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Quivers and the Adams spectral sequence

Robert Burklund, Piotr Pstrągowski

Abstract

In this paper, we describe a novel way of identifying Adams spectral sequence $E_2$-terms in terms of homological algebra of quiver representations. Our method applies much more broadly than the standard techniques based on descent-flatness, bearing on a varied array of ring spectra. In the particular case of $p$-local integral homology, we are able to give a decomposition of the $E_2$-term, describing it completely in terms of the classical Adams spectral sequence. In the appendix, which can be read independently from the main body of the text, we develop functoriality of deformations of $\infty$-categories of the second author and Patchkoria.

Quivers and the Adams spectral sequence

Abstract

In this paper, we describe a novel way of identifying Adams spectral sequence -terms in terms of homological algebra of quiver representations. Our method applies much more broadly than the standard techniques based on descent-flatness, bearing on a varied array of ring spectra. In the particular case of -local integral homology, we are able to give a decomposition of the -term, describing it completely in terms of the classical Adams spectral sequence. In the appendix, which can be read independently from the main body of the text, we develop functoriality of deformations of -categories of the second author and Patchkoria.
Paper Structure (28 sections, 58 theorems, 205 equations)

This paper contains 28 sections, 58 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Integral homology
  3. Further examples
  4. The calculus of deformations
  5. Quiver homology theories
  6. Quivers of simple modules
  7. Homology operations
  8. Ind-perfect $\mathcal{Q}$-comodules
  9. Monoidal structure on quivers
  10. Synthetic spectra based on a quiver homology theory
  11. Construction of synthetic modules
  12. Comparison of deformations
  13. Integral homology
  14. Bockstein couples
  15. Integral homology cooperations
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $R$ be $\EuScript{P}$-Adams-type. Then there exists a unique exact, cocontinuous comonad $\mathcal{Q}$ on $\mathop{\mathrm{Rep}}\nolimits(\EuScript{P})$ such that for any spectrum $A$. Moreover, for any spectrum $A$ the homology $\mathsf{H}(A)$ has a natural structure of a $\mathcal{Q}$-comodule and there is a canonical identification between the $E_{2}$-term of the $R$-based Adams spectral

Theorems & Definitions (164)

  • Definition 1.1
  • Theorem 1.2: \ref{['theorem:comodule_homology_theory_adapted_in_pcat_flat_case']}
  • Theorem 1.4
  • Theorem 1.5: \ref{['thm:pullback-Z']}
  • Proposition 1.6: \ref{['theorem:integral_todas_obstructions_groups_for_bp_vanish']}
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.7: Values in graded abelian groups
  • ...and 154 more