A note on weight filtrations at the characteristic

TL;DR

This paper establishes a canonical weight filtration for certain cohomology theories over affine Dedekind schemes, connecting it to known filtrations and invariants of schemes, and providing explicit computations and foundational results.

Contribution

It introduces a canonical weight filtration for $ ext{kgl}$-linear cohomology theories on resolvable motives over schemes, extending known filtrations to positive and mixed characteristic.

Findings

Weight filtration recovers Deligne's pole-order filtration on de Rham cohomology.

Spectral sequence from pole-order filtration is an invariant of the open scheme.

Explicit examples show invariance of dual complex cohomology under compactification.

Abstract

We show that $\kgl$-linear cohomology theories over an affine Dedekind scheme $S$ admit a canonical weight filtration on resolvable motives without inverting residual characteristics. Combined with upcoming work of Annala--Hoyois--Iwasa, this endows essentially all known logarithmic cohomology theories with weight filtrations when evaluated on projective sncd pairs $(X,D)$ over $S$. Furthermore, the weight-filtered cohomology is an invariant of the open part $U = X-D$. On variants of de Rham cohomology, we show that our weight filtration recovers the d\'ecalaged pole-order filtration defined by Deligne. One interpretation of this is that the spectral sequence associated to the pole-order filtration is an invariant of $U$ from the $E_2$-page onwards, which generalizes a result of Deligne from characteristic 0 to positive and mixed characteristic, and suggests that ``mixed Hodge theory'' is a useful invariant of $S$-schemes. Finally, we compute explicit examples of weight filtered pieces of cohomology theories. One of the computations reproves a slight weakening of a result of Thuillier stating that the singular cohomology of the dual complex associated to the boundary divisor of a good projective compactification does not depend on the chosen compactification. In the appendix, we prove the folklore results that the Whitehead tower functor is fully faithful and that perfect bivariant pairings with respect to the twisted arrow category correspond to duality.