Séminaire d'Homotopie et Géométrie Algébrique
# Ultrasolid algebra and deformation theory

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IMT 1R2 207 (Salle Pellos)
### IMT 1R2 207

#### Salle Pellos

Description

Solid modules over Q or 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 Q or 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 E_{∞} 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, and interpret this result in terms of Koszul duality.