An introduction to solid algebra

When working with infinitely generated modules and their duals, it is typically necessary to introduce various linear or adic topologies in order to control duality and various completed tensor products. Such questions arise, for example, when studying p-adic analysis in number theory or formal schemes in algebraic geometry. Unfortunately, there is no reasonable abelian category of topological modules, and so traditionally people have used various ad hoc tools to do homological algebra in the topologized setting. Recently, Clausen-Scholze have found a solution to this foundational problem, a good abelian category of “solid modules” (more generally, “condensed modules”). We give an elementary user guide to this theory, focusing on concrete calculations in the category of solid modules. If time permits, we discuss some applications to the geometry of arc and loop spaces of algebraic varieties.