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Throughout the literature, one can find many definitions of the Hochschild homology of various objects. However, the precise relationship between these definitions is not always so clear, at least to me. Here are a few of those definitions (throughout, all my rings and algebras will be commutative):

Definition 1: Let $k$ be a commutative ring, and let $A$ be a commutative [dg or simplicial, if you like] $k$-algebra. The Hochschild homology of $A/k$ is defined to be $$ HH(A/k) := A\otimes^L_{A\otimes^L_k A}A $$ (considered as an object of either $D(A)$ or $D(k)$). The $n$th Hochschild homology group of $A/k$ is $$HH_n(A/k) := \pi_n(HH(A/k)).$$

Definition 2: Let $\mathcal{C}$ be a $k$-linear dg-category. Then the $n$th Hochschild homology group of $\mathcal{C}$ is defined to be $$ HH_n(\mathcal{C}/k) := \pi_n\left(\left|N^{cyc}_\bullet(\mathcal{C}/k)\right|\right), $$ where $N^{cyc}_\bullet(\mathcal{C}/k)$ a simplicial $k$-module with $$ N^{cyc}_n(\mathcal{C}/k) = \bigoplus_{(c_0,c_1,\dots, c_n)}\mathcal{C}(c_1,c_0)\otimes_k\mathcal{C}(c_2,c_1)\otimes_k\dots\otimes_k\mathcal{C}(c_0,c_n). $$ See [1]. (I have also heard that if $\mathcal{C}$ does not consist of compact objects, we should replace $\mathcal{C}$ by the full subcategory of compact objects before forming $N^{cyc}_\bullet(\mathcal{C}/k)$, although this issue is not mentioned in [1].)

Definition 3: Let $\mathcal{A}$ be an associative algebra object in a symmetric monoidal $\infty$-category $\mathcal{C}$. The Hochschild homology or trace of $\mathcal{A}$ is $$ HH(\mathcal{A}/\mathcal{C}) := \mathcal{A}\otimes_{\mathcal{A}\otimes \mathcal{A}^{op}}\mathcal{A}. $$ See [2].

Now, it's a theorem of McCarthy that if $A$ is a commutative $k$-algebra, $BA$ is the one-object category with endomorphism ring $A,$ and $\mathsf{Perf}_A$ is the category of finitely generated projective $A$-modules, then the natural inclusion $BA\to\mathsf{Perf}_A$ induces a quasi-isomorphism $$ \left|N^{cyc}_\bullet(BA/k)\right|\to\left|N^{cyc}_\bullet(\mathsf{Perf}_A/k)\right|, $$ and hence an isomorphism $$ HH_n(BA/k)\to HH_n(\mathsf{Perf}_A/k) $$ for all $n.$ Moreover, if $A/k$ is flat, one sees immediately that the chain complex $\left|N^{cyc}_\bullet(BA/k)\right|$ associated to $N^{cyc}_\bullet(BA/k)$ is precisely the usual bar complex for $A,$ which computes the Hochschild homology groups of $A/k$ in the sense of definition 1.

Ben-Zvi, Francis, and Nadler also show that $HH(\mathcal{A}/\mathcal{C})$ can be computed using an appropriate cyclic bar complex $N^{cyc}_\bullet(\mathcal{A})$ with $N^{cyc}_n(\mathcal{A})\simeq \mathcal{A}^{\otimes n+1}.$

Presumably, if $A$ is not flat over $k,$ then definition 2 must be modified in order to get isomorphisms $$ HH_n(A/k)\xrightarrow{\sim} HH_n(\mathsf{Perf}_A/k). $$

These definitions are quite similar, and the objects which compute the Hochschild homology groups/objects are also quite similar -- in each case (at least, if things are flat) we have a bar complex. However, I'm somewhat confused about the precise relationship between all these objects.

In particular, we have inclusions of $\infty$-categories $$ \mathsf{dgAlg}_k\xrightarrow{i}\mathsf{dgCat}_k\xrightarrow{j}\mathsf{Pr}^L, $$ where $\mathsf{Pr}^L$ is the $\infty$-category of presentable $\infty$-categories, with left adjoints as morphisms (as in [2]). Under these inclusions, $A\in\mathsf{dgAlg}_k$ maps to $BA\in\mathsf{dgCat}_k,$ which includes into $\mathsf{Perf}_A$ and $\mathsf{Mod}_A.$ Then we may map these dg-categories to $D(\mathsf{Perf}_A),D(\mathsf{Mod}_A)\in\mathsf{Pr}^L.$

