The following result appears in the book 'Commutative Algebra with a View Toward Algebraic Geometry' by David Eisenbud:
Theorem 18.16* Let $(R,P)$ be a regular local ring, and let $(A,Q)$ be a local Noetherian $R$-algebra, with $PA \subset Q$.
a. $A$ is flat over $R$ iff $\operatorname{depth}(PA,A) = \dim R$.
b. If $A$ is Cohen-Macaulay, then $A$ is flat over $R$ iff $\dim A = \dim R + \dim A/PA$.
(The above result appears here).
Is there a similar result without 'local'? (probably not?).
Thank you very much!