Top cohomology of projective space
= Top cohomology of projective space
For a field $k$,
$$
H^n(\mathbb P_k^n,\mathcal O(-n-1))\cong k.
$$
This nonvanishing obstructs affine covers of $\mathbb P_k^n$ having only $n$ members.
= Top cohomology of projective space
For a field $k$,
$$
H^n(\mathbb P_k^n,\mathcal O(-n-1))\cong k.
$$
This nonvanishing obstructs affine covers of $\mathbb P_k^n$ having only $n$ members.