Commutative $\mathbb{S}$–algebras of prime characteristics and applications to unoriented bordism

Abstract

The notion of highly structured ring spectra of prime characteristic is made precise and is studied via the versal examples $\mathbb{S}∕∕p$ for prime numbers $p$. These can be realized as Thom spectra, and therefore relate to other Thom spectra such as the unoriented bordism spectrum $MO$. We compute the Hochschild and André–Quillen invariants of the $\mathbb{S}∕∕p$. Among other applications, we show that $\mathbb{S}∕∕p$ is not a commutative algebra over the Eilenberg–Mac Lane spectrum $H{\mathbb{F}}_p$, although the converse is clearly true, and that $MO$ is not a polynomial algebra over $\mathbb{S}∕∕2$.