Quadratic forms classify products on quotient ring spectra (Q441083)

In this paper the authors consider the question: Given a commutative ring spectrum \(R\) and an \(R\)-module \(F\), how many maps \(\mu : F \wedge F \to F\) are there such that \((F, \mu)\) is an \(R\)-algebra? Such a \(\mu\) will be called an ``\(R\)-product''.NEWLINENEWLINEThe authors work in the following context. The graded ring \(R_*\) of the homotopy groups of \(R\) is assumed to be concentrated in even degrees and is required to be a domain. The spectrum \(F\) will be an \(R\)-module, with a map \(R \to F\) inducing a surjection on homotopy groups, so \(F_* \cong R_*/I\) for some ideal \(I\). Furthermore, the ideal \(I\) is assumed to be generated by a regular sequence.NEWLINENEWLINEOne can view the set of \(R\)-products \(\mu : F \wedge F \to F\) as a subset of the set of maps from \(F \wedge F\) to \(F\) in the derived category of \(R\). Two \(R\)-products are said to be equivalent if they produce \(R\)-algebras which are isomorphic in the derived category of \(R\).NEWLINENEWLINEThe primary result of this paper is the construction of a (natural) free and transitive action of the abelian group of bilinear forms on \(I/I^2[1]\) on this set. This action then induces a free and transitive action of the abelian group of quadratic forms on \(I/I^2[1]\) on the set of equivalence classes of \(R\)-products.NEWLINENEWLINEThis result leads to a number of useful immediate consequences, such as: if \(2 \in F_*\) is invertible then there is a unique commutative product on \(F\), up to equivalence. The authors end the paper with a discussion of how this theorem applies to the Morava \(K\)-theories \(K(n)\) and their two-periodic version \(K_n\).