A spectrum-level Hodge filtration on topological Hochschild homology

Abstract

We define a functorial spectrum-level filtration on the topological Hochschild homology of any commutative ring spectrum R, and more generally the factorization homology $$R \otimes X$$ for any space X, echoing algebraic constructions of Loday and Pirashvili. We give a geometric description of this filtration, investigate its multiplicative properties, and show that it breaks $${\text {THH}}$$ up into common eigenspectra of the Adams operations.