Quadratic forms classify products on quotient ring spectra

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 $K_n$.