# How exactly is Hochschild homology a monad homology?

Many texts which praise the generality of the bar construction associated to a monad, say that Hochschild homology is an example of this.

What exactly is in this case the underlying endofunctor of the monad, on which category is it an endofunctor, what are the monad structure maps and, most important (since I think my confusion lies here), why do then the face maps look as on the wikipedia page?

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Interesting addition: If k is not a field, apparently Hochschild cohomology is in fact not a cotriple cohomology! -- See the introduction to Jibladze/Pirashvili: Cohomology of Algebraic Theories, JOURNAL OF ALGEBRA 137, 253-296 (1991) – Peter Arndt Oct 28 '09 at 20:58

## 3 Answers

Tyler, are you sure about this? I thought the bar construction comes from the adjunction between R-modules and k-modules for R a given k-algebra (i.e. relative Tor). Besides, what you say only makes sense if we're taking coefficients in R itself and not a general bimodule M.

If memory serves correctly, starting with a k-algebra R and looking at a simplicial resolution for it via the adjunction k-modules -- k-algebras leads to cyclic homology as in the paper of Feigin-Tsygan.

That wiki page also looks off to me: the Loday construction is for the Hochschild homology and decomposition for commutative algebras, and this isn't made very clear in the wiki.

quick edit: There's a good if terse discussion of the bar construction via monads in Weibel Chapter 8 (p.283). I suspect you could also extract the desired information out of the much more general machinery in Jon Beck's thesis, modulo some possible struggle with notation.

encore une fois: Consider the adjunction between k-mod and R-mod. If M is an object of R-mod then the simplicial construction provided by the adjunction looks like this

M <--- R\otimes M <--- R \otimes R \otimes M <----- etc

where I've not been able to draw in all the face maps, but hopefully you get what I mean. Now by taking the alternating sum of face maps in each degree, we get a split exact sequence of R-module maps

M <--- R\otimes M <--- R \otimes R \otimes M <----- etc

which is a resolution in the classical sense of M in R-mod-R by R-mod projectives - I'm assuming k is a field for sake of convenience. (So you can use it to calculate Tor^R if you wish.)

Now take M=R and note that we have a resolution of R by R^e-projectives. Apply Hom{R^e}(__, X) where X is your coefficient module, and you get precisely the Hochschild chain complex as in the original papers.

Of course, we didn't have to take sums of face maps before applying the Hom functor. So, if we start with R regarded as an object of R-mod, the canonical simplicial construction (for M=R) would give us a contractible simplicial object in R-mod with M=R at the bottom, this object would in fact live in R^e-mod, and so is eligible to be hit with HomR^e(__,X). If we do this, we get a simplicial object in k-mod, and said object should be the one described in the wiki article, corresponding to the Hochschild chain complex.

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The bar construction you're describing would be computing Tor over R rather than something involving Hochschild homology. To get cyclic homology as I usually understand it you usually need to make use of Connes' cyclic operator, and I'm pretty sure that cyclic homology is not arising from the adjunction between k-modules and k-algebras. What I was describing was indeed not with coefficients in a general module, but only the ring itself. I may have made a mistake but it should be checkable on a free algebra. – Tyler Lawson Oct 28 '09 at 17:33
Oops, I meant Tor^{R^e/k} (R, coefficient) - which is Hochschild homology of R with coefficients. The construction I allude to with cyclic homology as in Tsygan's paper (not Connes') is got, I think, by taking a simplicial resolution of your algebra and then hitting the simplicial object with the functor A --> A /[A,A] (or, if you want cohomology, taking traces). Besides, the original question was about the bar resolution usually used to compute Hochschild homology, and that is the bar resolution of an R-module for fixed R, and it does come from k-mod vs R-mod (Weibel 8.6.12, 9.1.3) – Yemon Choi Oct 28 '09 at 17:44
Right. The bar construction from k-mod vs R-mod computes Tor over R. If you use the bar construction from R-mod vs R^e-mod I think that then you will probably get the cyclic bar complex. – Tyler Lawson Oct 28 '09 at 18:06
While doubtless correct, this answer isn't very satisfying as a demonstration of the "generality of the bar construction associated to a monad", since the bar construction for the k-mod / R-mod adjunction is the first one most people learn about. – Reid Barton Oct 28 '09 at 19:57
Well, I am still confused by both answers - do they contradict each other? Shouldn't the monad stay the same if we change coefficients from the ring itself to some module? And tensoring with R gives the free what exactly? Shouldn't it be a bimodule? But shouldn't I tensor with two copies of R then? ... – Peter Arndt Oct 28 '09 at 20:11

This is partly in response to Reid, but also intended as general clarification.

As I understand it, Peter's original question was:

-- here is the Hochschild chain complex for an algebra $A$ and bimodule $M$, as defined in Hochschild's original papers; -- it is the chain complex associated to a certain simplicial object as defined on the Wikipedia page; -- one is told that this object comes from the bar construction (or standard resolution) associated to some monad; -- where/what is the monad?

