The Supertrace Defect of a Graded Endomorphism

Source abstract

Motivated by Igusa's relation between relative cyclic homology and the logarithm of the graded Cartan determinant, we isolate the homological step behind the passage from logarithmic determinants to traces of powers. For a bounded chain complex $(C_*,d)$ and a degree-zero graded endomorphism $f$, not assumed to commute with $d$, we construct the universal quotient on which $f$ becomes a chain map. The kernel of this reflection iterates to a canonical functorial minimal filtration, whose Rees module gives a flat degeneration to an uncurved associated graded. For finite-dimensional chain groups all power supertraces of $f$ are represented by Lefschetz traces on this associated graded, so the resulting logarithmic-determinant and zeta series admit a canonical homological representation.