Is the cohomology groups of a finite CW complex $X=X^{n}$ are zero for $k>n$?

Question

Let $X$ be a finite CW complex of dimension n, $X=X^{n}$, We know that the singular homology groups $H_{k}$ are trivial for all $k>n$. I tried to show the same for Cohomology groups $H^{k}$ By applying the universal coefficients theorem. I arrived to prove that it's true for $k>n+1$ but for $k=n+1$, l cannot prove it. Because I have $$H^{n+1}(X^{n},R)=Ext(H_{n}(X,R),R)$$ Can someone tell me how to do it?

Answer 1

Yes, the proof that the homology groups vanish is either a corollary of or a lite version of the fact that cellular homology is isomorphic to singular homology. Since singular cohomology is isomorphic to singular cohomology, the same reasoning applies, i.e. there are no nontrivial cochain groups above dimension n.