Slices of the special linear algebraic cobordism spectrum

Abstract

Let $F$ be a field of exponential characteristic $e$. We compute the slices of $\mathbf{MSL}[e^{-1}]$, where $\mathbf{MSL}$ is the special linear algebraic cobordism spectrum defined by Panin and Walter. The answer is expressed in terms of the second page of the Adams-Novikov spectral sequence for the special unitary cobordism spectrum, which was explicitly determined by Novikov. Its applicability is demonstrated by computations with the slice spectral sequence for $\mathbf{MSL}$, which determine the first few Milnor-Witt stems of its homotopy groups (up to the third) in terms of very effective hermitian $K$-theory. We also establish a decomposition of the rational special linear algebraic cobordism spectrum over an arbitrary qcqs scheme.