Here's a direct reference:

Hochster Math 615 - 2004, February 18th

I found this by tracking references from page 35 of Mel Hochster's notes here:

Hochster Math 615 - 2010

Here's a direct reference:

Hochster Math 615 - 2004, February 18th

It's a corollary of the characterization of regularity by the projective dimension of $R/m$.

I found this by tracking references from page 35 of Mel Hochster's notes here:

Hochster Math 615 - 2010