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Quadratic forms classify products on quotient ring spectra. (English) Zbl 1250.55004
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”.
The 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.
One 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 $$R$$.
The 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.
This 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 $$K_n$$.

##### MSC:
 55P43 Spectra with additional structure ($$E_\infty$$, $$A_\infty$$, ring spectra, etc.) 55P42 Stable homotopy theory, spectra 55U20 Universal coefficient theorems, Bockstein operator 18E30 Derived categories, triangulated categories (MSC2010)
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