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In many expositions and discussions of Hochschild homology, it is stated that the classical definition via the cyclic bar complex (with underived tensor products) is incorrect (call the obtained Hochschild homology $\mathrm{HH}^{\textrm{cl}}$). To obtain the correct definition, everything in the classical definition must be sufficiently derived (call the obtained Hochschild homology $\mathrm{HH}$). One example which is often cited is that $\mathrm{HH}^{\textrm{cl}}(\Bbb{F}_p/\Bbb{Z})\simeq\Bbb{F}_p,$ while $\mathrm{HH}(\Bbb{F}_p/\Bbb{Z})$ is a divided power algebra over $\Bbb{F}_p$ in one generator. (It is often pointed out that this is also in some sense incorrect, and one should really want $\mathrm{THH}(\Bbb{F}_p) = \mathrm{HH}(\Bbb{F}_p/\Bbb{S}),$ which is polynomial over $\Bbb{F}_p,$ though this is not the main topic of this question.)

From a philosophical perspective, replacing the definition of Hochschild homology with a version where "everything is derived" makes sense. If we want to work with derived/$\infty$-categories, it makes sense to demand that all constructions be suitably homotopy-invariant. However, are there non-philosophical reasons one might object to $\mathrm{HH}^{\textrm{cl}}$?

In particular, is there a concrete reason that the calculation $\mathrm{HH}^{\textrm{cl}}(\Bbb{F}_p/\Bbb{Z})\simeq\Bbb{F}_p$ (or a similar one) is not the answer we want? I'm hoping that there is an answer which is not simply a restatement of the philosophical objection I pointed to above; perhaps something like "$\mathrm{HH}(A/k)$ should do X, but this computation shows that it does not." Ideally, this answer should be convincing to those who were originally studying $\mathrm{HH}$ and demonstrate why it is insufficient, or even just to someone who is not already derived/$\infty$-categorically minded.

Edit: The provided discussions are useful -- if we care about $\mathrm{THH},$ then it makes sense that we would want to consider Shukla homology $\mathrm{HH}$ as $\mathrm{THH}$ is a special instance of this. At the same time, changing "base" from $\Bbb{S}\to\Bbb{Z}$ provides a relationship between $\mathrm{THH}(A)$ and $\mathrm{HH}(A/\Bbb{Z})$ which Qi Zhu points out is an isomorphism in degree $2.$

However, it appears Shukla/derived Hochschild homology came prior to $\mathrm{THH},$ which points to another source as being the motivation for extending the original un-derived Hochschild homology to the "fully derived" setting, and I'm curious what a "minimal" calculation/motivation (if one exists) would be which would lead one to define this. I'm not opposed to considering $\mathrm{THH}$ as motivation for why one would construct Shukla homology, but if we can motivate the construction without it simply by recognizing that some calculation(s) are somehow "wrong" without appealing to $\mathrm{THH},$ that would be ideal.

Tankut Beygu
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Stahl
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  • I'm not sure what you mean. Hochschild homology is pretty unambiguous via a cyclic bar construction or via the specific cyclic complex you mean. The two are equivalent under the Dold-Kan correspondence. The classical reason for it was to develop a notion of kahler differentials for schemes that are not smooth. It also happens to contain group cohomology as a special case. Topological hochschild homology was originally conjectured to exist by the computation that $THH(\Omega X) \simeq \mathcal{L}(X)$. The latter had a simplicial model due to Kan and so Bokstedt went on to define THH following – Andres Mejia Mar 03 '25 at 19:09
  • work of Connes on the cyclic category, but in a setting (FSP/spectra) where everything was suitably derived. The importance of this was because of its relationship to algebraic K-theory, namely that it receives a "trace map" from $K$-theory (and is in some sense universal among things that do) but it is the first Goodwillie derivative of both $K$-theory and $TC$, which tells us that the difference between these two theories is constant. Therefore "relative K-theory" can be reduced to $TC$ computations which are less insane. – Andres Mejia Mar 03 '25 at 19:12
  • It's worth mentioning that the classical Dennis trace $K(R) \to HH(R)$ is often trivial, while $THH$ still detects many elements. I recommend these notes of Matthew Morrow for an introduction that's fairly algebraic and concrete and these talbot notes for a survey for a modern $\infty$-categorical perspective. – Andres Mejia Mar 03 '25 at 19:16
  • @AndresMejia The "classical vs. modern" issue I'm trying to get at here is the issue of flatness of $k\to A.$ That is, $\mathrm{HH}^{\textrm{cl}}(A/k) = A\otimes^{\bf{L}}{A\otimes_k A}A$ and $\mathrm{HH}(A/k) = A\otimes^{\bf{L}}{A\otimes^{\bf{L}}_k A}A $ ($A$ commutative). The trace map being trivial for usual $\mathrm{HH}$ is sort of the type of answer I want, but I want something which points to why one would make the jump to derive the bar complex, not necessarily go all the way to $\mathrm{THH}.$ Perhaps this would be anachronistic, and we knew that we should use $\mathrm{THH}$ – Stahl Mar 03 '25 at 21:05
  • even before developing "Shukla homology," I'd be interested to know if that is the case as well. In a sense, that still leaves the issue of why one would want to consider the intermediate $\mathrm{HH}(A/k)$ rather than just $\mathrm{THH}$ or $\mathrm{HH}^{\textrm{cl}},$ although perhaps it becomes more obvious that $\mathrm{THH}$ should be compared to Shukla homology to get a sensible comparison. – Stahl Mar 03 '25 at 21:07
  • Personally I would put Shukla homology and $\mathsf{THH}$ into the same bucket: One can define $\mathsf{HH}(A/\mathsf{C})$ in any symmetric monoidal $\infty$-category $\mathsf{C}$ together with an algebra $A$ therein. Then, Shukla homology is $\mathsf{HH}(A/\mathcal{D}(\mathbb{Z}))$ and $\mathsf{THH}(R) = \mathsf{HH}(R/\mathsf{Sp})$. But I would guess that historically $\mathsf{THH}$ must come later since brave new algebra was only initiated later and it took everyone so long to find a convenient symmetric monoidal model category for $\mathsf{Sp}$. – Qi Zhu Mar 04 '25 at 08:04
  • @QiZhu topo,ogical hochschild homology was defined before there was a suitable symmetric monoidal structure on spectra. It was done with the more primitive notion “functors with smash product” – Andres Mejia Mar 05 '25 at 13:24
  • @Stahl see section 2.4 here https://swc-math.github.io/aws/2019/2019MorrowNotes.pdf. Especially the failure of 2.28 in the underived setting – Andres Mejia Mar 05 '25 at 13:31
  • @AndresMejia Ah, good to know, thanks for the comment! – Qi Zhu Mar 05 '25 at 13:51
  • @QiZhu no problem :))). History is cool. – Andres Mejia Mar 06 '25 at 08:45

