Example of excision in Hochschild homology

Question

The excision theorem for Hochschild homology introduced by Wodzicki seems like a very powerful tool (as scision was hyper-useful in topology).

However, I cannot actually seem to think of a result where it yields interesting computations.

So my question is, can someone show me a case where the Wodzicki excision theorem actually helps compute the Hochschild homology of an algebra in a useful way?

Answer 1

This is in Cyclic Homology by Loday, theorem 1.4.14:

Theorem: Let $I$ be a H-unital $k$-algebra. Then $M_r(I)$ (matrices of size $r$) is $H$-unital for all $r$ (including $r = > \infty$), and the maps (resp. induced by the trace and the inclusion): $$\begin{align} \mathrm{tr}_* & : HH_*(M_r(I), M_r(M)) \to HH_*(I,M) \\ \mathrm{inc}_* & : HH_*(I, M) \to HH_*(M_r(I), M_r(M)) \end{align}$$ are isomorphisms and inverse to each other.

This theorem is proven using excision (I am not courageous enough to copy the proof here). It's called Morita invariance. Excision is used to go from unital algebras (where you can prove Morita invariance "easily") to merely H-unital algebras.

---

Another example is Mayer-Vietoris (exercise E.1.4.5 in Loday's book): if $$\begin{array}{ccc} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D \end{array}$$ is a pullback square of unital $k$-algebras and $B \to D$ is surjective and $k$-split, then there is a long exact sequence: $$\dots \to HH_n(A) \to HH_n(B) \oplus HH_n(C) \to HH_n(D) \to HH_{n-1}(A) \to \dots$$