Topological Hochschild homology

In mathematics, Topological Hochschild homology is a topological refinement of Hochschild homology which rectifies some technical issues with computations in characteristic $$p$$. For instance, if we consider the $$\mathbb{Z}$$-algebra $$\mathbb{F}_p$$ then $$HH_k(\mathbb{F}_p/\mathbb{Z}) \cong \begin{cases} \mathbb{F}_p & k \text{ even} \\ 0 & k \text{ odd} \end{cases}$$ but if we consider the ring structure on $$\begin{align} HH_*(\mathbb{F}_p/\mathbb{Z}) &= \mathbb{F}_p\langle u \rangle \\ &= \mathbb{F}_p[u,u^2/2!, u^3/3!,\ldots]

\end{align}$$ (as a divided power algebra structure) then there is a significant technical issue: if we set $$u \in HH_2(\mathbb{F}_p/\mathbb{Z})$$, so $$u^2 \in HH_4(\mathbb{F}_p/\mathbb{Z})$$, and so on, we have $$u^p = 0$$ from the resolution of $$\mathbb{F}_p$$ as an algebra over $$\mathbb{F}_p\otimes^\mathbf{L}\mathbb{F}_p$$, i.e. $$HH_k(\mathbb{F}_p/\mathbb{Z}) = H_k(\mathbb{F}_p\otimes_{   \mathbb{F}_p\otimes^\mathbf{L}\mathbb{F}_p }\mathbb{F}_p)$$ This calculation is further elaborated on the Hochschild homology page, but the key point is the pathological behavior of the ring structure on the Hochschild homology of $$\mathbb{F}_p$$. In contrast, the Topological Hochschild Homology ring has the isomorphism "$THH_*(\mathbb{F}_p) = \mathbb{F}_p[u]$"giving a less pathological theory. Moreover, this calculation forms the basis of many other THH calculations, such as for smooth algebras $$A/\mathbb{F}_p$$

Construction
Recall that the Eilenberg–MacLane spectrum can be embed ring objects in the derived category of the integers $$D(\mathbb{Z})$$ into ring spectrum over the ring spectrum of the stable homotopy group of spheres. This makes it possible to take a commutative ring $$A$$ and constructing a complex analogous to the Hochschild complex using the monoidal product in ring spectra, namely, $$\wedge_\mathbb{S}$$ acts formally like the derived tensor product $$\otimes^\mathbf{L}$$ over the integers. We define the Topological Hochschild complex of $$A$$ (which could be a commutative differential graded algebra, or just a commutative algebra) as the simplicial complex, pg 33-34 called the Bar complex $$\cdots \to HA\wedge_\mathbb{S}HA\wedge_\mathbb{S}HA \to HA\wedge_\mathbb{S}HA \to HA $$ of spectra (note that the arrows are incorrect because of Wikipedia formatting...). Because simplicial objects in spectra have a realization as a spectrum, we form the spectrum"$THH(A) \in \text{Spectra}$"which has homotopy groups $$\pi_i(THH(A)) $$ defining the topological Hochschild homology of the ring object $$A$$.