Motivation behind the definition of hochschild cohomology

Question

For an associative algebra $A$ one can define the Hochschild cohomology of $A$ as $ HH^n(A,A):= Hom_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$ (this definition also works for the graded and dg cases as well). There is a lot of information contained in this gadget: when $ n=0$ it is the center and when $n=1$ it is the lie algebra $\text{Der}(A)/\text{Inn}(A)$ and when $n=2$ it says something about the deformation theory of the algebra. Also there is a cup product and Gerstenhaber bracket on $HH^*(A,A)$.

My question is: What is the reason/motivation for the definition $ HH^n(A,A):= Hom_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$, if you didnt know about all this higher structure what would lead you to study it in the first place?

Answer 1

I think you will have different answers to your questions. As you noticed there is more about Hochschild Homology in n-lab page and you also pointed out the interpretation of (positive) low dimension of Hochschild cohomology. I would say that the Hochschild cohomology is "The" cohomology theory for differential graded algebras in the following sense. Let $\mathsf{dgAlg}_{k}^{\geq 0}= \mathsf{A}$ be the category of (connective) differential graded algebras over a commutative ring $k$, which is a model category . Fix a morphism of DG $k$-algebras $f:R\rightarrow S $ with $R$ cofibrant DGA, then the homotopy groups of the mapping space $Map_{\mathsf{A}}(R,S)_{f}$ are closely related to the (negative) Hochschild cohomology $HH^{-\ast}(R,S)$ where you see $S$ as an $R$-bimodule via $f$. More precisely, for all $i>1$: $$\pi_{i}Map_{\mathsf{A}}(R,S)_{f}\cong HH^{1-i}(R,S). $$ In the particular case $id:R\rightarrow R$, we can say more i.e., $\pi_{1}Map_{\mathsf{A}}(R,R)_{id} $ is a subgroup of the units group of the algebra $HH^{0}(R,R)$ (i.e., $HH^{0}(R,R)^{\times}$).

Block and Lazarev gave an interpretation of the (negative) Andre-Quillen cohomology in the case of commutative differential graded $\mathbf{Q}$-algebras denoted by $\mathsf{C}$. $$ \pi_{i}Map_{\mathsf{C}}(R,S)_{f}\cong AQ^{-i}(R,S), i>1. $$