Consider the complex $A_{*}: A _ {i \in \{0,1,2\}} = \mathbb{R}^2$, where all other $A_i = 0$. Find all relevant homologies.

Question

I should include the maps here:

$f_{0\to 1} (x,y) = (x,0)$, and $f_{1 \to 2} (x,y) = (0,y)$.

The first thing we are asked is to show this is indeed a complex, which is really trivial for me. (simple composition of maps).

The follow up is to find all relevant homologies.

$A_{*}: 0 \to A_0 \to A_1 \to A_2 \to 0$. I know that the definition of a homology is

$H_i (A_{*}) = \dfrac{ker f_i}{im f_{i-1}}$

Using this, I think there are only 3 relevant homologies (meaning non-trivial). Is this correct? I also need to say what they are.

For example, I think that $H_1 (A_{*}) = \dfrac{ker f_{0 \to 1}}{0} = A_1$. Is this correct?

Answer 1

Note that $H_0(A_*)=\ker f_{0\to 1}= 0\times \mathbb R\cong\mathbb R$.

The complex is exact at $A_1$, i.e. $\ker f_{1\to 2}=\operatorname{im} f_{0\to 1}$, hence $H_1(A_*) = 0$.

Finally $H_2(A_*)=A_2/\operatorname{im}f_{1\to 2}= \mathbb R^2/(0\times \mathbb R)\cong \mathbb R$.