Papers
Topics
Authors
Recent
Search
2000 character limit reached

Maps between spherical group rings

Published 10 May 2024 in math.AT and math.AC | (2405.06448v1)

Abstract: We prove that for finitely generated abelian groups $A$ and $B$, the space of $\mathbb{E}_\infty$-ring maps between the spherical groups rings $\mathbb{S}[A] \to \mathbb{S}[B]$ is equivalent to the discrete set of group homomorphisms $A \to B$. We also prove generalizations where the sphere is replaced by other ring spectra, e.g. we give a formula for the strict units in group rings of the form $R[A]$ for $A$ a finite $p$-group and $R$ $p$-completely chromatically complete.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (41)
  1. Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map. Geom. Topol., 22(7):3761–3825, 2018.
  2. Prismatic cohomology relative to δ𝛿\deltaitalic_δ-rings. arXiv preprint arXiv:2310.12770, 2023.
  3. Tobias Barthel. Chromatic completion. Proceedings of the American Mathematical Society, 144(5):2263–2274, 2016.
  4. Tobias Barthel and A Bousfield. On the comparison of stable and unstable p𝑝pitalic_p-completion. Proceedings of the American Mathematical Society, 147(2):897–908, 2019.
  5. The Chromatic Fourier Transform. arXiv preprint arXiv:2210.12822, 2022.
  6. Bhargav Bhatt. Lecture II: δ𝛿\deltaitalic_δ-rings. https://public.websites.umich.edu/ bhattb/teaching/prismatic-columbia/lecture2-delta-rings.pdf.
  7. Prasit Bhattacharya. Higher associativity of Moore spectra. Advances in Mathematics, 402:108319, 2022.
  8. K𝐾Kitalic_K-theoretic counterexamples to Ravenel’s telescope conjecture. arXiv preprint arXiv:2310.17459, 2023.
  9. The cyclotomic trace and algebraic K𝐾Kitalic_K-theory of spaces. Invent. Math., 111(3):465–539, 1993.
  10. Topological Hochschild homology and integral p𝑝pitalic_p-adic Hodge theory. Publ. Math. Inst. Hautes Études Sci., 129:199–310, 2019.
  11. Projectivity of the Witt vector affine Grassmannian. Invent. Math., 209(2):329–423, 2017.
  12. Prisms and prismatic cohomology. Annals of Mathematics, 196(3):1135–1275, 2022.
  13. A note on the Segal conjecture for large objects. arXiv preprint arXiv:2403.06724, 2024.
  14. The Chromatic Nullstellensatz. arXiv preprint arXiv:2207.09929, 2022.
  15. Robert Burklund. Multiplicative structures on Moore spectra. arXiv preprint arXiv:2203.14787, 2022.
  16. Shachar Carmeli. On the Strict Picard Spectrum of Commutative Ring Spectra. arXiv preprint arXiv:2208.03073, 2022.
  17. Purity for flat cohomology. arXiv preprint arXiv:1912.10932, 2019.
  18. Ambidexterity in chromatic homotopy theory. arXiv preprint arXiv:1811.02057, 2018.
  19. Ambidexterity and Height. arXiv preprint arXiv:2007.13089, 2020.
  20. Chromatic cyclotomic extensions. arXiv preprint arXiv:2103.02471, 2021.
  21. Sanath Devalapurkar. Roots of unity in K⁢(n)𝐾𝑛K(n)italic_K ( italic_n )-local rings. Proc. Amer. Math. Soc., 148(7):3187–3194, 2020.
  22. Jun Hou Fung. Strict units of commutative ring spectra. PhD thesis, Harvard University, 2020.
  23. Giles Gardam. A counterexample to the unit conjecture for group rings. Ann. of Math. (2), 194(3):967–979, 2021.
  24. Graham Higman. The units of group-rings. Proceedings of the London Mathematical Society, 2(1):231–248, 1940.
  25. Ambidexterity in K⁢(n)𝐾𝑛K(n)italic_K ( italic_n )-local stable homotopy theory. preprint, available at https://www.math.ias.edu/ lurie/, 2013.
  26. Ambidexterity in K⁢(n)𝐾𝑛K(n)italic_K ( italic_n )-local stable homotopy theory. preprint, 2013.
  27. Lars Hesselholt and Ib Madsen. On the K𝐾Kitalic_K-theory of local fields. Ann. of Math. (2), 158(1):1–113, 2003.
  28. Adam Holeman. Derived δ𝛿\deltaitalic_δ-rings and relative prismatic cohomology. arXiv preprint arXiv:2303.17447, 2023.
  29. Units of group rings. Journal of Pure and Applied Algebra, 107(2-3):233–251, 1996.
  30. G Karpilovsky. On finite generation of unit groups of commutative group rings. Archiv der Mathematik, 40:503–508, 1983.
  31. Achim Krause. Flat H∞subscript𝐻H_{\infty}italic_H start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT Moore spectra are δ𝛿\deltaitalic_δ-rings. In preparation.
  32. Tyler Lawson. Adjoining roots in homotopy theory. arXiv preprint arXiv:2002.01997, 2020.
  33. Jacob Lurie. Elliptic cohomology II: Orientations. https://www.math.ias.edu/ lurie/papers/Elliptic-II.pdf, 2018.
  34. Akhil Mathew. The galois group of a stable homotopy theory. Advances in Mathematics, 291:403–541, 2016.
  35. J. P. May and J. Sigurdsson. Parametrized homotopy theory, volume 132 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2006.
  36. On topological cyclic homology. Acta Mathematica, 221(2):203–409, 2018.
  37. Shichirô Oka. Multiplications on the Moore spectrum. Mem. Fac. Sci. Kyushu Univ. Ser. A, 38(2):257–276, 1984.
  38. Peter Scholze. What are the potential applications of perfectoid spaces to homotopy theory? MathOverflow. URL:https://mathoverflow.net/q/386521 (version: 2021-03-15).
  39. The Stacks Project Authors. Stacks Project. https://stacks.math.columbia.edu, 2018.
  40. Twisted Morava K-theory and E-theory. J. Topol., 8(4):887–916, 2015.
  41. Allen Yuan. Integral models for spaces via the higher frobenius. Journal of the American Mathematical Society, 2022.

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 2 tweets with 1 like about this paper.

Sign Up for Free