I've got a question about explicitly lifting regular sequences. Let $I$ be an ideal in a polynomial ring $S$ with some term order. We'll denote the initial ideal by $in(I)$. It is false in general that a regular sequence on $S/in(I)$ is regular on $S/I$. For example consider $I=(x+y)$, with $x>y$ Then $x+y$ is a regular element on $S/in(I)$ but is not regular on $S/I$. However, $3x-y$ IS a regular element mod $I$.
My question is: Can we can do this in general? i.e. Given a regular sequence on $S/in(I)$, can we obtain a regular sequence on $S/I$ by just replacing all the coefficients in all the elements with generic coefficients?
We know that the depth of $S/in(I)$ is at most the depth of $S/I$, but I haven't actually seen too many proofs of this written down. The ones I've seen first show a bound on Betti numbers and then use the Auslander-Buchsbaum formula. I was wondering if one could prove this fact by answering the question above, and if anyone has a reference. I think one might be able to use a flat family argument. In general it would be nice to have an explicit way of going back and form between regular sequences on $I$ and $in(I)$. Any reference or suggestions would be greatly appreciated.
Thanks so much for your help!