Modified Traces and the Nakayama Functor

We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor Σ on a finite abelian category M, we introduce the notion of a Σ-twisted trace on the class Proj (M) of projective objects of M. In our framework, there is a one-to-one correspondence between the set of Σ-twisted traces on Proj (M) and the set of natural transformations from Σ to the Nakayama functor of M. Non-degeneracy and compatibility with the module structure (when M is a module category over a finite tensor category) of a Σ-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular.