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 $\mathbb$-algebra $\mathbb_p$ then

$HH_k(\mathbb_p/\mathbb) \cong \begin \mathbb_p & k \text \\ 0 & k \text \end$

but if we consider the ring structure on

\begin
HH_*(\mathbb_p/\mathbb) &= \mathbb_p\langle u \rangle \\
&= \mathbb_p ,u^2/2!, u^3/3!,\ldots
\end

(as a divided power algebra structure) then there is a significant technical issue: if we set $u \in HH_2(\mathbb_p/\mathbb)$, so $u^2 \in HH_4(\mathbb_p/\mathbb)$, and so on, we have $u^p = 0$ from the resolution of $\mathbb_p$ as an algebra over $\mathbb_p\otimes^\mathbf\mathbb_p$, i.e.

$HH_k(\mathbb_p/\mathbb) = H_k(\mathbb_p\otimes_\mathbb_p)$

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 $\mathbb_p$. In contrast, the Topological Hochschild Homology ring has the isomorphism

THH_*(\mathbb_p) = \mathbb_p /math>

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

Construction

Recall that the Eilenberg–MacLane spectrum can be embed ring objects in the derived category of the integers $D(\mathbb)$ 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, $\wedge_\mathbb$ acts formally like the derived tensor product $\otimes^\mathbf$ 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

$\cdots \to HA\wedge_\mathbbHA\wedge_\mathbbHA \to HA\wedge_\mathbbHA \to HA$

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

Homological algebra
Algebraic topology