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.