## Algebraic & Geometric Topology

### Quadratic forms classify products on quotient ring spectra

#### Abstract

We construct a free and transitive action of the group of bilinear forms $Bil(I∕I2[1])$ on the set of $R$–products on $F$, a regular quotient of an even $E∞$–ring spectrum $R$ with $F∗≅R∗∕I$. We show that this action induces a free and transitive action of the group of quadratic forms $QF(I∕I2[1])$ 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 $Kn$.

#### Article information

Source
Algebr. Geom. Topol., Volume 12, Number 3 (2012), 1405-1441.

Dates
Received: 9 March 2011
Accepted: 24 February 2012
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715403

Digital Object Identifier
doi:10.2140/agt.2012.12.1405

Mathematical Reviews number (MathSciNet)
MR2966691

Zentralblatt MATH identifier
1250.55004

#### Citation

Jeanneret, Alain; Wüthrich, Samuel. Quadratic forms classify products on quotient ring spectra. Algebr. Geom. Topol. 12 (2012), no. 3, 1405--1441. doi:10.2140/agt.2012.12.1405. https://projecteuclid.org/euclid.agt/1513715403

#### References

• V Angeltveit, Topological Hochschild homology and cohomology of $A_ \infty$ ring spectra, Geom. Topol. 12 (2008) 987–1032
• J M Boardman, Stable operations in generalized cohomology, from: “Handbook of algebraic topology”, (I M James, editor), North-Holland, Amsterdam (1995) 585–686
• N Bourbaki, Éléments de mathématique. Première partie: Les structures fondamentales de l'analyse. Livre II: Algèbre. Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind. 1272, Hermann, Paris (1959)
• A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Math. Surveys and Monogr. 47, Amer. Math. Soc. (1997) With an appendix by M Cole
• A Jeanneret, S Wüthrich, Clifford algebras from quotient ring spectra, Manuscripta Math. 136 (2011) 33–63
• C Nassau, On the structure of $P(n)_ \ast P((n))$ for $p=2$. [On the structure of $P(n)_ *(P(n))$ for $p=2$], Trans. Amer. Math. Soc. 354 (2002) 1749–1757
• J Rognes, Galois extensions of structured ring spectra. Stably dualizable groups, Mem. Amer. Math. Soc. 192, no. 898, Amer. Math. Soc. (2008)
• J-P Serre, Local algebra, Springer Monogr. in Math., Springer, Berlin (2000) Translated from the French by C Chin and revised by the author
• N P Strickland, Products on ${\rm MU}$–modules, Trans. Amer. Math. Soc. 351 (1999) 2569–2606
• S Wüthrich, $I$–adic towers in topology, Algebr. Geom. Topol. 5 (2005) 1589–1635