Reading through Bhatt-Morrow-Scholze's ["Topological Hochschild Homology and Integral p-adic Hodge Theory"](https://arxiv.org/abs/1802.03261) I encountered the following statement.



> 
> We use only “formal” properties of THH throughout the paper, with the
>  one exception of Bökstedt’s computation of $THH(\mathbb{F}\_p)$ ...
> 
> 
> 


As a reminder, Bökstedt (and Breen?) computed $\pi\_{\*}THH(\mathbb{F}\_p)$ as $\mathbb{F\_p}[\sigma]$, where $\sigma$ is a polynomial generator in degree 2. Whereas ordinary (derived) Hochschild homology of $\mathbb{F\_p}$ is easily seen to be a divided power algebra generated in the same degree, $\mathbb{F}\_p\langle x\rangle.$ Note that in the latter algebra, writing $x^p = p! \frac{x^p}{p!}$ gives $x^p = 0,$ quite a different flavor than the topological theory.


At some point I looked this up and the computation appeared to rest on some not particularly formal spectral sequence manipulations which I was unable to follow. Is this still the situation? Has anyone revisited these computations recently?



revisiting $THH(\mathbb{F}_p)$
Alright, here is the promised answer. First, the Hopkins-Mahowald theorem states that $\mathbb{F}\_p$ is the free $\mathbb{E}\_2$-ring with $p=0$, i.e. it is the homotopy pushout in $\mathbb{E}\_2$-rings of $S^0 \leftarrow \mathrm{Free}\_{\mathbb{E}\_2}(x) \rightarrow S^0$ where we have $x \mapsto 0$ for the first arrow and $x \mapsto p$ for the second arrow. 
Equivalently (after $p$-localization), if we let $S^1 \to \mathrm{BGL}\_1(S^0\_{(p)})$ be adjoint to $1-p \in \pi\_0(S^0)\_{(p)}^{\times}$, extend to a double loop map $\Omega^2S^3 \to \mathrm{BGL}\_1(S^0\_{(p)})$ and take the Thom spectrum, we get $\mathrm{H}\mathbb{F}\_p$. At the prime 2 there are many references, and the original proof was due to Mahowald- it appears, for example, in his papers "A new infinite family in $\pi\_\*S^0$" and in "Ring spectra which are Thom complexes". At odd primes this is due to Hopkins-Mahowald, but it's hard to trace down the "original" reference- it appears, for example, in Mahowald-Ravenel-Shick "The Thomified Eilenberg-Moore Spectral Sequence" as Lemma 3.3. In both cases the result is stated in the Thom spectrum language. And, again, let me reiterate: this is a **non-formal result** and essentially equivalent to Bokstedt's original computation. One always needs to know Steinberger's result (proved independently in Bokstedt's manuscript) that $Q\_1\tau\_i = \tau\_{i+1}$ mod decomposables in $\mathcal{A}\_\*$, where $Q\_1$ is the top $\mathbb{E}\_2$-operation. (I suppose one could get away with slightly less: one needs to know that $\tau\_0$ generates $\mathcal{A}\_\*$ with $\mathbb{E}\_2$-Dyer-Lashof operations).


Anyway, given this result, let's see how to compute THH. In Blumberg-Cohen-Schlichtkrull (<https://arxiv.org/pdf/0811.0553.pdf>) they explain how to compute THH of any $\mathbb{E}\_1$-Thom spectrum, with increasingly nice answers as we add multiplicative structure. For an $\mathbb{E}\_2$-Thom spectrum (i.e. one arising from a double loop map) we have that $\mathrm{THH}(X^{\xi}) \simeq X^{\xi} \wedge BX^{\eta B\xi}$ where $\eta B\xi$ is the composite $BX \to B(BGL\_1(S^0)) \stackrel{\eta}{\to} BGL\_1(S^0)$. So, in our case we have that $THH(\mathbb{F}\_p) \simeq \mathbb{F}\_p \wedge (\Omega S^3)^{\eta B\xi}$. Thom spectra are colimits, and smash products commute with those, so we can compute this smash product as the Thom spectrum of $BX \to B(BGL\_1(S^0)) \to BGL\_1(S^0) \to BGL\_1(\mathbb{F}\_p)$. But $\eta \mapsto 0$ in $\pi\_\*\mathrm{H}\mathbb{F}\_p$ so this is the trivial bundle, and we deduce the result we're after:


$\mathrm{THH}(\mathbb{F}\_p) \simeq \mathbb{F}\_p \wedge \Omega S^3\_+ \simeq \mathbb{F}\_p[x\_2]$. 




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It's worth giving the same argument again but in a more algebraic way, and, as a bonus, you can basically see why the Blumberg-Cohen-Schlichtkrull result holds. 


The point (which is very well explained in a more general setting in Theorem 5.7 of Klang's paper here: <https://arxiv.org/pdf/1606.03805.pdf>) is that $\mathbb{F}\_p$, as a module over $\mathbb{F}\_p \wedge \mathbb{F}\_p^{op}$, is obtained by extending scalars from $S^0$ as an $S^0[\Omega S^3\_+]$-module. 


Indeed, start with the pushout of $\mathbb{E}\_2$-algebras $S^0 \leftarrow \mathrm{Free}\_{\mathbb{E}\_2}(x) \to S^0$ and smash with $\mathbb{F}\_p$ to get a pushout of $\mathbb{E}\_2$-$\mathbb{F}\_p$-algebras. But now both maps in the pushout are the augmentation, and we see that the map $\mathbb{F}\_p \wedge \mathbb{F}\_p \simeq \mathbb{F}\_p \wedge \mathbb{F}\_p^{op} \to \mathbb{F}\_p$ is equivalent to the map $\mathrm{Free}\_{\mathbb{E}\_2-\mathbb{F}\_p}(\Sigma x) \to \mathbb{F}\_p$ which is just the augmentation. This is tensored up from the map $\mathrm{Free}\_{\mathbb{E}\_2}(\Sigma x) \to S^0$, so we get that $\mathrm{THH}(\mathbb{F}\_p) \simeq \mathbb{F}\_p \otimes\_{\mathbb{F}\_p \wedge \mathbb{F}\_p} \mathbb{F}\_p \simeq S^0 \otimes\_{\mathrm{Free}\_{\mathbb{E}\_2}(\Sigma x)} \mathbb{F}\_p \simeq S^0 \otimes\_{\mathrm{Free}\_{\mathbb{E}\_2}(\Sigma x)} S^0 \otimes\_{S^0} \mathbb{F}\_p$.


The left hand factor is given by $\mathrm{Free}\_{\mathbb{E}\_1}(\Sigma^2x)$, which you can see in various ways. For example, this bar construction is the suspension spectrum of the relative tensor product in spaces $\* \otimes\_{\Omega^2 S^3} \*$, which is the classifying space construction and yields $B(\Omega^2S^3) \simeq \Omega S^3$.




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