not complete solution in computation of singular cohomology

Question

I know how to compute singular cohomology of single point. Then I want to compute cohomology of $\mathbb{Z}$. Consider $C_{k}(\mathbb{Z})=\oplus_{i\in Z} C_{k}(point)$. $$\text{Hom}(A\oplus B, G)= \text{Hom}(A,G)\oplus \text{Hom}(B,G).$$ Then $$H^{k}(A\oplus B)=H^{k}(A)\oplus H^{k}(B).$$ And finally we have $H^{0}(Z)=\oplus_{i\in Z}\mathbb{Z}$. My answer is in what I'm wrong? Why is it not complete solution?

What is meant by 'point'? – o0BlueBeast0o Jun 18 '14 at 19:02 — o0BlueBeast0o, Jun 18 '14 at 19:02

score 1 · Answer 1 · answered Jun 18 '14 at 19:07

1

You should have that $$\operatorname{Hom}(\oplus A_i,B)\cong \prod\operatorname{Hom}(A_i,B).$$

answered Jun 18 '14 at 19:07

J126

17,451

Why the product? If it is obvious let's send me to the textbook.. – boris1993 Jun 18 '14 at 19:10
@boris1993 The proof is contained in the Wikipedia page here: http://en.wikipedia.org/wiki/Direct_sum_of_modules – J126 Jun 18 '14 at 19:11

not complete solution in computation of singular cohomology

1 Answers1