Sheaf (mathematics)

A sheaf of sets is a presheaf in which compatible sections on any open cover glue to one unique section on the union.

A sheaf may additionally carry abelian group or module structure on its sections, with restriction and gluing preserving that structure. The underlying sheaf of sets retains the same local-to-global axiom.

For inclusion of a point $i:\{x\}\to X$, this is the direct-image sheaf of a set. It is right adjoint to the stalk functor. It uses the singleton terminal set on opens omitting $x$, rather than the zero group used for abelian-group skyscraper sheaves.

The space of germs of a presheaf has basic opens given by the germs of one section over an open set. Projection to its base point is a local homeomorphism. Continuous sections of this projection give sheafification. For the constant presheaf of sets, the basic opens are $U\times \{t\}$, giving the ordinary product with a discrete fibre.

The sections are locally constant functions into a discrete set $T$, with ordinary restriction. It is the sheafification of the constant presheaf of sets. Its value on an empty open set is a singleton, even if $T$ is empty. For a topological space, this construction is left adjoint to the global sections functor: a map from it to a sheaf is a family of global sections indexed by $T$.

A morphism of sheaves assigns a map $\mathcal{F}(U)\to \mathcal{G}(U)$ on every open set, commuting with all restriction maps. For sheaves of abelian groups, sheaves of rings or sheaves of modules, the maps must also preserve the relevant algebraic structure. It induces maps on every stalk, and equality of sheaf morphisms can be checked there.

For a morphism of sheaves $\varphi :\mathcal{F}\to \mathcal{G}$ of abelian groups or modules, the cokernel sheaf is the sheafification of the presheaf $U\mapsto \mathcal{G}(U)/\varphi (\mathcal{F}(U))$. Its stalk at $x$ is $\mathrm{coker}({\mathcal{F}}_x\to {\mathcal{G}}_x)$. The presheaf quotient itself need not satisfy the sheaf gluing axiom, which explains why a surjective morphism of sheaves need not be surjective on global sections.

For a continuous map $f:X\to Y$, an f-morphism from a sheaf on $Y$ to a sheaf on $X$ is a family of the displayed maps commuting with restrictions. It is equivalently a morphism of sheaves $\mathcal{G}\to f_{\ast }\mathcal{F}$. For each $x\in X$ it induces a map ${\mathcal{G}}_{f(x)}\to {\mathcal{F}}_x$ on stalks, by pulling back the neighbourhood representing a germ.

A morphism of sheaves of modules is surjective when the induced map on every stalk is onto. Equivalently every target section can locally be lifted. It need not be onto on global sections; the long exact sequence in sheaf cohomology describes that obstruction.

For a morphism of sheaves of abelian groups or modules, the kernel sheaf has sections $\ker (\mathcal{F}(U)\to \mathcal{G}(U))$. These already satisfy the sheaf gluing axiom. Its stalk at $x$ is the kernel of ${\mathcal{F}}_x\to {\mathcal{G}}_x$, since filtered colimits of modules preserve exactness. A sheaf morphism is injective exactly when its kernel sheaf is zero.

A sheaf of abelian groups is a sheaf of sets whose sections on each open set form an abelian group and whose restriction maps are group homomorphisms. Its stalk of a sheaf is the corresponding group of germs. Kernels are computed on sections, whereas cokernels are obtained by sheafification of the sectionwise cokernel. An exact sequence can therefore be checked on stalks.

The direct sum of sheaves of abelian groups is the sheafification of the presheaf $U\mapsto {⨁}_i{\mathcal{F}}_i(U)$. It need not equal that presheaf: sections are locally represented using finitely many summands, while globally infinitely many can occur. For point-supported skyscraper sheaves at closed points, allowable point supports are a locally finite family of subsets.

A sheaf of abelian groups is injective if every morphism to it from a subsheaf extends to the containing sheaf. Every such sheaf embeds in an injective sheaf. Injective sheaves are flasque: the inclusion of extension by zero constant-integer sheaves associated to $U\subseteq V$ turns the extension property into surjectivity of restriction from $V$ to $U$.

An exact sequence of sheaves of abelian groups is a sequence of sheaf homomorphisms for which the image of each map equals the kernel of the next. Exactness can be checked on every stalk of a sheaf. Exactness of the associated sequences of sections on every open set is a stronger condition: a surjective sheaf homomorphism need only admit lifts locally.

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.

For an abelian group $A$, the constant sheaf has sections over an open set given by its locally constant $A$-valued functions. It is the sheafification of the constant presheaf, and its sections on a disconnected open set need not have a single common value.

Every nonempty open subset of an irreducible space is connected. A locally constant function into a fixed abelian group $D$ is therefore constant on that open set. Restrictions between nonempty opens identify with the identity of $D$, and restrictions to the empty set are surjective onto zero. Thus the constant sheaf is flasque, with global sections $D$ and zero higher sheaf cohomology on a nonempty irreducible space. For an integral scheme its constant function-field sheaf has its natural structure-sheaf action by regular functions viewed in the function field.

The sheaf gluing axiom says that sections $s_i$ on an open cover $(U_i)$ that agree on every overlap $U_i\cap U_j$ glue to a unique section on ${\bigcup }_i{U}_i$.