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

${\begin{aligned}HH_{*}(\mathbb {F} _{p}/\mathbb {Z} )&=\mathbb {F} _{p}\langle u\rangle \\&=\mathbb {F} _{p}[u,u^{2}/2!,u^{3}/3!,\ldots ]\end{aligned}}$

(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 $\mathbb {F} _{p}\otimes ^{\mathbf {L} }\mathbb {F} _{p}$ ,^[1] 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,^[2] ^{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 $A$ .

Construction

See also