In its various forms, the The New Intersection Theorem is concerned with the length of a finite free complex, that is, a complex F=0↔Fn↔⋯↔F1↔F0↔0 of finitely generated free modules, over a local ring (R,m). The classic version, due to Peskine and Szpiro (1973) asserts that if HH(F) is non-zero and each homology module HHi(F) is of finite length, then n≥dimR holds.
In the talk I will discuss the history of this a recent result up to a recent improvement obtained in joint work with Luigi Ferraro.