  Archimedean cohomology provides a cohomological interpretation for the
calculation of the local L-factors at archimedean places as zeta regularized
determinant of a log of Frobenius. In this paper we investigate further the
properties of the Lefschetz and log of monodromy operators on this cohomology.
We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian
connection whose residue is the log of the monodromy. We also present a
dictionary of analogies between the geometry of a tubular neighborhood of the
``fiber at arithmetic infinity'' of an arithmetic variety and the complex of
nearby cycles in the geometry of a degeneration over a disk, and we recall
Deninger's approach to the archimedean cohomology through an interpretation as
global sections of a analytic Rees sheaf. We show that action of the Lefschetz,
the log of monodromy and the log of Frobenius on the archimedean cohomology
combine to determine a spectral triple in the sense of Connes. The archimedean
part of the Hasse-Weil L-function appears as a zeta function of this spectral
triple. We also outline some formal analogies between this cohomological theory
at arithmetic infinity and Givental's homological geometry on loop spaces.
