We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category
$\operatorname {Sing}(R)$
, and that the support of an object in
$\operatorname {Sing}(R)$
vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz’ support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.