Coherent six-functor formalisms: Pro vs Solid

TL;DR

This paper compares two advanced constructions of the missing $j_!$ functor in the Grothendieck six-functor formalism, showing they are equivalent via a natural functor.

Contribution

It proves the equivalence of Deligne's pro sheaf extension and Clausen-Scholze's solid modules extension for the $j_!$ functor.

Findings

Deligne's and Clausen-Scholze's constructions coincide.

The natural functor between the two constructions is fully faithful on Mittag-Leffler pro-systems.

The work bridges two approaches in the six-functor formalism.

Abstract

In the classical theory for coherent sheaves, the only missing piece in the Grothendieck six-functor formalism picture is $j_!$ for an open immersion $j$. Towards fixing this gap, Deligne proposed a construction of $j_!$ by extending the sheaf class to pro sheaves, while Clausen-Scholze provided another solution by extending the sheaf class to solid modules. In this work, we prove that Deligne's construction coincides with the Clausen-Scholze construction via a natural functor, whose restriction to the full subcategory of Mittag-Leffler pro-systems is fully faithful.