Sign lemma for dimension shifting

TL;DR

This paper investigates the minus signs in dimension shifting isomorphisms in homological algebra, explicitly determining their values to prevent contradictions, with applications to de Rham cohomology and Chern classes.

Contribution

It explicitly determines the signs involved in dimension shifting isomorphisms, clarifying a subtle but crucial aspect of homological algebra techniques.

Findings

Explicit sign determination in dimension shifting isomorphisms

Application to de Rham cohomology of Riemann surfaces

Application to Chern class considerations

Abstract

There is a surprising occurrence of some minus signs in the isomorphisms produced in the well-known technique of dimension shifting in calculating derived functors in homological algebra. We explicitly determine these signs. Getting these signs right is important in order to avoid basic contradictions. We illustrate the lemma by some de Rham cohomology and Chern class considerations for compact Riemann surfaces.