Sheaf of modules

For a ringed space $(X,{\mathcal{O}}_X)$, a sheaf of ${\mathcal{O}}_X$-modules is a sheaf $\mathcal{F}$ such that every $\mathcal{F}(U)$ is an ${\mathcal{O}}_X(U)$-module and restriction maps preserve scalar multiplication.

The stalk ${\mathcal{F}}_x$ is the direct limit of $\mathcal{F}(U)$ over neighborhoods $U$ of $x$. A morphism of sheaves is an isomorphism or forms an exact sequence exactly when it does so on every stalk.

A skyscraper sheaf supported at $x$ has a prescribed nonzero stalk at $x$ and zero stalks elsewhere.

Sheafification associates a sheaf to a presheaf without changing its stalks and is universal among morphisms from that presheaf to sheaves.

For an open inclusion $j:U\hookrightarrow X$, extension by zero is the sheaf $j_{!}\mathcal{F}$ whose stalk equals ${\mathcal{F}}_x$ on $U$ and zero outside $U$. Its sections are local sections on $U$ whose support is closed in the ambient open set.

A sequence $0\to {\mathcal{F}}^{\prime }\to \mathcal{F}\to {\mathcal{F}}^{\prime \prime }\to 0$ of sheaves is exact exactly when the induced sequence on every stalk is exact.

A sheaf of modules on a scheme is quasi-coherent when every affine chart $\mathrm{Spec}A$ restricts it to the sheaf associated with an $A$-module. On an affine scheme it is determined by its module of global sections.

The tensor product of two sheaves of ${\mathcal{O}}_X$-modules is the sheafification of $U\mapsto \mathcal{F}(U){\otimes }_{{\mathcal{O}}_X(U)}\mathcal{G}(U)$. Its stalk at $x$ is ${\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x$.

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 inverse image sheaf $f^{-1}\mathcal{E}$ is the sheaf associated with the presheaf whose sections near $U\subseteq X$ are obtained as a colimit of $\mathcal{E}(V)$ over open sets $V\supseteq f(U)$.

For a morphism of ringed spaces, the pullback of an ${\mathcal{O}}_Y$-module is

For a morphism of ringed spaces, there is a natural morphismIt is an isomorphism when $\mathcal{E}$ is a locally free sheaf of finite rank, because the claim is local and then reduces to distributivity over a finite direct sum.

A sheaf of ${\mathcal{O}}_X$-modules is locally free of finite rank when every point has an open neighborhood $U$ on which it is isomorphic to ${\mathcal{O}}_U^{\oplus r}$ for some finite $r$. If $X$ is connected, the rank is constant.

A line bundle on a scheme is a locally free sheaf of rank one. Its global sections can define a morphism to projective space when they have no common zero.

The Picard group is the group of isomorphism classes of line bundles under tensor product.

A line bundle is very ample when its global sections define a closed embedding into projective space.

A basepoint-free vector space of global sections $V\subseteq H^0(X,L)$ defines the Kodaira map $X\to \mathbb{P}(V^{\ast })$ by evaluating the sections at each point.

Sheaf cohomology consists of the right derived functors of global sections. The zeroth group is $H^0(X,\mathcal{F})=\Gamma (X,\mathcal{F})$, and higher groups measure obstructions to gluing local sections.

A sheaf is flasque when every restriction map is surjective. Flasque sheaves are acyclic for global sections, so $H^i(X,\mathcal{F})=0$ for $i>0$.

For an open cover $\mathcal{U}=(U_i)$, the Čech cochain group iswith the alternating sum of restrictions as differential. Its cohomology is the Čech cohomology of $\mathcal{F}$ with respect to $\mathcal{U}$.

The Čech cochain complex places sections on $(p+1)$-fold intersections in degree $p$ and uses the alternating sum of restriction maps as its differential.

A Čech cochain is a cocycle when its alternating coboundary vanishes. For a multiplicative one-cochain $(g_{ij})$, this says $g_{ij}{g}_{jk}=g_{ik}$ on triple intersections.

A Čech coboundary is the image of a cochain in the preceding degree. Multiplicatively, a zero-cochain $(h_i)$ changes a one-cocycle by $g_{ij}\mapsto h_i^{-1}{g}_{ij}{h}_j$.

If every nonempty finite intersection of members of an open cover has vanishing higher sheaf cohomology for $\mathcal{F}$, then the cover's Čech cohomology computes $H^p(X,\mathcal{F})$. An affine open cover of a separated scheme satisfies this condition for a quasi-coherent sheaf because its finite intersections are affine.

For $X=U\cup V$, the short exact sequence of sheaves obtained by restricting to $U$, $V$, and $U\cap V$ induces a long exact sequence