In its various forms, the The New Intersection Theorem is concerned with the length of a finite free complex, that is, a complex \begin{equation*} F = 0 \leftrightarrow F_n \leftrightarrow \cdots \leftrightarrow F_1 \leftrightarrow F_0 \leftrightarrow 0 \end{equation*} of finitely generated free modules, over a local ring $(R,\mathfrak m)$. The classic version, due to Peskine and Szpiro (1973) asserts that if $HH(F)$ is non-zero and each homology module $HH_i(F)$ is of finite length, then $n \geq \dim R$ 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.