the following general result:

\[commalg\] Let $R$ and $S$ be two Noetherian \*local rings and $\varphi : R \to S$ a ring homomorphism which is homogeneous of degree 0 and maps the maximal graded ideal ${\mathfrak m}$ of $R$ into the maximal graded ideal ${\mathfrak n}$ of $S$. Assume that both $R/{\mathfrak m}$ and $S/{\mathfrak n}$ are fields. Let $A$ be a finitely generated non-zero $R$-module and $B$ a finitely generated non-zero $S$-module, which is flat over $R$. Then:

\(a) ${\rm depth}_S \ \! A\otimes_R B = {\rm depth}_R \ \! A + {\rm depth}_S \ \! B/{\mathfrak m}B$; (b) $\dim_S A\otimes_R B = \dim_R  A + \dim_S  B/{\mathfrak m}B$;

\(c) The module $A\otimes_R B$ is Cohen-Macaulay over $S$ if and only if $A$ and $B/{\mathfrak m}B$ are Cohen-Macaulay over $R$ and $S$, respectively.

In the special case when $R$ and $S$ are just local rings, both (a) and (b) are standard facts in commutative algebra, cf. e.g. Prop. 1.2.16 and Thm. A.11 in [@BrunsHerzog]. The general case can be reduced to this special situation by using localization at the maximal graded ideals. One takes into account that if $I$ is a graded ideal of $R$ then $ {\rm grade} 