Power operations modulo Lubin-Tate parameters

Abstract

Power operations provide an algebraic approach to understanding $\E_{\infty}$-rings. In this thesis, we work in the chromatic setting and consider power operations on $K(h)$-local $\E_{\infty}$-$E$-algebras wbere $E$ is a Morava $E$-theory for height $h$. For each $0\leq i\leq h$, we consider a subquotient of power operations acting on $\pi_*(-/p,\cdots, u_{i-1})$ of $K(h)$-local $\E_{\infty}$-$E$-algebras and show that it is Koszul of length $h-i+1$ i.e. that its Koszul complex has length $h-i+1$. We show that at the prime 2, there are cofiber sequences relating power operations mod $p,\cdots, u_{i-1}$ for different $i$'s, which allows an inductive understanding of the structure of operations. We use this to gain insight into the $E$-theory of configuration spaces on $\R^n$, equivalently, of free $\E_n$-algebras on spheres. In particular, we inductively show that certain Tor groups over the algebra of power operations vanish in nonzero degrees. These Tor groups are the linearization of the $E_2$-page of a bar spectral sequence converging to the graded $E$-cohomology of configuration spaces on $\R^n$. This is joint work with Andrew Senger.