The Stacks project

Example 37.73.8. Let $k$ be a field and let $X = \mathbf{P}^ n_ k$ be $n$-dimensional projective space over $k$. Lemma 37.73.5 applies to this by Constructions, Lemma 27.13.3. Hence the affine stratification number of $\mathbf{P}^ n_ k$ is at most $n$. On the other hand, we have nonzero cohomology in degree $n$ for some quasi-coherent modules on $\mathbf{P}^ n_ k$, see Cohomology of Schemes, Lemma 30.8.1. Using Proposition 37.73.7 we conclude that the affine stratification number of $\mathbf{P}^ n_ k$ is equal to $n$.