Hochschild (co)homology

Stream: new members

[Sarah El Boustany]

Hi! I am currently trying to formalize Hochschild homology and cohomology. For this, I need the notion of enveloping algebra. I see that it is already present in mathlib in the setting of Lie algebras. I would like to use it in a more general setting. What would be the best way to go about this?

[Kevin Buzzard]

Can you clarify what you mean by "more general setting"? I think that might be a good way to start. We might have what you are looking for already.

[Sarah El Boustany]

I would like to look at the enveloping algebra of $A$ a $k$-algebra with $k$ a commutative ring.

[Joël Riou]

Out of context, I am afraid this is not specific enough. Is there a precise definition in references like https://ncatlab.org/nlab/show/HomePage ?

[Kevin Buzzard]

What is the definition (in maths) of this enveloping algebra? It seems that "enveloping algebra" can mean several things.

[Edison Xie]

Sarah El Boustany said:

I would like to look at the enveloping algebra of $A$ a $k$-algebra with $k$ a commutative ring.

are you talking about A ⊗[k] Aᵐᵒᵖ ?

[Sarah El Boustany]

Yes, that is the one. I'm using the definition on page 2 of Sarah Witherspoon's book https://people.tamu.edu/~sjw/pub/gsm204.pdf

[Sarah El Boustany]

ncatlab has this definition https://ncatlab.org/nlab/show/enveloping+algebra but it's even more general than what I am looking at.

[Sarah El Boustany]

Here is a quote from Witherspoon's book with the definition.

For now, let $k$ be a commutative associative ring (with 1), and let $A$ be a $k$-algebra. That is, $A$ is an associative ring (with multiplicative identity) that is also a $k$-module for which multiplication is a $k$-bilinear map. Denote the multiplicative identity of $A$ also by 1, identified with the multiplicative identity of $k$ via the unit map $k \rightarrow A$ given by $c \rightarrow c \cdot 1$ for all $c \in k$. Denote by $A^{op}$ the opposite algebra of $A$; this is $A$ as a module over $k$, with multiplication $a \cdot_{op} b = ba$ for all $a, b \in A$. Tensor products will often be taken over $k$, and unless otherwise indicated, $\otimes = \otimes_k$. Let $A^e = A \otimes A^{op}$, with the tensor product multiplication: $(a_1 \otimes b1) · (a_2 \otimes b_2) = a_1a_2 \otimes b_2b_1$ for all $a_1, a_2, b_1, b_2 \in A$. (Technically, we are really taking $b_1, b_2$ to be elements of $A^{op}$, but since the underlying $k$-modules are the same, we write $b_1, b_2 \in A$ where convenient.) We call $A^e$ the enveloping algebra of $A.$

[Joël Riou]

For the iterated tensor products which appear in the definition of Hochschild (co)homology, it may be possible to use the work by @Robin Carlier #mathlib4 > Coherence of higher arity tensor products in mon. categories @ 💬 (not in mathlib). Alternatively, we have the notion of tensor product of a family of modules docs#PiTensorProduct
In any case, it seems that the formalization will be more intricate as compared to the formalization of the standard resolutions which appear in the context of group (co)homology.

[Sarah El Boustany]

Thank you, I will look into this, I'm sorry it took me a while to provide enough details

[Robin Carlier]

I would be glad to upstream the code linked here if needed, though I admitedly have not so much time for this particular project right now (I might need a few weeks at least, but I most certainly will come back to this since I’m tackling the symmetric case).

As a "fun story": the entire genesis of that "coherence/infinity-operads"-project was that I wanted (and still want) to have general "bar resolutions" that describes "relative tensor products" in general monoidal categories (and also, the case of endofunctors, which should give the "bar resolutions" for a monad, which is a nice abstract framework that gives a lot of "standard resolutions", not only for group cohomology but also e.g for simplicial rings etc.).

That being said, a lot of this is made easier (in mathlib) in the case of modules, k-algebras etc because there is no need (a priori) to invoke general theorems like this to at least write the iterated tensor products. The faces and degeneracies of the simplicial resolutions are an other story though, and I expect that to be involved.

[Joël Riou]

As far as I understand, the faces and degeneracies will be mostly given by the unit and the multiplication in the algebra A. Thanks to @Robin Carlier's work again, it would be possible to define simplicial operations by using a presentation of the simplex category by generators and relations #25740 (probably in mathlib in a relatively near future). But I agree that the verification of the relations will probably be quite involved.

[Robin Carlier]

Right, I also need to take the time to fix that project as well :sweat_smile: This should provide some incentive for fixing this.

[Robin Carlier]

(everything is indeed given by units and multiplications, but in general monoidal categories the difficulty is that you a priori have to add a whole mess of unitors and associators to make things work out, but in the explicit case of "honests" modules and algebras, there are probably shortcuts just like there are shortcuts to the iterated tensor products)

[Sarah El Boustany]

Thank you all very much, I will look into this, and come back later if I have other questions.