Special Values without Semi-Simplicity Via K-Theory

Motivated by studying special values of zeta functions attached to finite type $\mathbf{F}_p$-schemes, we introduce a category of ``arithmetic $C(S^1,R)$-modules'' attached to any Dedekind ring R, and compute the 0th K-group of this category. Specializing to the case of $R=\mathbf{Z}_\ell$ for some prime $\ell \neq p$ (resp. $R=\mathbf{Z}_p$), we prove that there is a natural functorial lift of the etale cohomology of perfect etale $\mathbf{Z}_\ell$ sheaves (resp. syntomic cohomology of perfect prismatic F-gauges) on a point to arithmetic $C(S^1,\mathbf{Z}_\ell)$-modules (resp. arithmetic $C(S^1,\mathbf{Z}_p)$-modules). This allows us to define a notion of the multiplicative Euler characteristic via a map from the $K_0$-group which makes sense without assuming Tate's semi-simplicity conjecture. In particular, we can remove this hypothesis from a theorem of Milne proving a cohomological formula for zeta values attached to smooth proper $\mathbf{F}_p$-schemes.