=\mathrm{Frac}(k)$ extends to a finite surjective morphism $X_V\to V$ for some dense open subset $V$ of $\mathrm{Spec}(k)$. This is the geometric interpretation of the statement in M. Hochster's note in Harry Gindi's post.

In general, such a morphism can not exist because it would imply that the fibers of $X\to \mathrm{Spec}(k)$ all have the same dimension.

Suppose that this condition is satisfied: the fibers of $X\to\mathrm{Spec}(k)$ all have the same dimension $d$. Then I think that a reasonable statement (Noether's normalization lemma over a ring $k$) would be: there exists a quasi-finite and surjective morphism $X\to \mathbb{A}^d_k$. If $k$ is noetherian, then by Zariski's Main Theorem, this implies that $X$ is an open subscheme of scheme which is finite surjective over $\mathbb{A}^d_k$. In general one can not expect better result than quasi-finite (consider the case $d=0$).

The above "reasonable statement" should be easy to prove when $k$ is a local ring.

http://www.math.lsa.umich.edu/~hochster/615W10/supNoeth.pdf

Supplementary notes from Mel Hochster's commutative algebra class. They discuss, in particular, a generalization of Noether normalization to integral domains.

I recently came to want this generalization of Noether normalization for my own commutative algebra course and notes. So I just wanted to report that I found what seems to me to be the optimally efficient and clear treatment of this result, at the beginning of Chapter 8 of these commutative algebra notes of K.M. Sampath. All in all I highly recommend Sampath's notes: they are excellent.

I am starting to find it surprising that this simple and useful generalization of Noether Normalization is not the standard version: it has some important applications, e.g. finiteness of integral closure of domains which are finitely generated over $\mathbb{Z}$. Does anyone know who first came up with this version (Hochster, perhaps)?