It seems like there should be comparison maps $$ ji(HH(A/k))\to ji(HH(A/\mathsf{dgAlg}_k))\to j(HH(BA/\mathsf{dgCat}_k))\to HH(BA/\mathsf{Pr}^L), $$ $$ j(HH(\mathsf{Perf}_A/\mathsf{dgCat}_k))\to HH(D(\mathsf{Perf}_A)/\mathsf{Pr}^L), $$ $$ j(HH(\mathsf{Mod}_A/\mathsf{dgCat}_k))\to HH(D(A)/\mathsf{Pr}^L), $$ as well as comparison maps $$ HH(BA/\mathsf{dgCat}_k)\to HH(\mathsf{Perf}_A/\mathsf{dgCat}_k)\to HH(\mathsf{Mod}_A/\mathsf{dgCat}_k) $$ and $$ HH(D(\mathsf{Perf}_A)/\mathsf{Pr}^L)\to HH(D(\mathsf{Mod}_A)/\mathsf{Pr}^L). $$

My questions about this situation are the following:

  1. Which, if any, of these comparison maps are equivalences?
  2. What model structures on the ordinary categories $\mathsf{dgAlg}_k$ and $\mathsf{dgCat}_k$ (if any) can we use to compute $HH(-/\mathsf{dgAlg}_k)$ and $HH(-/\mathsf{dgCat}_k)$? That is, are there symmetric monoidal model structures on these categories such that $HH(A/\mathsf{dgAlg}_k)\simeq A\otimes^L_{A\otimes^L A}A$ and $HH(\mathcal{A}/\mathsf{dgCat}_k)\simeq\mathcal{A}\otimes^L_{\mathcal{A}\otimes^L\mathcal{A}^{op}}\mathcal{A}$? (I have been told that the Dwyer-Kan model structure and symmetric monoidal $\otimes^L$ defined in [3] on $\mathsf{dgCat}_k$ may not be appropriate to work with for this purpose, and I have also heard that the model structures on $\mathsf{dgAlg}_k$ and $\mathsf{dgCat}_k$ are not compatible in the way one would want.)
  3. What is the "correct" way to modify definition 2 in the case when $A$ is not flat over $k$, so that we still have isomorphisms $HH_n(A/k)\cong HH_n(\mathsf{Perf}_A/k)$?
  4. Finally, where, if anywhere, are these comparisons (or even others which I have not mentioned here) discussed in the literature? The only reference I have found which discusses these issues to any extent is [1], although it seems that comparison isomorphisms like these are common folklore statements.

My apologies if these questions are elementary; my experience with $\infty$-categories is rather minimal and I am not yet accustomed to working out technicalities like these. I also may have made mistakes in my presentation of this story, and corrections to my understanding of the overall picture would be greatly appreciated.

[1] McCarthy, R. (1994). The cyclic homology of an exact category. Journal of Pure and Applied Algebra, 93(3), 251–296. doi:10.1016/0022-4049(94)90091-4

[2] Ben-Zvi, D., Francis, J., Nadler, D. (2010). Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry. Journal of the American Mathematical Society, 23(4), 909-966.

[3] Toën, B. (2004). The homotopy theory of dg-categories and derived Morita theory.

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    $\begingroup$ So you're always assuming $A$ to be commutative, right ? Otherwise $ji \mathrm{HH}(A/k)$ need not make sense. Can you also clarify what you call $D(Perf_A)$ and $D(Mod_A)$ ? Because there are versions of these where these would be the same $\infty$-categories, so the question would become rather moot. As for your comparison maps, a general thing to note (I might try to answer your question after you've clarified, but in the meantime let me say that) is that anything over $k$ will be ... over $k$, while things in $Pr^L$ will be over $Sp$, i.e. over the sphere spectrum (1/2) $\endgroup$ Feb 18, 2021 at 23:17
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    $\begingroup$ So things like $HH(D(A)/Pr^L)$ will be closely related to topological Hochschild homology. In fact, for a commutative algebra $A$, I'm pretty sure this is exactly modules (in $Sp$) over $THH(A)$, so that should be a general principle to tell which of the comparisons aren't equivalences. More generally, if you derive everything, then most likely most things over $k$ will be the same, and most things over $Sp$ will be the same (2/2) $\endgroup$ Feb 18, 2021 at 23:18
  • $\begingroup$ @MaximeRamzi Thanks for your comments. Yes, I mean for all my rings/algebras/etc. to be commutative. As for $D(\mathsf{Perf}_A)$ and $D(\mathsf{Mod}_A),$ it may very well be that these are the same -- I'm not fluent in the $\infty$-categorical language, but I believe that for both of these I want the $\infty$-category which is the dg-nerve of the corresponding dg-categories. (If there are alternative definitions that I should be considering, I would be interested in hearing about them.) $\endgroup$
    – Stahl
    Feb 19, 2021 at 0:10
  • $\begingroup$ You mean the dg-nerve of the category of chain complexes, right ? $\endgroup$ Feb 19, 2021 at 9:10
  • $\begingroup$ @MaximeRamzi Well, $A$ for me is a commutative dg-algebra over (an ordinary ring) $k,$ so modules over $A$ are already chain complexes in some sense. I don't think I want to take chain complexes of dg-modules/chain complexes (although perhaps I'm mistaken), although I might mean take the dg-nerve of the full subcategory of fibrant-cofibrant objects in an appropriate model structure. $\endgroup$
    – Stahl
    Feb 20, 2021 at 10:36

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