## Algebraic $K$-theory of $\operatorname {THH}(\mathbb {F}_p)$

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- by Haldun Özgür Bayındır and Tasos Moulinos PDF
- Trans. Amer. Math. Soc.
**375**(2022), 4177-4207 Request permission

## Abstract:

In this work we study the $E_{\infty }$-ring $\operatorname {THH}(\mathbb {F}_p)$ as a graded spectrum. Following an identification at the level of $E_2$-algebras with $\mathbb {F}_p[\Omega S^3]$, the group ring of the $E_1$-group $\Omega S^3$ over $\mathbb {F}_p$, we show that the grading on $\operatorname {THH}(\mathbb {F}_p)$ arises from decomposition on the cyclic bar construction of the pointed monoid $\Omega S^3$. This allows us to use trace methods to compute the algebraic $K$-theory of $\operatorname {THH}(\mathbb {F}_p)$. We also show that as an $E_2$ $H\mathbb {F}_p$-ring, $\operatorname {THH}(\mathbb {F}_p)$ is uniquely determined by its homotopy groups. These results hold in fact for $\operatorname {THH}(k)$, where $k$ is any perfect field of characteristic $p$. Along the way we expand on some of the methods used by Hesselholt-Madsen and later by Speirs to develop certain tools to study the THH of graded ring spectra and the algebraic $K$-theory of formal DGAs.## References

