2024/04/29 by Hyslop, Logan
#11G25 (Secondary) #14G15 (Primary) 14G10 #Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2404.18367
In this paper, motivated by studying special values of zeta functions attached to finite type Fp-schemes, we introduce a category of ``arithmetic C(S1,R)-modules'' attached to any Dedekind ring R, and compute the 0th K-group of this category. Specializing to the case of R=Zl for some prime l neq p (resp. R=Zp), we prove that there is a natural functorial lift of the etale cohomology of perfect etale Zl sheaves (resp. syntomic cohomology of perfect prismatic F-gauges) on a point to arithmetic C(S1,Zl)-modules (resp. arithmetic C(S1,Zp)-modules). This allows us to define a notion of the multiplicative Euler characteristic via a map from the K0-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 Fp-schemes. We also discuss extensions of these zeta value formulae to finite type Fp-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.