  We prove a generalized version of Evans and Griffith's Improved New
Intersection Theorem: Let I be an ideal in a local ring R. If a finite free
R-complex, concentrated in nonnegative degrees, has I-torsion homology in
positive degrees, and the homology in degree 0 has an I-torsion minimal
generator, then the length of the complex is at least dim R - dim R/I. This
improves the bound ht I obtained by Avramov, Iyengar, and Neeman in 2018.
