Background: the Hochschild homology of an associative algebra is the homology of the complex
... --> A (x) A (x) A --> A (x) A --> A
where the last two differentials are
a(x)b(x)c \mapsto ab(x)c-a(x)bc+ca(x)b and
a(x)b \mapsto ab-ba, and you can guess the rest. More generally, it's "derived coinvariants": take a projective resolution of your algebra, then coinvariants of that.
k[t], the Hochschild homology is concentrated in degrees 0 and 1, and in both of those degrees it's
k[t]. I know that I can go look up Loday (\S 3.2.2) and find a calculation, but I'd like a better explanation.
I know that the zero-th Hochschild homology
HH_0(k[t]) must just be
k[t], because the zero-th Hochschild homology is just coinvariants, and k[t] is commutative.
What I'd like is a "good" explanation for
Edit: Ben has a simple explanation below. Let me also rephrase the question, hoping for more. Here are a few things: If
A is semisimple, then
HH_*(A) is concentrated in degree 0. Is there something about
k[t] that ensures it's concentrated in degrees 0 and 1. Conversely, can I conclude anything about
A from the fact that
HH_*(A) zero above