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There is a well known link from the first topic in the title to the thirdone. In this paper we thread that link through the second topic. The centralresult is a criterion for the tensor nilpotence of morphisms of perfectcomplexes over commutative noetherian rings, in terms of a numerical invariantof the complexes known as their level. Applications to local rings include astrengthening of the Improved New Intersection Theorem, short direct proofs ofseveral results equivalent to it, and lower bounds on the ranks of the modulesin every finite free complex that admits a structure of differential gradedmodule over the Koszul complex on some system of parameters.

Big Cohen-Macaulay modules, morphisms of perfect complexes, and intersection theorems in local algebra