# Calculate the hochschild homology

Could you help me to calculate the hochschild homology of the following chain complex: $0 \longleftarrow M \longleftarrow M\otimes Z[i] \longleftarrow M\otimes Z[i]\otimes Z[i]\longleftarrow$

where $Z[i]$ is Gaussian intgers and $M$ is $Z[i]$-module.

I know that $Z[i] \cong Z^2$ and hence i can rewrite the chain complex as following:

$0 \longleftarrow M \longleftarrow M\otimes Z^2 \longleftarrow M\otimes Z^2\otimes Z^2\longleftarrow$

and i know that $H_n = ker/Im$. However i am struggling to calculate $H_0$, $H_1$ and $H_2$.

• I don't think you should rewrite $\mathbb{Z}[i]$ as $\mathbb{Z}^2$ so quickly, because you'll need to use the multiplication in the Hochschild differentials. I also don't think it's possible to say much specific in this case unless we know what $M$ is. – JHF Mar 13 '16 at 0:20