## Algebraic & Geometric Topology

### A May-type spectral sequence for higher topological Hochschild homology

#### Abstract

Given a filtration of a commutative monoid $A$ in a symmetric monoidal stable model category $C$, we construct a spectral sequence analogous to the May spectral sequence whose input is the higher order topological Hochschild homology of the associated graded commutative monoid of $A$, and whose output is the higher order topological Hochschild homology of $A$. We then construct examples of such filtrations and derive some consequences: for example, given a connective commutative graded ring $R$, we get an upper bound on the size of the $THH$–groups of $E ∞$–ring spectra $A$ such that $π ∗ ( A ) ≅ R$.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 5 (2018), 2593-2660.

Dates
Revised: 28 January 2018
Accepted: 3 March 2018
First available in Project Euclid: 30 August 2018

https://projecteuclid.org/euclid.agt/1535594418

Digital Object Identifier
doi:10.2140/agt.2018.18.2593

Mathematical Reviews number (MathSciNet)
MR3848395

Zentralblatt MATH identifier
06935816

#### Citation

Angelini-Knoll, Gabe; Salch, Andrew. A May-type spectral sequence for higher topological Hochschild homology. Algebr. Geom. Topol. 18 (2018), no. 5, 2593--2660. doi:10.2140/agt.2018.18.2593. https://projecteuclid.org/euclid.agt/1535594418

#### References

• G Angelini-Knoll, On topological Hochschild homology of the $K(1)$–local sphere, preprint (2016)
• G Angelini-Knoll, A Salch, Maps of simplicial spectra whose realizations are cofibrations, preprint (2016)
• V Angeltveit, On the algebraic $K$–theory of Witt vectors of finite length, preprint (2011)
• V Angeltveit, M A Hill, T Lawson, Topological Hochschild homology of $\ell$ and $ko$, Amer. J. Math. 132 (2010) 297–330
• C Ausoni, Topological Hochschild homology of connective complex $K$–theory, Amer. J. Math. 127 (2005) 1261–1313
• M Basterra, André–Quillen cohomology of commutative $S$–algebras, J. Pure Appl. Algebra 144 (1999) 111–143
• M Bayeh, K Hess, V Karpova, M Kędziorek, E Riehl, B Shipley, Left-induced model structures and diagram categories, from “Women in topology: collaborations in homotopy theory” (M Basterra, K Bauer, K Hess, B Johnson, editors), Contemp. Math. 641, Amer. Math. Soc., Providence, RI (2015) 49–81
• J M Boardman, Conditionally convergent spectral sequences, from “Homotopy invariant algebraic structures” (J-P Meyer, J Morava, W S Wilson, editors), Contemp. Math. 239, Amer. Math. Soc., Providence, RI (1999) 49–84
• M Bökstedt, The topological Hochschild homology of $\mathbb{Z}$ and of $\mathbb{Z}/p\mathbb{Z}$, unpublished manuscript (1987)
• A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972)
• M Brun, Topological Hochschild homology of ${\bf Z}/p^n$, J. Pure Appl. Algebra 148 (2000) 29–76
• M Brun, Filtered topological cyclic homology and relative $K$–theory of nilpotent ideals, Algebr. Geom. Topol. 1 (2001) 201–230
• H Cartan, S Eilenberg, Homological algebra, Princeton Univ. Press (1956)
• B Day, Construction of biclosed categories, PhD thesis, University of New South Wales (1970) Available at \setbox0\makeatletter\@url http://www.math.mq.edu.au/~street/DayPhD.pdf {\unhbox0
• D Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 177–201
• D Dugger, Multiplicative structures on homotopy spectral sequences, II, preprint (2003)
• D Dugger, A primer on homotopy colimits, unpublished manuscript (2008) Available at \setbox0\makeatletter\@url http://math.uoregon.edu/~ddugger/hocolim.pdf {\unhbox0
• W G Dwyer, D M Kan, Function complexes in homotopical algebra, Topology 19 (1980) 427–440
• S Eilenberg, N E Steenrod, Axiomatic approach to homology theory, Proc. Nat. Acad. Sci. U. S. A. 31 (1945) 117–120
• A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, Amer. Math. Soc., Providence, RI (1997)
• N Gambino, Weighted limits in simplicial homotopy theory, J. Pure Appl. Algebra 214 (2010) 1193–1199
• O Gwilliam, D Pavlov, Enhancing the filtered derived category, preprint (2016)
• J E Harper, Bar constructions and Quillen homology of modules over operads, Algebr. Geom. Topol. 10 (2010) 87–136
• M Hovey, Model categories, Mathematical Surveys and Monographs 63, Amer. Math. Soc., Providence, RI (1999)
• M Hovey, Some spectral sequences in Morava $E$–theory, unpublished manuscript (2004) Available at \setbox0\makeatletter\@url http://hopf.math.purdue.edu/Hovey/morava-E-SS.pdf {\unhbox0
• M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149–208
• S B Isaacson, Cubical homotopy theory and monoidal model categories, PhD thesis, Harvard University (2009) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/304891860 {\unhbox0
• G M Kelly, Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series 64, Cambridge Univ. Press (1982)
• J-L Loday, Opérations sur l'homologie cyclique des algèbres commutatives, Invent. Math. 96 (1989) 205–230
• S Mac Lane, Homology, Grundl. Math. Wissen. 114, Academic, New York (1963)
• S Mac Lane, Categories for the working mathematician, 2nd edition, Graduate Texts in Mathematics 5, Springer (1998)
• M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441–512
• J P May, The cohomology of restricted Lie algebras and of Hopf algebras: application to the Steenrod algebra, PhD thesis, Princeton University (1964) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/302273947 {\unhbox0
• J P May, The additivity of traces in triangulated categories, Adv. Math. 163 (2001) 34–73
• J McClure, R Schwänzl, R Vogt, $\mathrm{THH}(R)\cong R\otimes S^1$ for $E_\infty$ ring spectra, J. Pure Appl. Algebra 121 (1997) 137–159
• J E McClure, R E Staffeldt, On the topological Hochschild homology of $bu$, I, Amer. J. Math. 115 (1993) 1–45
• T Pirashvili, Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup. 33 (2000) 151–179
• D G Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer (1967)
• C Reedy, Homotopy theory of model categories, unpublished manuscript (1973) Available at \setbox0\makeatletter\@url http://www-math.mit.edu/~psh/reedy.pdf {\unhbox0
• R Schwänzl, R M Vogt, F Waldhausen, Topological Hochschild homology, J. London Math. Soc. 62 (2000) 345–356
• S Schwede, $S$–modules and symmetric spectra, Math. Ann. 319 (2001) 517–532
• S Schwede, An untitled book project about symmetric spectra, book project (2007) Available at \setbox0\makeatletter\@url http://www.math.uni-bonn.de/people/schwede/SymSpec.pdf {\unhbox0
• S Schwede, B E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000) 491–511
• D White, Model structures on commutative monoids in general model categories, J. Pure Appl. Algebra 221 (2017) 3124–3168