Relative Hochschild cohomology

Question

Gerstenhaber's paper Algebraic Cohomology and Deformation Theory introduces relative Hochschild cohomology, which I have never seen before:

Let $k$ be a field. Let $A$ be an associative, unital $k$-algebra, $S\subseteq A$ a subalgebra. Let $M$ be an $A$-bimodule. We define $C^n(A,S;M)$ the $S$-relative Hochschild $n$-cochains with coefficients in $M$: Let $C^0(A,S;M):=\{m\in M \vert sm=ms \ \forall s\in S\}$ be the commutant. For $n>0$, we let $C^n(A,S;M)$ be those $k$-linear maps $A^{\otimes_k n}\rightarrow M$ satisfying $$f(sa_1,\ldots,a_n)=sf(a_1,\ldots,a_n)$$ $$f(a_1,\ldots,a_ns)=f(a_1,\ldots,a_n)s$$ $$f(\ldots,a_is,a_{i+1},\ldots)= f(\ldots,a_i,sa_{i+1},\ldots)$$ for all $a_i\in A$, $s\in S$. Then we consider the usual Hochschild coboundary map, turning $C^{\ast}(A,S;M)$ into a cochain complex. The cohomology of this complex is called $S$-relative Hochschild cohomology

In what contexts is the above generalization of Hochschild cohomology used/useful? Why is this a good definition? Is there a more conceptual description of this cohomology? Why is this called relative cohomology? Is there a relationship to relative singular cohomology in algebraic topology? Doesa similar generalization of Hochschild homology exist?

Answer 1

This is useful, for example, when the subalgebra S is separable. In that case it actually coincides with the normal Hochschild cohomology of A.

You could ask... why bother then? Well, the complex is much much smaller in general, so it is easier to deal with.

A good example of that is when A is an admissible quotient of the path algebra on a quiver and S is the subalgebra generated by the vertices, which is separable. Then the theory tells you that you can compute the cohomology of A using the bar complex as a resolution but with the tensor products taken over S — this is the complex Najonathan calls the relative bar resolution in their answer.

If you write down what exactly is that complex in this case you will appreciate what changes.

Answer 2

Relative Hochschild cohomology was first defined by Hochschild in 1956 in "Relative homological algebra". Let $k$ be a field (for ease) and let $A$ be a $k$-algebra and let $S \subseteq A$ be a $k$-subalgebra. Then, as I think about it, the rough idea of relative homological algebra is that we want to mimic how homological algebra behaves over a field $k$ to instead behaving the same over any subalgebra $S$.

Every exact sequence of $A$-modules split over $k$ as $k$ is a field. We can define an exact sequence of $A$-modules to be $(A|S)$-relative exact if the sequence is exact and it splits as $B$-modules. Using this we can defined relative projective and injective modules akin to the normal case, and relative derived functors (see the article by Hochschild).

In the normal case, we have the bar resolution $$ ... \rightarrow A^{\otimes_k n} \rightarrow A^{\otimes_k (n-1)} \rightarrow ... \rightarrow A \otimes_k A \rightarrow A \rightarrow 0 $$ as the standard resolution of $A$ as an $A$-bimodule (or $A^e$-module) which we can use to define Hochschild cohomology. In the case of relative Hochschild cohomology, we instead have the relative bar resolution which is $$ ... \rightarrow A^{\otimes_S n} \rightarrow A^{\otimes_S (n-1)} \rightarrow ... \rightarrow A \otimes_S A \rightarrow A \rightarrow 0 $$ The cochain that Gerstenhaber-Schack describe one gets by applying $\operatorname{Hom}_{A^e}(-,M)$ to the relative bar resolution. One can define relative Hochschild homology by instead applying $- \otimes_{A^e} M$ to the relative bar resolution.

The relative Hochschild cohomology $\text{HH}^*(A|S)$ can be computed from the relative bar resolution as being $$ \text{Degree 0}: Z(A) $$ $$ \text{Degree 1}: \text{Der}_S(A,A)/\text{Inn}_B(A,A) $$ where $\text{Der}_S(A,A)$ are all derivations from $A$ to $A$ which vanish on $S$ and $\text{Inn}_S(A,A)$ are all inner derivation given by elements which commute with all elements in $S$, i.e all inner derivations that vanish on $S$. If $k$ is a field and $S$ is separable then we have that $\operatorname{HH}^*(A|S) \cong \operatorname{HH}^*(A)$, see Theorem 1.2 in "Relative Hochschild cohomology, rigid algebras, and the Bockstein" by Gerstenhaber-Schack.

I don't know enough algebraic topology to say if it similar to relative singular cohomology unfortunately.

Regarding applications, Gerstenhaber-Schack used it in the article "Relative Hochschild cohomology, rigid algebras, and the Bockstein" in the context of deformations theory. It has also been used by Cibils-Lanzilotta-Marcos-Solotar to study what happens to Hochschild (co)homology when adding or deleting an arrow in a quiver, see "Deleting or adding arrows of a bound quiver algebra and Hochschild (co)homology".