Direct image sheaf ($f_{\ast }\mathcal{F}$)

For a continuous map $f:X\to Y$ and a sheaf $\mathcal{F}$ on $X$, the direct image is defined by $(f_{\ast }\mathcal{F})(V)=\mathcal{F}(f^{-1}V)$ for every open $V\subseteq Y$.

The direct image sheaf is the result of the direct image functor: $(f_{\ast }\mathcal{F})(V)=\mathcal{F}(f^{-1}V)$.

For a continuous map $f:X\to Y$, the derived direct image $R{f}_{\ast }$ is obtained by applying $f_{\ast }$ to an injective resolution. Composition with derived global sections gives $R\Gamma (Y,R{f}_{\ast }E)\cong R\Gamma (X,E)$.

For a proper map of locally compact spaces and a constructible complex $E$, the derived stalk satisfies $(R{f}_{\ast }E{)}_y\cong R\Gamma (f^{-1}(y),E{|}_{f^{-1}(y)})$. In particular $(R^i{f}_{\ast }\mathbb{Q}{)}_y\cong H^i(f^{-1}(y);\mathbb{Q})$. Properness is what prevents fibre cohomology from missing nearby sections.

For $f:{\mathbb{P}}_k^1\to \mathrm{Spec}k$, let $D$ consist of $0$ and infinity. Its ideal sheaf is $\mathcal{O}(-2)$, giving an exact coherent sequence $0\to \mathcal{O}(-2)\to \mathcal{O}\to {\mathcal{O}}_D\to 0$. The map on global sections is the diagonal $k\to k\oplus k$, which is not surjective. Since direct image to the point is global sections, projectivity does not make direct image right exact.

For a finite morphism between Noetherian schemes, a quasi-coherent sheaf is coherent exactly when its direct image sheaf is coherent. Over an affine target chart, let $B$ be the finite algebra over the target ring $A$, and let $M$ represent the sheaf on $\mathrm{Spec}B$. Finite generation of $M$ over $A$ implies finite generation over $B$, using the same generators; finite generation over $B$ implies finite generation over $A$ because $B$ is finite over $A$.

For an open inclusion $j:U\hookrightarrow X$, this sheaf has sections $\mathcal{F}(V\cap U)$ on an open set $V$. Restriction defines $\mathcal{F}\to {}_U\mathcal{F}$. Its stalks outside $U$ may be nonzero, so it differs from extension by zero.

Tensoring sections on $f^{-1}V$ gives a balanced pairing of the two direct image sheaves on $V$. The universal property of the tensor product of modules and sheafification produce the displayed morphism. Composing with the pullback-direct-image adjunction unit gives the projection formula for sheaves. The comparison itself need not be an isomorphism.

For a closed inclusion $i:X\hookrightarrow Y$, direct image sheaf formation is exact: stalks on $X$ are unchanged and stalks outside $X$ are zero. It also preserves flasque sheaves. Pushing forward a flasque resolution thus gives a resolution computing cohomology on $Y$, and its complex of global sections is identical to the original one on $X$. This proves the displayed isomorphism in every nonnegative degree.

For an affine morphism $f:X\to Y$, the direct image of every quasi-coherent sheaf on $X$ is quasi-coherent. On affine charts this is the fact that restriction of scalars turns the module sheaf $\tilde{M}$ into the module sheaf associated with the same underlying module over the target ring.

Let $j:{\mathbb{A}}_k^1\hookrightarrow {\mathbb{P}}_k^1$ be a standard affine chart and $\mathcal{G}=\tilde{M}$ a quasi-coherent sheaf there. Its direct image is quasi-coherent because $j$ is an affine morphism. The standard two-chart acyclic cover has Čech cochain complex $M\oplus M_x\to M_x$ in degrees zero and one, with differential $(m,n)\mapsto n-m/1$. Surjectivity gives $H^i({\mathbb{P}}_k^1,j_{\ast }\mathcal{G})=0$ for $i>0$. This does not assert that the direct image itself is flasque.

If $f:X\to Y$ is quasi-compact and quasi-separated, then the direct image of every quasi-coherent sheaf on $X$ is again a quasi-coherent sheaf. A finite affine cover of each inverse image of an affine target chart expresses the direct image as the kernel of a morphism between finite sums of quasi-coherent direct images from affine charts.