  We give sufficient conditions which ensure that a functor of finite length
from an additive category to finite-dimensional vector spaces has a projective
resolution whose terms are finitely generated. For polynomial functors, we
study also a weaker homological finiteness property, which applies to twisted
homological stability for matrix monoids. This is inspired by works by Schwartz
and Betley-Pirashvili, which are generalised; this also uses decompositions
{\`a} la Steinberg over an additive category that we recently got with Vespa.
We show also, as an application, a finiteness property for stable homology of
linear groups on suitable rings.

K-Theory and Homology