## Geometry & Topology

### Formal groups and stable homotopy of commutative rings

Stefan Schwede

#### Abstract

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 $F∗:Hℤ→DB$ 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 $Hℤ$ and $DB$. That description involves formal group law data and the homotopy units of the ring spectrum $DB$.

#### Article information

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

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

https://projecteuclid.org/euclid.gt/1513883370

Digital Object Identifier
doi:10.2140/gt.2004.8.335

Mathematical Reviews number (MathSciNet)
MR2033483

Zentralblatt MATH identifier
1056.55012

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

#### Citation

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. https://projecteuclid.org/euclid.gt/1513883370

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