Ultrasolid Homotopical Algebra

Solid modules over $\mathbb{Q}$ or $\mathbb{F}_p$, introduced by Clausen and Scholze, are a well-behaved variant of complete topological vector spaces that forms a symmetric monoidal Grothendieck abelian category. For a discrete field $k$, we construct the category of ultrasolid $k$-modules, which specialises to solid modules over $\mathbb{Q}$ or $\mathbb{F}_p$. In this setting, we show some commutative algebra results like an ultrasolid variant of Nakayama's lemma. We also explore higher algebra in the form of animated and $\mathbb{E}_\infty$ ultrasolid $k$-algebras, and their deformation theory. We focus on the subcategory of complete profinite $k$-algebras, which we prove is contravariantly equivalent to equal characteristic formal moduli problems with coconnective tangent complex.