Algebraic & Geometric Topology

Operad bimodules and composition products on André–Quillen filtrations of algebras

Nicholas Kuhn and Luís Pereira

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

If O is a reduced operad in a symmetric monoidal category of spectra (S–modules), an O–algebra I can be viewed as analogous to the augmentation ideal of an augmented algebra. From the literature on topological André–Quillen homology, one can see that such an I admits a canonical (and homotopically meaningful) decreasing O–algebra filtration I I1 I2 I3 satisfying various nice properties analogous to powers of an ideal in a ring.

We more fully develop such constructions in a manner allowing for more flexibility and revealing new structure. With R a commutative S–algebra, an O–bimodule M defines an endofunctor of the category of O–algebras in R–modules by sending such an O–algebra I to M OI. We explore the use of the bar construction as a derived version of this. Letting M run through a decreasing O–bimodule filtration of O itself then yields the augmentation ideal filtration as above. The composition structure of the operad then induces pairings among these bimodules, which in turn induce natural transformations (Ii)j Iij, fitting nicely with previously studied structure.

As a formal consequence, an O–algebra map I Jd induces compatible maps In Jdn for all n. This is an essential tool in the first author’s study of Hurewicz maps for infinite loop spaces, and its utility is illustrated here with a lifting theorem.

Article information

Source
Algebr. Geom. Topol., Volume 17, Number 2 (2017), 1105-1130.

Dates
Received: 21 February 2016
Revised: 20 June 2016
Accepted: 1 August 2016
First available in Project Euclid: 19 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1508431455

Digital Object Identifier
doi:10.2140/agt.2017.17.1105

Mathematical Reviews number (MathSciNet)
MR3623683

Zentralblatt MATH identifier
1362.55008

Subjects
Primary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)
Secondary: 18D50: Operads [See also 55P48]

Keywords
operads Andre–Quillen homology

Citation

Kuhn, Nicholas; Pereira, Luís. Operad bimodules and composition products on André–Quillen filtrations of algebras. Algebr. Geom. Topol. 17 (2017), no. 2, 1105--1130. doi:10.2140/agt.2017.17.1105. https://projecteuclid.org/euclid.agt/1508431455


Export citation

References

  • M Basterra, André–Quillen cohomology of commutative $S$–algebras, J. Pure Appl. Algebra 144 (1999) 111–143
  • A,D Elmendorf, M,A Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006) 163–228
  • J,E Harper, Homotopy theory of modules over operads in symmetric spectra, Algebr. Geom. Topol. 9 (2009) 1637–1680
  • J,E Harper, Bar constructions and Quillen homology of modules over operads, Algebr. Geom. Topol. 10 (2010) 87–136
  • J,E Harper, K Hess, Homotopy completion and topological Quillen homology of structured ring spectra, Geom. Topol. 17 (2013) 1325–1416
  • P,S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, Amer. Math. Soc., Providence, RI (2003)
  • M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149–208
  • N,J Kuhn, Localization of André–Quillen–Goodwillie towers, and the periodic homology of infinite loopspaces, Adv. Math. 201 (2006) 318–378
  • N,J Kuhn, Adams filtration and generalized Hurewicz maps for infinite loopspaces, preprint (2014)
  • N Kuhn, J McCarty, The mod $2$ homology of infinite loopspaces, Algebr. Geom. Topol. 13 (2013) 687–745
  • M,A Mandell, J,P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441–512
  • V Minasian, André–Quillen spectral sequence for $\mathrm{THH}$, Topology Appl. 129 (2003) 273–280
  • L,A Pereira, Goodwillie calculus in the category of operads over a spectral operad, preprint (2015) Available at \setbox0\makeatletter\@url http://www.faculty.virginia.edu/luisalex/Papers/GoodwillieCalculusAlgO.pdf {\unhbox0
  • L,A Pereira, Cofibrancy of operadic constructions in positive symmetric spectra, Homology Homotopy Appl. 18 (2016) 133–168
  • B Shipley, A convenient model category for commutative ring spectra, from “Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$–theory” (P Goerss, S Priddy, editors), Contemp. Math. 346, Amer. Math. Soc., Providence, RI (2004) 473–483