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As I am not so proficient with the theory of (topological) Hochschild homology, there are probably many reasons without passing to the homotopical setting, but here is one important (computational) reason that I can think of.

Let $R$ be a classical ring. Then, one can show that $\mathsf{THH}(R) \to \mathsf{HH}(R)$ is $3$-connected. In particular, $\mathsf{THH}_2(R) \to \mathsf{HH}_2(R)$ is an isomorphism. In the case $R = \mathbb{F}_p$ we deduce $\mathsf{THH}_2(\mathbb{F}_p) \cong \mathsf{HH}_2(\mathbb{F}_p) \cong \mathbb{F}_p$ which could not have been available in the non-derived setting. This yields a generator $x \in \mathsf{THH}_2(\mathbb{F}_p)$ of utmost importance: It allows us to state (and later prove) Bökstedt Periodicity!

Theorem (Bökstedt). There is an isomorphism $\mathsf{THH}_{\bullet}(\mathbb{F}_p) \cong \mathbb{F}_p[x]$.

This theorem then allows us to access $\mathsf{TP}(\mathbb{F}_p), \mathsf{TC}^-(\mathbb{F}_p), \mathsf{TC}(\mathbb{F}_p)$ and so on and is the beginning of the trace method story to attack $K$-theory which is one of the only available promising tools to compute $K$-theory.

In fact, at least according to the experts of the theory like Thomas Nikolaus, this is essentially the only computational result currently available in the theory - every computation in some way comes from this single result. Bökstedt's theorem is the ultimate computational tool in the theory and the beginning of the story was $\mathsf{HH}_2(\mathbb{F}_p)$.

Qi Zhu
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  • Thanks for the answer! Perhaps I need to acquaint myself with the history of the topic a bit better, but do you know which of $\mathrm{THH}$ vs. Shukla homology came first? If it was $\mathrm{THH},$ then the comparison results between $\mathrm{THH}$ and $\mathrm{HH}$ very much make sense as a reason to derive things in the non-flat situation. If Shukla homology came first, then it seems that maybe there's another motivation (or perhaps $\mathrm{THH}$ was secretly always around, but under a different name). – Stahl Mar 03 '25 at 21:11
  • @Stahl I don't really know but it seems like Shukla homology was defined in 1961 in a paper from Shukla while $\mathsf{THH}$ was defined in 1985, 1988 by Bökstedt. I don't really understand your comment why it would make more sense if $\mathsf{THH}$ was defined first it would make more sense, could you elaborate on that? Morally, there is not really a difference between these, as Shukla homology is Hochschild homology in the $\infty$-category $\mathcal{D}(\mathbb{Z})$ over $\mathbb{Z}$ while $\mathsf{THH}$ is Hochschild homology in $\mathsf{Sp}$ over $\mathbb{S}$. – Qi Zhu Mar 04 '25 at 07:56
  • I'm thinking about the motivation: if $\mathrm{THH}$ came first, then having some sort of a comparison with $\mathrm{HH}$ would feel natural, and the ideal comparison would be the one with Shukla homology rather than classical $\mathrm{HH}$. I'm viewing this question partly historically: why did Shukla develop the "derived" variant of $\mathrm{HH},$ and are there "minimal" calculations of $\mathrm{HH}^{\textrm{cl}}$ which one can recognize are deficient? – Stahl Mar 04 '25 at 16:03
  • If $\mathrm{THH}$ comes later, then a comparison between $\mathrm{THH}$ and $\mathrm{HH}$ wouldn't be the original motivation, although I do like this perspective as a reason to consider Shukla homology. – Stahl Mar 04 '25 at 16:03
  • @Stahl Ah, that's what you mean. It's a good question, I do not know the original motivation for Shukla homology. I wonder if he writes something about it in his original paper? – Qi Zhu Mar 04 '25 at 18:25
  • @Stahl the comparison is more or less induced by changing the base ring along the Hurewicz map $\mathbb S \to H(\mathbb Z)$. Even spectrally, flatness is a $\pi_0$ condition – Andres Mejia Mar 05 '25 at 13:37