#### Volume 15, issue 4 (2015)

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Completed power operations for Morava $E$–theory

### Tobias Barthel and Martin Frankland

Algebraic & Geometric Topology 15 (2015) 2065–2131
##### Abstract

We construct and study an algebraic theory which closely approximates the theory of power operations for Morava $E\phantom{\rule{0.3em}{0ex}}$–theory, extending previous work of Charles Rezk in a way that takes completions into account. These algebraic structures are made explicit in the case of $K\phantom{\rule{0.3em}{0ex}}$–theory. Methodologically, we emphasize the utility of flat modules in this context, and prove a general version of Lazard’s flatness criterion for module spectra over associative ring spectra.

##### Keywords
power operation, Morava $E$–theory, Dyer–Lashof, completion, $L$–complete
##### Mathematical Subject Classification 2010
Primary: 55S25
Secondary: 55S12, 13B35
##### Publication
Received: 7 March 2014
Revised: 16 October 2014
Accepted: 24 November 2014
Published: 10 September 2015
##### Authors
 Tobias Barthel Max-Planck-Institut für Mathematik Vivatsgasse 7 53111 Bonn Germany Martin Frankland Department of Mathematics University of Western Ontario Middlesex College London, ON N6A 5B7 Canada