{"title":"Ultrasolid Homotopical Algebra","authors":"Sofía Marlasca Aparicio","doi":"arxiv-2406.04063","DOIUrl":null,"url":null,"abstract":"Solid modules over $\\mathbb{Q}$ or $\\mathbb{F}_p$, introduced by Clausen and\nScholze, are a well-behaved variant of complete topological vector spaces that\nforms a symmetric monoidal Grothendieck abelian category. For a discrete field\n$k$, we construct the category of ultrasolid $k$-modules, which specialises to\nsolid modules over $\\mathbb{Q}$ or $\\mathbb{F}_p$. In this setting, we show\nsome commutative algebra results like an ultrasolid variant of Nakayama's\nlemma. We also explore higher algebra in the form of animated and\n$\\mathbb{E}_\\infty$ ultrasolid $k$-algebras, and their deformation theory. We\nfocus on the subcategory of complete profinite $k$-algebras, which we prove is\ncontravariantly equivalent to equal characteristic formal moduli problems with\ncoconnective tangent complex.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"28 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2406.04063","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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.