Topological Hochschild homology

In mathematics, Topological Hochschild homology is a topological refinement of Hochschild homology which rectifies some technical issues with computations in characteristic. For instance, if we consider the -algebra then

but if we consider the ring structure on

then there is a significant technical issue: if we set, so, and so on, we have from the resolution of as an algebra over, i.e.

This calculation is further elaborated on the Hochschild homology page, but the key point is the pathological behavior of the ring structure on the Hochschild homology of. In contrast, the Topological Hochschild Homology ring has the isomorphism

giving a less pathological theory. Moreover, this calculation forms the basis of many other THH calculations, such as for smooth algebras

Construction

Recall that the Eilenberg–MacLane spectrum can be embed ring objects in the derived category of the integers into ring spectrum over the ring spectrum of the stable homotopy group of spheres. This makes it possible to take a commutative ring and constructing a complex analogous to the Hochschild complex using the monoidal product in ring spectra, namely, acts formally like the derived tensor product over the integers. We define the Topological Hochschild complex of as the simplicial complex, ^{pg 33-34} called the Bar complex

of spectra. Because simplicial objects in spectra have a realization as a spectrum, we form the spectrum

which has homotopy groups defining the topological Hochschild homology of the ring object.