Topological Hochschild homology

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

${\displaystyle HH_{k}(\mathbb {F} _{p}/\mathbb {Z} )\cong {\begin{cases}\mathbb {F} _{p}&k{\text{ even}}\\0&k{\text{ odd}}\end{cases}}}$

but if we consider the ring structure on

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

(as a divided power algebra structure) then there is a significant technical issue: if we set ${\displaystyle u\in HH_{2}(\mathbb {F} _{p}/\mathbb {Z} )}$, so ${\displaystyle u^{2}\in HH_{4}(\mathbb {F} _{p}/\mathbb {Z} )}$, and so on, we have ${\displaystyle u^{p}=0}$ from the resolution of ${\displaystyle \mathbb {F} _{p}}$ as an algebra over ${\displaystyle \mathbb {F} _{p}\otimes ^{\mathbf {L} }\mathbb {F} _{p}}$, ^[1] i.e.

${\displaystyle HH_{k}(\mathbb {F} _{p}/\mathbb {Z} )=H_{k}(\mathbb {F} _{p}\otimes _{\mathbb {F} _{p}\otimes ^{\mathbf {L} }\mathbb {F} _{p}}\mathbb {F} _{p})}$

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 ${\displaystyle \mathbb {F} _{p}}$. In contrast, the Topological Hochschild Homology ring has the isomorphism

${\displaystyle THH_{*}(\mathbb {F} _{p})=\mathbb {F} _{p}[u]}$

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

Construction

Recall that the Eilenberg–MacLane spectrum can be embed ring objects in the derived category of the integers ${\displaystyle D(\mathbb {Z} )}$ into ring spectrum over the ring spectrum of the stable homotopy group of spheres. This makes it possible to take a commutative ring ${\displaystyle A}$ and constructing a complex analogous to the Hochschild complex using the monoidal product in ring spectra, namely, ${\displaystyle \wedge _{\mathbb {S} }}$ acts formally like the derived tensor product ${\displaystyle \otimes ^{\mathbf {L} }}$ over the integers. We define the Topological Hochschild complex of ${\displaystyle A}$ (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

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

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

${\displaystyle THH(A)\in {\text{Spectra}}}$

which has homotopy groups ${\displaystyle \pi _{i}(THH(A))}$ defining the topological Hochschild homology of the ring object ${\displaystyle A}$.

See also

1. Hesselholt, Lars; Nikolaus, Thomas. "Lectures on Topological Hochschild Homology and Cyclotomic Spectra".
2. Morrow, Matthew. "Topological Hochschild homology in arithmetic geometry" (PDF). Archived (PDF) from the original on 24 Dec 2020.

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