Topological Hochschild homology

Summary

Topological Hochschild homology, written $THH$, is an analogue of Hochschild homology. In it, tensor products over the integers are replaced by smash products over the sphere spectrum. It is defined for ring spectra, and in particular for ordinary rings. It carries an action of the circle group, and from it one builds topological cyclic homology. Through the cyclotomic trace it is one of the main tools to compute algebraic K-theory.