#### Volume 12, issue 3 (2012)

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Quadratic forms classify products on quotient ring spectra

### Alain Jeanneret and Samuel Wüthrich

Algebraic & Geometric Topology 12 (2012) 1405–1441
##### Abstract

We construct a free and transitive action of the group of bilinear forms $Bil\left(I∕{I}^{2}\left[1\right]\right)$ on the set of $R$–products on $F$, a regular quotient of an even ${E}_{\infty }$–ring spectrum $R$ with ${F}_{\ast }\cong {R}_{\ast }∕I$. We show that this action induces a free and transitive action of the group of quadratic forms $QF\left(I∕{I}^{2}\left[1\right]\right)$ 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 $K\left(n\right)$ and the $2$–periodic Morava $K$–theories ${K}_{n}$.

##### Keywords
structured ring spectra, Bockstein operation, Morava $K$–theory, stable homotopy theory, derived category
##### Mathematical Subject Classification 2010
Primary: 55P42, 55P43, 55U20
Secondary: 18E30