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Real syntomic cohomology

Gabriel Angelini-Knoll, Hana Jia Kong, J. D. Quigley

Abstract

We introduce a theory of syntomic cohomology for ring spectra with involution, which we call Real syntomic cohomology. We show that our construction extends the theory of syntomic cohomology for rings with involution due to Park. Our construction also refines syntomic cohomology as developed by Bhatt--Morrow--Scholze, Morin, Bhatt--Lurie, and Hahn--Raksit--Wilson. We compute the Real syntomic cohomology of Real topological K-theory and topological modular forms with level structure.

Real syntomic cohomology

Abstract

We introduce a theory of syntomic cohomology for ring spectra with involution, which we call Real syntomic cohomology. We show that our construction extends the theory of syntomic cohomology for rings with involution due to Park. Our construction also refines syntomic cohomology as developed by Bhatt--Morrow--Scholze, Morin, Bhatt--Lurie, and Hahn--Raksit--Wilson. We compute the Real syntomic cohomology of Real topological K-theory and topological modular forms with level structure.

Paper Structure

This paper contains 55 sections, 85 theorems, 422 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Results
  3. The strongly even filtration and descent
  4. Artin--Tate Real motivic cohomology
  5. The motivic filtration
  6. Real syntomic cohomology of Real truncated Brown--Peterson spectra
  7. Lichtenbaum--Quillen for Real syntomic cohomology
  8. Outline
  9. Conventions
  10. Acknowledgments
  11. The strongly even filtration
  12. The regular slice filtration
  13. Strongly even $C_2$-spectra
  14. The strongly even filtration
  15. A $p$-complete variant
  16. ...and 40 more sections

Key Result

Theorem A

If $A\to B$ is a seff map of $C_2$-commutative ring spectra and $B$ is strongly even, then there is an equivalence of filtered $C_2$-spectra.

Figures (8)

  • Figure 1: The $\mathrm{RO}(C_2)$-graded homotopy groups of $H\underline{\mathop{\mathrm{\mathbb{F}}}\nolimits}_2$. The group $\pi^{C_2}_{m+n\sigma}(H\underline{\mathop{\mathrm{\mathbb{F}}}\nolimits}_2)$ is displayed in bidegree $(m,n)$. Each bullet $\bullet$ represents a copy of $\mathop{\mathrm{\mathbb{F}}}\nolimits_2$. The blue dashed lines indicate the gap, which implies that the $C_2$-spectrum $H\underline{\mathop{\mathrm{\mathbb{F}}}\nolimits}_2$ is strongly even.
  • Figure 2: The $\mathrm{E}_2=\mathrm{E}_{\infty}$-page of the periodic $\overline{t}$-Bockstein spectral sequence computing $\pi_{\star}^{C_2}\mathrm{gr}_{\textup{mot}}^*\mathop{\mathrm{\mathrm{TPR}}}\nolimits(\underline{\mathop{\mathrm{\mathbb{F}}}\nolimits}_2)$. The group $E_2^{m+n\sigma,a,b}$ appears in bidegree $(m,n)$. Each bullet $\bullet$ represents a copy of $\mathbb{F}_2$.
  • Figure 3: The motivic spectral sequence computing $\pi_{\star}^{C_2}\mathop{\mathrm{\mathrm{TCR}}}\nolimits(\underline{\mathop{\mathrm{\mathbb{F}}}\nolimits}_2)/2$. Each bullet $\bullet$ represents a copy of $\pi_{\star}^{C_2}\underline{\mathop{\mathrm{\mathbb{F}}}\nolimits}_2$. The horizontal axis represents the stem (a $C_2$-representation) and the vertical axis represents the Adams weight.
  • Figure 4: The $\mathrm{E}_3=\mathrm{E}_{\infty}$-page of periodic $\overline{t}$-Bockstein spectral sequence computing $\pi_{\star}^{C_2}\mathrm{gr}_{\textup{mot}}^*\mathop{\mathrm{\mathrm{TPR}}}\nolimits(\underline{\mathop{\mathrm{\mathbb{Z}}}\nolimits}_2)/(\overline{v}_0,\overline{v}_1)$. The group $E_3^{m+n\sigma,a,b}$ appears in bidegree $(m,n)$. Each bullet $\bullet$ represents a copy of $\mathop{\mathrm{\mathbb{F}}}\nolimits_2$. Black bullets indicate classes in motivic filtration $0$ and black lines indicate multiplications in motivic filtration $0$. Blue bullets indicate classes in motivic filtration $1$ and blue dashed lines indicate multiplications in motivic filtration $1$.
  • Figure 5: The motivic spectral sequence computing $\pi_{\star}^{C_2}\mathop{\mathrm{\mathrm{TCR}}}\nolimits(\underline{\mathop{\mathrm{\mathbb{Z}}}\nolimits}_2)/(2)$. Each bullet $\bullet$ represents a copy of $\pi_{\star}^{C_2}\underline{\mathop{\mathrm{\mathbb{F}}}\nolimits}_2[\overline{v}_1]$. The horizontal axis represents the stem (a $C_2$-representation) and the vertical axis represents the Adams weight. Lines of slope $1$ indicate multiplication by $\eta_{C_2}$. Lines of slope $-1$ indicate multiplication by $\partial$.
  • ...and 3 more figures

Theorems & Definitions (265)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Remark 1.2
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 255 more