Solid realization of motives with modulus

Abstract

We construct a covariant realization functor, denoted \textsc{Solidm}, from the category of motives with modulus to the derived category of solid modules in the sense of Clausen--Scholze. For any smooth modulus pair (X, D), the dual of Solidm(X, D) recovers the Hodge realization of Kelly--Miyazaki for (X, D). Using Ren's pro-solid comparison theorem, we give an explicit description of Solidm(X, D) and compute Solidm of the cone of M(U, D restricted to U) $\to$ M(X, D), in the setting where X is a smooth proper variety over a field, D $\subset$ X is a simple normal crossings divisor, and U $\subset$ X is an open immersion. We identify the result via the formal completion of X along the complement X $\setminus$ U.

This version implicitly used a global form of Ren's pro-solid comparison theorem. At present, only the affine-local case is established, and the required gluing for the global case remains open (we thank Fei Ren for pointing this out). Until the general case is proved, we withdraw this version and will resubmit a revised manuscript