The last one seems to be Reid's underlying point/question. Tyler says you can get it, up to a dimension-shift, from the adjunction between k-modules and k-algebras (at least when $A=M$). My earlier recollection was that this more naturally leads to cyclic homology a.k.a. additive K-theory as defined by Feigin and Tsygan, but I have yet to check this against a copy of their paper. (The point is that in characteristic zero, the cyclic homology of a free tensor algebra on a given k-module, coincides with the cyclic homology of the ground field, so one can take free resolutions of a given $k$-algebra and then use spectral sequence arguments.) On reflecting a bit more, because the Hochschild homology of a free (=tensor) algebra is confined to degrees 0 and 1, perhaps one can also obtain $H_n(A,M)$ as Tyler suggests, by taking the free algebra resolution of A (in the category of k-algebras) and then hitting the resulting simplicial object with a suitable functor - but this seems trickier than in the commutative case (Andre-Quillen) and I can't get hold of a copy of Quillen's paper at the moment.

Alors. As I understand it, following Weibel's book (and the papers of Barr & Beck et al), the simplicial object (in the category of $k$-modules) that yields the Hochschild chain complex, arises by applying a certain Hom-functor (namely $\{\}\_A{\rm Hom}_A(\ \cdot\ ,X)$ ) to another simplicial object, say $\beta(A)$, in the category of $A$-bimodules.

Now $\beta(A)$ is not contractible in the category of $A$-bimodules, in general, and doesn't come from a (co)monad on that category. However, $\beta(A)$ can be identified with another simplicial object $F(A)$, which lives in the category of $A$-modules.

What is $F(A)$?

Well, take a step back and consider the adjunction between $k$-modules and $A$-modules (maybe you need $k$ to be a field at this point, maybe not). That gives rise to a bar construction in $A$-mod, namely for any given $M$ in $A$-mod one obtains a simplicial object $F(M)$ which is given in each degree by

$$F_{-1}(M)=M\quad,\quad F_n(M) = M \otimes A^{\otimes n+1} \ {\rm for }\ n \geq 0.$$

Note that this is contractible in $A$-mod by the general machinery of the bar resolution associated to a monad. There was nothing to stop us taking $M=A$, that's a perfectly good $A$-module; and on doing so, lo and behold, we get the same simplicial object $F(A)$.

Thus, Hochschild homology, regardless of the choice of coefficients, can be thought of as "coming from" a comonad - namely, that induced on $A$-mod by the forgetful functor from $A$-mod to $k$-mod. In my opinion, that is probably the (co)monad they are talking about.

It so happens that, since $F(A)$ is contractible in $A$-mod and hence a fotiori in $k$-mod, the "chain-complex-ification" of $\beta(A)$ is, as a chain complex in $R$-bimod, a resolution of $R$ by $k$-relatively projective $R$-bimodules -- and hence applying ${}_R{\rm Hom}_R(\ \cdot \ ,X)$ to it and taking homology coincides with taking $k$-relative Tor of $R$ and $X$ as R-bimodules. Hence the point of view that Hochschild homology is a special case of relative Tor.

Finally, I actually agree with Reid that this is not the best example to motivate (co)monad (co)homology. Group cohomology with coefficients in the ground field; or indeed André-Quillen cohomology, which is given by a "free algebra" adjunction but only for commutative algebras, or sheaf cohomology, would be better. (No originality in my choices; I've cribbed them out of Weibel Section 8.6).

(Apologies for the length and the tediousness, by the way.)

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Wow, thanks. By the way I am not looking for a motivation for comonad cohomology but am trying to put Hochschild homology into a framework I know. But maybe that's not the best idea here...:-) – Peter Arndt Oct 28 '09 at 23:33

Assuming that the base ring $k$ is a field (or that the algebra is projective over $k$), a functor whose monad computes Hochschild homology is the one which maps $k$-modules $M$ to the $A$-bimodule $A\otimes M\otimes A$, which is adjoint to the forgetful functor in the other direction.

The bar complex corresponding to this adjunction is not exactly the one used by Hochschild to define his cohomology, but it is easily seen to give naturally isomorphic results, since it more or less evidently constructs gives projective resolutions of bimodules (not only of $A$, as is the case with the monad coming from one-sided extension of scalars)

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While I may be guilty of being a bit dogmatic on this one: I still maintain that the answer to the original question is the monad from k-modules to one-sided A-modules. The original question seemed to concern a particular simplicial object, and IIRC it's the one underlying the Hochschild chain complex, not the one of the monad you describe. (Of course, once you interpret it as relative Ext for A-bimodules then this is often a better way to do things) – Yemon Choi Dec 16 '09 at 6:05