Formal groups and stable homotopy of commutative rings

Schwede, Stefan

doi:10.2140/gt.2004.8.335

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Formal groups and stable homotopy of commutative rings

Stefan Schwede

Geometry & Topology 8 (2004) 335–412

DOI: 10.2140/gt.2004.8.335

arXiv: math.AT/0402372

Abstract

We explain a new relationship between formal group laws and ring spectra in stable homotopy theory. We study a ring spectrum denoted $D B$ which depends on a commutative ring $B$ and is closely related to the topological André–Quillen homology of $B$ . We present an explicit construction which to every 1–dimensional and commutative formal group law $F$ over $B$ associates a morphism of ring spectra $F_{*} : H ℤ \to D B$ from the Eilenberg–MacLane ring spectrum of the integers. We show that formal group laws account for all such ring spectrum maps, and we identify the space of ring spectrum maps between $H ℤ$ and $D B$ . That description involves formal group law data and the homotopy units of the ring spectrum $D B$ .

Keywords

ring spectrum, formal group law, André–Quillen homology

Mathematical Subject Classification 2000

Primary: 55U35

Secondary: 14L05

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Publication

Received: 12 July 2003

Revised: 12 February 2004

Accepted: 30 January 2004

Published: 14 February 2004

Proposed: Bill Dwyer

Seconded: Thomas Goodwillie, Haynes Miller

Authors

Stefan Schwede
Mathematisches Institut Universität Bonn 53115 Bonn Germany

Volume 8, issue 1 (2004)