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Multiplication by the homogeneous equation $f_d$ and restriction to its zero scheme give the structure-sheaf sequence of a hypersurfaceBy cohomology under a closed immersion, $H^q({\mathbb{P}}^3,i_{\ast }{\mathcal{O}}_X)\cong H^q(X,{\mathcal{O}}_X)$. The associated long exact sequence in cohomology containsBoth outer groups vanish by the cohomology of twisting sheaves on projective space, because they are intermediate cohomology groups on ${\mathbb{P}}^3$. Therefore

Choose a finite affine cover $\mathcal{U}=(U_j)$ of the Noetherian scheme $X$. Because $X$ is separated, every finite intersection $U_J$ is affine. Its inverse image $V_J=Z\cap U_J$ under the closed immersion is also affine. By the definition of the direct image sheaf,The Čech complexes for $i_{\ast }\mathcal{F}$ on $\mathcal{U}$ and for $\mathcal{F}$ on the induced cover $(Z\cap U_j)$ are therefore identical, including their restriction maps. Both affine covers are acyclic for the relevant quasi-coherent sheaves, so the acyclic cover theorem givesfor every $q$. This is cohomology under a closed immersion.

For $X={\mathbb{P}}_k^n$, use its $n+1$ standard affine opens. The induced cover of $Z$ is still acyclic, and its Čech complex has no cochains in degrees greater than $n$. The cohomological dimension bound from an affine cover therefore yields