In mathematics, Topological Hochschild homology is a topological refinement of Hochschild homology which rectifies some technical issues with computations in characteristic
For instance, if we consider the
odd
{\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
from the resolution of
as an algebra over
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
In contrast, the Topological Hochschild Homology ring has the isomorphism
giving a less pathological theory.
Moreover, this calculation forms the basis of many other THH calculations, such as for smooth algebras
Recall that the Eilenberg–MacLane spectrum can be embed ring objects in the derived category of the integers
into ring spectrum over the ring spectrum of the stable homotopy group of spheres.
This makes it possible to take a commutative ring
and constructing a complex analogous to the Hochschild complex using the monoidal product in ring spectra, namely,
acts formally like the derived tensor product
over the integers.
We define the Topological Hochschild complex of
(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
Spectra
{\displaystyle THH(A)\in {\text{Spectra}}}
which has homotopy groups
π
{\displaystyle \pi _{i}(THH(A))}
defining the topological Hochschild homology of the ring object