Summary: Let \(R\) be the quotient of a local domain \((Q,{\mathfrak n})\) by a proper ideal minimally generated by \(f_1,\dots, f_c\). Assume \(Q/{\mathfrak n}\) is algebraically closed, and let \(M\) and \(N\) be finitely generated \(R\)-modules. We show there is an algebraic set in \(c\)-dimensional affine space, called the support set of the pair \((M, N)\), which describes those hypersurfaces \(h\in(f_1,\dots, f_c)-{\mathfrak n}(f_1,\dots, f_c)\) over which there are infinitely many nonzero \(\text{Ext}^i_{Q/(h)}(M, N)\). This generalizes to arbitrary quotients of regular local rings the notion of support variety for modules over complete intersections.