A note on weight filtrations at the characteristic

Abstract: We show that logarithmic variants of ($p$-adic) cohomology theories admit a canonical weight filtration on resolvable motives, for example on the motive associated to a projective relative strict normal crossing divisor pair. Our methods apply to $\kgl$-linear cohomology theories on smooth schemes over Dedekind domains. Importantly, the produced filtrations are invariants of the open part of the log scheme, thus providing new, computable invariants of smooth schemes admitting a good compactification.