Dimension and depth inequalities over complete intersections

For a pair of finitely generated modules $M$ and $N$ over a codimension $c$ complete intersection ring $R$ with $\mathrm{\ell }(M{\otimes }_{R}N)$ finite, we pay special attention to the inequality $\dim M+\dim N\le \dim R+c$. In particular, we develop an extension of Hochster’s theta invariant whose nonvanishing detects equality. In addition, we consider a parallel theory where dimension and codimension are replaced by depth and complexity, respectively.