Let $A$ be an associative algebra over a commutative ring $k$. I've read statements saying that Hochschild (co)homology is the "right" notion of (co)homology for associative algebras. When $A$ is projective over $k$, the Hochschild cohomology, say, can be written as $Ext^\*\_{A \otimes A^{op}}(A,A)$, where $A^{op}$ is the opposite algebra, i.e., $A$ with $xy$ redefined to be $yx$.


On the other hand, when $A$ is augmented, the ext-group $Ext^\*\_A(k,k)$ is also referred to as the cohomology of $A$. What is the difference between these two notions of cohomology, and why would I choose one over the other?



Cohomology of associative algebras
For good algebras, Hochschild cohomology computes ‘all’ other interesting cohomologies. For example, it is already in Cartan-Eilenberg that if $M$ and $N$ are left $A$-modules, then $\mathrm{Ext}\_A^\bullet(M,N)=H^\bullet(A,\hom(M,N))$, where on the right $H^\bullet(A,\mathord-)$ is Hochschild cohomology with coefficients, and $\hom(M,N)$ is the space of homomorphisms over the base field turned into an $A$-bimodule using the left $A$-module structures of $M$ and $N$.


One general-nonsense explanation of the fact that Hochschild cohomology is somehow preferred is that an associative algebra is an algebra over the $\mathcal{A}ss$ operad, which is a Koszul operad, and a Koszul operad determines a canonical cohomology theory for its algebras: in the case of $\mathcal{A}ss$ you obtain in this way Hochschild cohomology. (Likewise, the operad $\mathcal{L}ie$ whose algebras are Lie algebras picks the usual Lie algebra cohomology, and $\mathcal{C}omm$, the operad of commutative algebras, picks the Harrison cohomology. This explanation breaks for, say, Hopf algebras—which are not the algebras of an operad—and then you do not have a clear winner among the cohomologies: then you have $\mathrm{Ext}\_H^\bullet(k,k)$ and the Gerstenhaber-Schack cohomology as alternatives, both quite ‘preferred’...)


