# Characteristic polynomial of the unique automorphism of the zero module

Is there any convention which makes sense of the characteristic polynomial of the unique automorphism of the zero module?

This might seem like an odd question but it matters to me. The background (for those who understand):

I am studying the K-theory of the category of pairs $(P,f)$ where $P$ is some projective $R$-module and $f$ is an automorphism. One can show* that this category is equivalent to the category of finitely generated $R[U]$-modules of projective dimension one and which are of $g$-torsion for some polynomial $g\in S$, where $S$ is the set of all monic polynomials which have a unit as a constant term. The set $S$ consists in fact of all characteristic polynomials of all possible projective modules. Since I want to localise the ring $R[U]$ at $S$ later it seems crucial whether $S$ contains 1 or 0.

I assume that I will find a unique way to make everything work, but it would be great to have an intrinsic explanation.

*The proof works for a more general setting and doesn't use all properties of $S$. So we can't really deduce an answer from here.

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2022-07-25 20:43:33
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