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Higher algebra in $t$-structured tensor triangulated $\infty$-categories

Jiacheng Liang

Abstract

We generalize fundamental notions of higher algebra, traditionally developed within the $\infty$-category of spectra, to the broader setting of $t$-structured tensor triangulated $\infty$-categories ($ttt$-$\infty$-categories). Under a natural structural condition, which we call "projective rigidity", we establish higher categorical analogues of Lazard's theorem and prove the existence and universal property of Cohn localizations. Furthermore, we generalize higher almost ring theory to the $ttt$-$\infty$-categorical setting, showing that $π_0$-epimorphic idempotent algebras are in natural bijection with idempotent ideals. By exploiting deformation theory, we establish a general étale rigidity theorem, proving that the $\infty$-category of étale algebras over a fixed connective base is completely determined by its discrete counterpart. Finally, we characterize the moduli of such projectively rigid $ttt$-$\infty$-categories, demonstrating that the presheaf $\infty$-category on the 1-dimensional framed cobordism $\infty$-category serves as the universal projectively rigid $ttt$-$\infty$-category.

Higher algebra in $t$-structured tensor triangulated $\infty$-categories

Abstract

We generalize fundamental notions of higher algebra, traditionally developed within the -category of spectra, to the broader setting of -structured tensor triangulated -categories (--categories). Under a natural structural condition, which we call "projective rigidity", we establish higher categorical analogues of Lazard's theorem and prove the existence and universal property of Cohn localizations. Furthermore, we generalize higher almost ring theory to the --categorical setting, showing that -epimorphic idempotent algebras are in natural bijection with idempotent ideals. By exploiting deformation theory, we establish a general étale rigidity theorem, proving that the -category of étale algebras over a fixed connective base is completely determined by its discrete counterpart. Finally, we characterize the moduli of such projectively rigid --categories, demonstrating that the presheaf -category on the 1-dimensional framed cobordism -category serves as the universal projectively rigid --category.

Paper Structure

This paper contains 31 sections, 133 theorems, 185 equations.

Table of Contents

  1. Preliminaries
  2. Prestable $\infty$-categories
  3. Algebra theory in Grothendieck abelian categories
  4. Flatness and faithful flatness
  5. $t$-structure on the category of modules
  6. Flat modules and algebras
  7. Projective modules
  8. Basic properties
  9. Projective rigidity and Lazard's theorem
  10. Modules over discrete algebras
  11. Cohn localizations of $\mathbf{E}_\infty$-algebras
  12. Cohn localizations in an abelian base
  13. Finiteness properties
  14. Perfect and almost perfect modules
  15. Finite presentation and almost of finite presentation
  16. ...and 16 more sections

Key Result

Theorem 2

Assume that $\mathcal{A}_{\geq 0}$ is projectively generated. Let $R \in \operatorname{Alg}(\mathcal{A}_{\geq 0})$. Then the following hold:

Theorems & Definitions (319)

  • Theorem 2: \ref{['cocompfil']}, and \ref{['cocompsimplicial']}
  • Theorem 3: \ref{['dicre']}
  • Theorem 4: \ref{['unilocal']}, Cohn localization
  • Theorem 5: \ref{['almostalg']}
  • Theorem 6: \ref{['etrig']}, Étale rigidity
  • Theorem 7: \ref{['uniexamp']}, Universal projectively rigid $ttt$-$\infty$-category
  • Definition 1.1: See sag C.1.2.2
  • Example 1.2
  • Proposition 1.3
  • Proposition 1.4: See sag C.1.4.1
  • ...and 309 more