Clifford algebras from quotient ring spectra

TL;DR

This paper describes how the homology and cohomology of regular quotient ring spectra of even E-infinity ring spectra can be characterized using Clifford and exterior algebras, with detailed examples including Morava K-theories.

Contribution

It provides natural descriptions of homology and cohomology algebras of quotient ring spectra, linking them to Clifford and exterior algebra structures, and includes detailed analysis of Morava K-theories.

Findings

Homology is a Clifford algebra with respect to a natural bilinear form.

Cohomology is isomorphic to an exterior algebra on derivations.

Detailed example analysis of Morava K-theories.

Abstract

We give natural descriptions of the homology and cohomology algebras of regular quotient ring spectra of even E-infinity ring spectra. We show that the homology is a Clifford algebra with respect to a certain bilinear form naturally associated to the quotient ring spectrum F. To identify the cohomology algebra, we first determine the derivations of F and then prove that the cohomology is isomorphic to the exterior algebra on the module of derivations. We treat the example of the Morava K-theories in detail.