- Benjamin Antieau, David Gepner, and Jeremiah Heller,
*$K$-theoretic obstructions to bounded $t$-structures*, Invent. Math.**216**(2019), no. 1, 241–300. MR**3935042**, DOI 10.1007/s00222-018-00847-0 - Gabe Angelini-Knoll,
*Detecting the $\beta$-family in iterated algebraic K-theory of finite fields*, Preprint, arXiv:1810.10088, 2018. - Benjamin Antieau, Akhil Mathew, Matthew Morrow, and Thomas Nikolaus,
*On the Beilinson fiber square*, Preprint, arXiv:2003.12541, 2020. - Haldun Özgür Bayındır,
*Topological equivalences of $E$-infinity differential graded algebras*, Algebr. Geom. Topol.**18**(2018), no. 2, 1115–1146. MR**3773750**, DOI 10.2140/agt.2018.18.1115 - Haldun Özgür Bayındır,
*Extension DGAs and topological Hochschild homology*, Preprint, arXiv:1911.13183, 2019. - Haldun Özgür Bayındır,
*DGAs with polynomial homology*, Adv. Math.**389**(2021), Paper No. 107907, 58. MR**4289043**, DOI 10.1016/j.aim.2021.107907 - M. A. Batanin and C. Berger,
*Homotopy theory for algebras over polynomial monads*, Theory Appl. Categ.**32**(2017), Paper No. 6, 148–253. MR**3607212** - M. Bökstedt, W. C. Hsiang, and I. Madsen,
*The cyclotomic trace and algebraic $K$-theory of spaces*, Invent. Math.**111**(1993), no. 3, 465–539. MR**1202133**, DOI 10.1007/BF01231296 - Maria Basterra and Michael A. Mandell,
*Homology of $E_n$ ring spectra and iterated $THH$*, Algebr. Geom. Topol.**11**(2011), no. 2, 939–981. MR**2782549**, DOI 10.2140/agt.2011.11.939 - Maria Basterra and Michael A. Mandell,
*The multiplication on BP*, J. Topol.**6**(2013), no. 2, 285–310. MR**3065177**, DOI 10.1112/jtopol/jts032 - Bhargav Bhatt, Matthew Morrow, and Peter Scholze,
*Topological Hochschild homology and integral $p$-adic Hodge theory*, Publ. Math. Inst. Hautes Études Sci.**129**(2019), 199–310. MR**3949030**, DOI 10.1007/s10240-019-00106-9 - Haldun Özgür Bayındır and Maximilien Péroux,
*Spanier-Whitehead duality for topological coHochschild homology*, Preprint, arXiv:2012.03966, 2020. - F. R. Cohen, J. P. May, and L. R. Taylor,
*Splitting of certain spaces $CX$*, Math. Proc. Cambridge Philos. Soc.**84**(1978), no. 3, 465–496. MR**503007**, DOI 10.1017/S0305004100055298 - Ralph L. Cohen,
*A model for the free loop space of a suspension*, Algebraic topology (Seattle, Wash., 1985) Lecture Notes in Math., vol. 1286, Springer, Berlin, 1987, pp. 193–207. MR**922928**, DOI 10.1007/BFb0078743 - Brian Day,
*On closed categories of functors*, Reports of the Midwest Category Seminar, IV, Lecture Notes in Mathematics, Vol. 137, Springer, Berlin, 1970, pp. 1–38. MR**0272852** - A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May,
*Rings, modules, and algebras in stable homotopy theory*, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole. MR**1417719**, DOI 10.1090/surv/047 - Lars Hesselholt,
*The big de Rham-Witt complex*, Acta Math.**214**(2015), no. 1, 135–207. MR**3316757**, DOI 10.1007/s11511-015-0124-y - Lars Hesselholt and Ib Madsen,
*Cyclic polytopes and the $K$-theory of truncated polynomial algebras*, Invent. Math.**130**(1997), no. 1, 73–97. MR**1471886**, DOI 10.1007/s002220050178 - Lars Hesselholt and Ib Madsen,
*On the $K$-theory of finite algebras over Witt vectors of perfect fields*, Topology**36**(1997), no. 1, 29–101. MR**1410465**, DOI 10.1016/0040-9383(96)00003-1 - Lars Hesselholt and Ib Madsen,
*On the $K$-theory of local fields*, Ann. of Math. (2)**158**(2003), no. 1, 1–113. MR**1998478**, DOI 10.4007/annals.2003.158.1 - Achim Krause and Thomas Nikolaus,
*Bökstedt periodicity and quotients of DVRs*, Preprint, arXiv:1907.03477, 2019. - Markus Land, Lennart Meier, and Georg Tamme,
*Vanishing results for chromatic localizations of algebraic $K$-theory*, Preprint, arXiv:2001.10425, 2020. - Jacob Lurie,
*Higher topos theory*, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR**2522659**, DOI 10.1515/9781400830558 - Jacob Lurie,
*Rotation invariance in algebraic K-theory*, Preprint (2015). - Jacob Lurie,
*Higher algebra*, Preprint https://www.math.ias.edu/~lurie/papers/HA.pdf (2017). - Tasos Moulinos,
*The geometry of filtrations*, Bull. Lond. Math. Soc.**53**(2021), no. 5, 1486–1499. MR**4335221**, DOI 10.1112/blms.12512 - Thomas Nikolaus and Peter Scholze,
*On topological cyclic homology*, Acta Math.**221**(2018), no. 2, 203–409. MR**3904731**, DOI 10.4310/ACTA.2018.v221.n2.a1 - Daniel Quillen,
*On the (co-) homology of commutative rings*, Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 65–87. MR**0257068** - Charles Rezk,
*Notes on the Hopkins-Miller theorem*, Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997) Contemp. Math., vol. 220, Amer. Math. Soc., Providence, RI, 1998, pp. 313–366. MR**1642902**, DOI 10.1090/conm/220/03107 - Noah Riggenbach,
*On the algebraic $K$-theory of double points*, Preprint, arXiv:2007.01227, 2020. - Stefan Schwede,
*Symmetric spectra*, Electronic book, http://www.math.uni-bonn.de/people/schwede/SymSpec-v3.pdf. - Martin Speirs,
*On the $K$-theory of truncated polynomial algebras, revisited*, Adv. Math.**366**(2020), 107083, 18. MR**4070307**, DOI 10.1016/j.aim.2020.107083 - Martin Speirs,
*On the $K$-theory of coordinate axes in affine space*, Algebr. Geom. Topol.**21**(2021), no. 1, 137–171. MR**4224738**, DOI 10.2140/agt.2021.21.137 - Torleif Veen,
*Detecting periodic elements in higher topological Hochschild homology*, Geom. Topol.**22**(2018), no. 2, 693–756. MR**3748678**, DOI 10.2140/gt.2018.22.693 - Charles A. Weibel,
*The $K$-book*, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, RI, 2013. An introduction to algebraic $K$-theory. MR**3076731**, DOI 10.1090/gsm/145

## Additional Information

**Haldun Özgür Bayındır**- Affiliation: Department of Mathematics, City, University of London, London EC1V 0HB, United Kingdom
**Tasos Moulinos**- Affiliation: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne 31062 Toulouse Cedex 9, France
- MR Author ID: 1137307
- Received by editor(s): May 20, 2021
- Received by editor(s) in revised form: October 7, 2021
- Published electronically: March 4, 2022
- Additional Notes: The first author acknowledges support from the project ANR-16-CE40-0003 ChroK. The second author was supported by grant NEDAG ERC-2016-ADG-741501 during the writing of this work
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 4177-4207 - MSC (2020): Primary 55P99, 19D99
- DOI: https://doi.org/10.1090/tran/8613
- MathSciNet review: 4419056