Topological Hochschild homology

In mathematics, Topological Hochschild homology is a topological refinement of Hochschild homology which rectifies some technical issues with computations in characteristic [math]\displaystyle{ p }[/math]. For instance, if we consider the [math]\displaystyle{ \mathbb{Z} }[/math]-algebra [math]\displaystyle{ \mathbb{F}_p }[/math] then

[math]\displaystyle{ HH_k(\mathbb{F}_p/\mathbb{Z}) \cong \begin{cases} \mathbb{F}_p & k \text{ even} \\ 0 & k \text{ odd} \end{cases} }[/math]

but if we consider the ring structure on

[math]\displaystyle{ \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} }[/math]

(as a divided power algebra structure) then there is a significant technical issue: if we set [math]\displaystyle{ u \in HH_2(\mathbb{F}_p/\mathbb{Z}) }[/math], so [math]\displaystyle{ u^2 \in HH_4(\mathbb{F}_p/\mathbb{Z}) }[/math], and so on, we have [math]\displaystyle{ u^p = 0 }[/math] from the resolution of [math]\displaystyle{ \mathbb{F}_p }[/math] as an algebra over [math]\displaystyle{ \mathbb{F}_p\otimes^\mathbf{L}\mathbb{F}_p }[/math], i.e.

[math]\displaystyle{ 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) }[/math]

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 [math]\displaystyle{ \mathbb{F}_p }[/math]. In contrast, the Topological Hochschild Homology ring has the isomorphism

[math]\displaystyle{ THH_*(\mathbb{F}_p) = \mathbb{F}_p[u] }[/math]

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

Construction

Recall that the Eilenberg–MacLane spectrum can be embed ring objects in the derived category of the integers [math]\displaystyle{ D(\mathbb{Z}) }[/math] into ring spectrum over the ring spectrum of the stable homotopy group of spheres. This makes it possible to take a commutative ring [math]\displaystyle{ A }[/math] and constructing a complex analogous to the Hochschild complex using the monoidal product in ring spectra, namely, [math]\displaystyle{ \wedge_\mathbb{S} }[/math] acts formally like the derived tensor product [math]\displaystyle{ \otimes^\mathbf{L} }[/math] over the integers. We define the Topological Hochschild complex of [math]\displaystyle{ A }[/math] (which could be a commutative differential graded algebra, or just a commutative algebra) as the simplicial complex, ^{pg 33-34} called the Bar complex

[math]\displaystyle{ \cdots \to HA\wedge_\mathbb{S}HA\wedge_\mathbb{S}HA \to HA\wedge_\mathbb{S}HA \to HA }[/math]

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

[math]\displaystyle{ THH(A) \in \text{Spectra} }[/math]

which has homotopy groups [math]\displaystyle{ \pi_i(THH(A)) }[/math] defining the topological Hochschild homology of the ring object [math]\displaystyle{ A }[/math].

See also

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