Topological Hochschild homology

In mathematics, Topological Hochschild homology is a topological refinement of Hochschild homology which rectifies some technical issues with computations in characteristic formula_1. For instance, if we consider the formula_2-algebra formula_3 then formula_4 but if we consider the ring structure on formula_5 (as a divided power algebra structure) then there is a significant technical issue: if we set formula_6, so formula_7, and so on, we have formula_8 from the resolution of formula_3 as an algebra over formula_10, i.e. formula_11 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 formula_3. In contrast, the Topological Hochschild Homology ring has the isomorphism formula_13 giving a less pathological theory. Moreover, this calculation forms the basis of many other THH calculations, such as for smooth algebras formula_14
Construction.
Recall that the Eilenberg–MacLane spectrum can be embed ring objects in the derived category of the integers formula_15 into ring spectrum over the ring spectrum of the stable homotopy group of spheres. This makes it possible to take a commutative ring formula_16 and constructing a complex analogous to the Hochschild complex using the monoidal product in ring spectra, namely, formula_17 acts formally like the derived tensor product formula_18 over the integers. We define the Topological Hochschild complex of formula_16 (which could be a commutative differential graded algebra, or just a commutative algebra) as the simplicial complex, pg 33-34 called the Bar complexformula_20of spectra (note the arrows are incorrect because of wikipedia formatting...). Because simplicial objects in spectra have a realization as a spectrum, we form the spectrumformula_21which has homotopy groups formula_22 defining the topological Hochschild homology of the ring object formula_16.