  A theory of cohomological support for pairs of DG modules over a Koszul
complex is investigated. These specialize to the support varieties of Avramov
and Buchweitz defined over a complete intersection ring, as well as support
varieties over an exterior algebra. The main objects of study are certain DG
modules over a polynomial ring; these determine the aforementioned
cohomological supports and are shown to encode (co)homological information
about pairs of DG modules over a Koszul complex. The perspective in this
article leads to new proofs of well-known results for pairs of complexes over a
complete intersection. Furthermore, these cohomological supports are used to
define a support theory for pairs of objects in the derived category of an
arbitrary commutative noetherian local ring. Finally, we calculate several
examples and provide an application by answering a question of D. Jorgensen in
the negative.
