Schur-finiteness and endomorphisms universally of trace zero via certain trace relations

We provide a su cient condition that ensures the nilpotency of endomorphisms universally of trace zero of Schur-fi nite objects in a category of homological type, i.e., a Q-linear-category with a tensor functor to super vector spaces. This generalizes previous results about finite-dimensional objects, in particular by Kimura in the category of motives. We also present some facts which suggest that this might be the best generalization possible of this line of proof. To get the result we prove an identity of trace relations on super vector spaces which has an independent interest in the eld of combinatorics. Our main tool is Berele-Regev's theory of Hook Schur functions. We use their generalization of the classic Schur-Weyl duality to the super case, together with their factorization formula.