  There is a well known link from the first topic in the title to the third
one. In this paper we thread that link through the second topic. The central
result is a criterion for the tensor nilpotence of morphisms of perfect
complexes over commutative noetherian rings, in terms of a numerical invariant
of the complexes known as their level. Applications to local rings include a
strengthening of the Improved New Intersection Theorem, short direct proofs of
several results equivalent to it, and lower bounds on the ranks of the modules
in every finite free complex that admits a structure of differential graded
module over the Koszul complex on some system of parameters.
