The talk concerns morphisms between perfect complexes over commutative noetherian rings.  The central result is a criterion for the tensor-nilpotence of such morphisms, in terms of numerical invariants of complexes known as levels. The proof uses the existence of big Cohen-Macaulay modules. Applications to local rings include a strengthening of the Improved New Intersection Theorem, and short direct proofs of several results equivalent to it.  The results come from recent joint work with Iyengar and Neeman; see https://arxiv.org/abs/1711.04052