  We consider a filtration on the cohomology of the structure sheaf indexed by
(not necessarily reduced) divisors ``at infinity''. We show that the filtered
pieces have transfers morphisms, fpqc descent, and are so called cube
invariant.
  In the presence of resolution of singularities and weak factorisation they
are invariant under blowup ``at infinity''. As such, they lead to a realisation
functor from Kahn, Miyazaki, Saito and Yamazaki's category of motives with
modulus over a characteristic zero base field.
