  We extend the support theory of Benson--Iyengar--Krause to the non-Noetherian
setting by introducing a new notion of small support for modules. This enables
us to prove that the stable module category of a finite group is canonically
stratified by the action of the Tate cohomology ring, despite the fact that
this ring is rarely Noetherian. In the tensor triangular context, we compare
the support theory proposed by W. Sanders (which extends the Balmer--Favi
support theory beyond the weakly Noetherian setting) with our generalized BIK
support theory. When the Balmer spectrum is homeomorphic to the Zariski
spectrum of the endomorphism ring of the unit, the two support theories
coincide as do their associated theories of stratification. We also prove a
negative result which states that the Balmer--Favi--Sanders support theory can
only stratify categories whose spectra are weakly Noetherian. This provides
additional justification for the weakly Noetherian hypothesis in the work of
Barthel, Heard and B. Sanders. On the other hand, the detection property and
the local-to-global principle remain interesting in the general setting.
