Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 113 / 3

The direct image sheaf is determined on the basis consisting of principal open subschemes $D(b)\subseteq \mathrm{Spec}B$. Their inverse images areand henceOn the other hand, localization of the restricted-scalar $B$-module givesThese isomorphisms commute with restriction to smaller principal opens, so they define the equality of quasi-coherent sheaves

Because $X$ is a Noetherian scheme, choose a finite affine cover $X={\bigcup }_i{U}_i$. Every intersection $U_i\cap U_j$ is quasi-compact, so choose a finite affine cover $U_i\cap U_j={\bigcup }_k{U}_{ijk}$. If $j_i:U_i\hookrightarrow X$ and $j_{ijk}:U_{ijk}\hookrightarrow X$ are the inclusions, the sheaf axiom gives an exact sequencewhere the last arrow is the difference of the two restrictions to each overlap chart.

Applying the left-exact direct image sheaf functor identifies $f_{\ast }\mathcal{F}$ with the kernel ofEvery map $U_i\to Y$ and $U_{ijk}\to Y$ is a morphism between affine schemes. Part (a) shows that all sheaves in the two finite sums are quasi-coherent sheaves. Kernels of morphisms of quasi-coherent sheaves on the affine scheme $Y$ are quasi-coherent, because they correspond to kernels of module homomorphisms. Therefore $f_{\ast }\mathcal{F}$ is quasi-coherent. This is the quasi-coherence of direct image under a quasi-compact quasi-separated morphism in the present Noetherian case.