Hochschild homology

Remarks

One checks that $b \circ b = 0$, so the homology is defined. In degree $0$, $b(m \otimes a) = ma - am$, so $HH_0(A, M)$ is the quotient of $M$ by the $k$-submodule spanned by all $ma - am$, and $HH_0(A) = A/[A, A]$. Example: for $A = M_n(k)$ with $k$ a field, $[A, A]$ is the space of trace-zero matrices, so the trace gives $HH_0(A) \cong k$. If $A$ is projective as a $k$-module, then $HH_n(A, M) \cong \operatorname{Tor}_n^{A^e}(A, M)$, where $A^e = A \otimes A^{\mathrm{op}}$ uses the opposite ring; see the Tor functor. Without such a flatness hypothesis the complex still makes sense, but this description by Tor can fail. Hochschild cohomology $HH^n(A, M)$ is defined dually from $\Hom_k(A^{\otimes n}, M)$, and it equals $\operatorname{Ext}^n_{A^e}(A, M)$ when $A$ is projective over $k$.