Homology of En ring spectra and iterated THH

Basterra, Maria¹ ; Mandell, Michael A²

¹Department of Mathematics, University of New Hampshire, Durham, NH, USA
²Department of Mathematics, Indiana University, Bloomington, IN, USA

Algebraic and Geometric Topology, Tome 11 (2011) no. 2, pp. 939-981

Résumé

We describe an iterable construction of THH for an En ring spectrum. The reduced version is an iterable bar construction and its nth iterate gives a model for the shifted cotangent complex at the augmentation, representing reduced topological Quillen homology of an augmented En algebra.

Détail

DOI : 10.2140/agt.2011.11.939

Keywords: $THH$, bar construction, $E_{n}$ ring spectrum, Quillen homology

Bibliographie

[1] C Ausoni, J Rognes, Algebraic K–theory of topological K–theory, Acta Math. 188 (2002) 1

[2] M Basterra, André–Quillen cohomology of commutative S–algebras, J. Pure Appl. Algebra 144 (1999) 111

[3] J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, 347, Springer (1973)

[4] M Bökstedt, W C Hsiang, I Madsen, The cyclotomic trace and algebraic K–theory of spaces, Invent. Math. 111 (1993) 465

[5] M Brun, G Carlsson, B I Dundas, Covering homology, Adv. Math. 225 (2010) 3166

[6] M Brun, Z Fiedorowicz, R M Vogt, On the multiplicative structure of topological Hochschild homology, Algebr. Geom. Topol. 7 (2007) 1633

[7] G Carlsson, C L Douglas, B I Dundas, Higher topological cyclic homology and the Segal conjecture for tori

[8] G Dunn, Uniqueness of n–fold delooping machines, J. Pure Appl. Algebra 113 (1996) 159

[9] A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, 47, American Mathematical Society (1997)

[10] J Francis, Derived algebraic geometry over En–rings, PhD thesis, MIT (2008)

[11] B Fresse, Iterated bar complexes of E∞ algebras and homology theories, Algebr. Geom. Topol. 11 (2011) 747

[12] L Hesselholt, I Madsen, Cyclic polytopes and the K–theory of truncated polynomial algebras, Invent. Math. 130 (1997) 73

[13] L Hesselholt, I Madsen, On the K–theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997) 29

[14] L Hesselholt, I Madsen, On the K–theory of local fields, Ann. of Math. (2) 158 (2003) 1

[15] M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149

[16] I Kříž, J P May, Operads, algebras, modules and motives, Astérisque (1995)

[17] T Leinster, Homotopy algebras for operads

[18] J Lurie, Derived algebraic geometry VI : Ek algebras

[19] S Macnbsp ;Lane, Categories for the working mathematician, 5, Springer (1971)

[20] M A Mandell, The smash product for derived categories in stable homotopy theory

[21] M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. (3) 82 (2001) 441

[22] J P May, The geometry of iterated loop spaces, 271, Springer (1972)

[23] J P May, Classifying spaces and fibrations, Mem. Amer. Math. Soc. 1 (1975)

[24] J P May, Multiplicative infinite loop space theory, J. Pure Appl. Algebra 26 (1982) 1

[25] T Pirashvili, Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup. (4) 33 (2000) 151

[26] F Waldhausen, Algebraic K–theory of topological spaces II, from: "Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978)", Lecture Notes in Math. 763, Springer (1979) 356