This work concerns finite free complexes over commutative noetherian rings, in particular over group algebras of elementary abelian groups. The main contribution is the construction of complexes such that the total rank of their underlying free modules, or the total length of their homology, is less than predicted by various conjectures in the theory of transformation groups and in local algebra.
"Examples of finite free complexes of small rank and small homology." Acta Math. 221 (1) 143 - 158, September 2018. https://doi.org/10.4310/ACTA.2018.v221.n1.a4