The Supertrace Defect of a Graded Endomorphism
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.
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BibTeX
@misc{erdal2026supertrace,
title = {{The Supertrace Defect of a Graded Endomorphism}},
author = {Esma Dirican Erdal and Atabey Kaygun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05096},
eprint = {2610.05096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05096},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Supertrace Defect of a Graded Endomorphism
AU - Esma Dirican Erdal
AU - Atabey Kaygun
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05096
UR - https://arxiv.org/abs/2610.05096
AB - 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.
ER -