Quadratic forms classify products on quotient ring spectra

We construct a free and transitive action of the group of bilinear forms $Bil (I ∕ I^{2} [1])$ on the set of $R$ –products on $F$ , a regular quotient of an even $E_{\infty}$ –ring spectrum $R$ with $F_{*} ≅ R_{*} ∕ I$ . We show that this action induces a free and transitive action of the group of quadratic forms $QF (I ∕ I^{2} [1])$ on the set of equivalence classes of $R$ –products on $F$ . The characteristic bilinear form of $F$ introduced by the authors in a previous paper is the natural obstruction to commutativity of $F$ . We discuss the examples of the Morava $K$ –theories $K (n)$ and the $2$ –periodic Morava $K$ –theories $K_{n}$ .

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Keywords

structured ring spectra, Bockstein operation, Morava $K$–theory, stable homotopy theory, derived category

Mathematical Subject Classification 2010

Primary: 55P42, 55P43, 55U20

Secondary: 18E30

References

Publication

Received: 9 March 2011

Accepted: 24 February 2012

Published: 23 June 2012

Authors

Alain Jeanneret
Mathematisches Institut Sidlerstrasse 5 CH-3012 Berne Switzerland

Samuel Wüthrich
SBB Brückfeldstrasse 16 CH-3000 Bern Switzerland

Volume 12, issue 3 (2012)

Alain Jeanneret and Samuel Wüthrich

Abstract