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 ${𝔽}_p$ then

$H{H}_k({𝔽}_p/\mathbb{Z})\cong \{\begin{array}{cc}{𝔽}_p& k\text{ even}\\ 0& k\text{ odd}\end{array}$

but if we consider the ring structure on

$\begin{array}{cc}H{H}_{\ast }({𝔽}_p/\mathbb{Z})& ={𝔽}_p⟨u⟩\\ & ={𝔽}_p[u,u^2/2!,u^3/3!,\dots ]\end{array}$

(as a divided power algebra structure) then there is a significant technical issue: if we set $u\in H{H}_2({𝔽}_p/\mathbb{Z})$, so $u^2\in H{H}_4({𝔽}_p/\mathbb{Z})$, and so on, we have $u^p=0$ from the resolution of ${𝔽}_p$ as an algebra over ${𝔽}_p{\otimes }^{𝐋}{𝔽}_p$, i.e.

$H{H}_k({𝔽}_p/\mathbb{Z})=H_k({𝔽}_p{\otimes }_{{𝔽}_p{\otimes }^{𝐋}{𝔽}_p}{𝔽}_p)$

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 ${𝔽}_p$. In contrast, the Topological Hochschild Homology ring has the isomorphism

$TH{H}_{\ast }({𝔽}_p)={𝔽}_p[u]$

giving a less pathological theory. Moreover, this calculation forms the basis of many other THH calculations, such as for smooth algebras $A/{𝔽}_p$

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, ${\wedge }_{𝕊}$ acts formally like the derived tensor product ${\otimes }^{𝐋}$ 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

$\cdots \to HA{\wedge }_{𝕊}HA{\wedge }_{𝕊}HA\to HA{\wedge }_{𝕊}HA\to HA$

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 $A$.

See also

• Revisiting THH(F_p)
• Topological cyclic homology of the integers