In this talk I will survey some results on the vanishing of (quantum) group representations, at the level of the stable category.  Equivalently, I will discuss effective ways to test projectivity of a given finite-dimensional G-representation, where G your favorite finite (quantum) group.  In the case of an elementary abelian p-group E, over k=\bar{F}_p,  for example, Carlson tells us that an object V in rep(E) is projective if and only if V has projective restriction along each flat algebra map \alpha: k[t]/(t^p) -> k[E] into the group ring.  One thus reduces a wild representation type calculation to a finite representation type calculation, via this P^{rank(E)}(k)-family of embeddings.  I will provide an analogous vanishing result for the small quantum group u_q(L), which involves the introduction of a G/B-family of small quantum Borels and an analysis of certain ``noncommutative complete intersections”.  This is joint work with Julia Pevtsova [arxiv:2012.15453, arxiv:2203.10764].