**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.

For `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 `HH_1(k[t])`

.

**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 `*=k`

?