In the paper under review, the author systematically develops a theory of cohomological support for pairs of dg modules over a Koszul complex. This idea specializes to the support varieties of Avramov and Buchweitz defined over a complete intersection ring, as well as the support varieties of Avramov and Iyengar defined over an exterior algebra. To be specific, let \(R\) be a commutative noetherian ring, \(\underline{a}=a_{1},\dots,a_{n}\) a sequence of elements of \(R\), and \(A=K^{R}(\underline{a})\) the associated Koszul complex regarded as a dg \(R\)-algebra. Given dg \(A\)-modules \(X\) and \(Y\) with finitely generated homologies, the author defines a Zariski-closed subset \(\mathcal{V}_{A}(X,Y)\) of the projective space, called the cohomological support of the pair \((X,Y)\), whose dimension records the polynomial growth rate of the minimal number of generators of \(\mathrm{Ext}_{A}^{\ast}(X,Y)\), the so-called complexity \(\mathrm{cx}_{A}(X,Y)\) of the pair \((X,Y)\). Then he proceeds to prove the identity \[\mathcal{V}_{A}(X,Y) \cap \mathcal{V}_{A}(X',Y')=\mathcal{V}_{A}(X',Y) \cap \mathcal{V}_{A}(X,Y'),\] from which he derives \(\mathrm{cx}_{A}(X,Y)=\mathrm{cx}_{A}(Y,X)\leq n\). This recovers the asymptotic theorems of Avramov and Buchweitz for local complete intersections, and Avramov and Iyengar for exterior algebras.