Special Values without Semi-Simplicity Via K-Theory

In this paper, motivated by studying special values of zeta functions attached to finite type 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=Z_l for some prime l neq p (resp. R=Z_p), we prove that there is a natural functorial lift of the etale cohomology of perfect etale Z_l sheaves (resp. syntomic cohomology of perfect prismatic F-gauges) on a point to arithmetic C(S^1,Z_l)-modules (resp. arithmetic C(S^1,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 F_p-schemes. We also discuss extensions of these zeta value formulae to finite type F_p-schemes, and how recent progress in motivic homotopy theory allows us to prove some results without any assumptions on resolution of singularities or Tate's semi-simplicity conjecture.