  We introduce an infinite variant of hypersurface support for
finite-dimensional, noncommutative complete intersections. By a noncommutative
complete intersection we mean an algebra R which admits a smooth deformation
$Q\to R$ by a Noetherian algebra $Q$ which is of finite global dimension. We
show that hypersurface support defines a support theory for the big singularity
category $Sing(R)$, and that the support of an object in $Sing(R)$ 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 a
companion piece, we employ hypersurface support, and the results of the present
paper, to classify thick ideals in stable categories for a number of families
of finite-dimensional Hopf algebras.
