Topological Hochschild homology explained

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 Z -algebra F _{p} then

HH _{k(F} _{p/Z)} \cong \begin{cases} F _{p} & k even \\ 0 & k odd \end{cases}

but if we consider the ring structure on

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

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

HH _{k(F} _{p/Z)} = H _{k(F} _{p ⊗} ^{F ^{L F} _{p}} 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 F _{p} . In contrast, the Topological Hochschild Homology ring has the isomorphism

THH _{*(F} _{p)} = 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/ F _{p}

Construction

Recall that the Eilenberg–MacLane spectrum can be embed ring objects in the derived category of the integers

D( 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 _{S} acts formally like the derived tensor product ⊗ ^{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

… \to HA\wedge _{SHA\wedge} _{SHA} \to HA\wedge _{SHA} \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 Spectra

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

See also

1. Web site: Hesselholt. Lars. Nikolaus. Thomas. Lectures on Topological Hochschild Homology and Cyclotomic Spectra.
2. Web site: Morrow. Matthew. Topological Hochschild homology in arithmetic geometry. live. https://web.archive.org/web/20201224194152/https://www.math.arizona.edu/~swc/aws/2019/2019MorrowNotes.pdf. 24 Dec 2020.

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