Geometry & Topology

Formal groups and stable homotopy of commutative rings

Stefan Schwede

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We explain a new relationship between formal group laws and ring spectra in stable homotopy theory. We study a ring spectrum denoted DB 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:HDB 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 and DB. That description involves formal group law data and the homotopy units of the ring spectrum DB.

Article information

Geom. Topol., Volume 8, Number 1 (2004), 335-412.

Received: 12 July 2003
Revised: 12 February 2004
Accepted: 30 January 2004
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55U35: Abstract and axiomatic homotopy theory
Secondary: 14L05: Formal groups, $p$-divisible groups [See also 55N22]

ring spectrum formal group law André–Quillen homology


Schwede, Stefan. Formal groups and stable homotopy of commutative rings. Geom. Topol. 8 (2004), no. 1, 335--412. doi:10.2140/gt.2004.8.335.

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