In mathematics, the stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point.
Motivation and definition Sheaves are defined on open sets, but the underlying topological space X {\displaystyle X} consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a single fixed point x {\displaystyle x} of X {\displaystyle X} . Conceptually speaking, we do this by looking at small neighborhoods of the point. If we look at a sufficiently small neighborhood of x {\displaystyle x} , the behavior of the sheaf F {\displaystyle {\mathcal {F}}} on that small neighborhood should be the same as the behavior of F {\displaystyle {\mathcal {F}}} at that point. Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort. The precise definition is as follows: the stalk of F {\displaystyle {\mathcal {F}}} at x {\displaystyle x} , usually denoted F x {\displaystyle {\mathcal {F}}_{x}} , is:
F x := lim → U ∋ x F ( U ) . {\displaystyle {\mathcal {F}}_{x}:=\varinjlim _{U\ni x}{\mathcal {F}}(U).}
Here the direct limit is indexed over all the open sets containing x {\displaystyle x} , with order relation induced by reverse inclusion ( U ≤ V {\displaystyle U\leq V} , if U ⊇ V {\displaystyle U\supseteq V} ). By definition (or universal property) of the direct limit, an element of the stalk is a germ of sections, an equivalence class of elements f U ∈ F ( U ) {\displaystyle f_{U}\in {\mathcal {F}}(U)} , where two such sections f U {\displaystyle f_{U}} and f V {\displaystyle f_{V}} are considered equivalent if the restrictions of the two sections coincide on some neighborhood of x {\displaystyle x} .
Alternative definition There is another approach to defining a stalk that is useful in some contexts. Choose a point x {\displaystyle x} of X {\displaystyle X} , and let i {\displaystyle i} be the inclusion of the one point space { x } {\displaystyle \{x\}} into X {\displaystyle X} . Then the stalk F x {\displaystyle {\mathcal {F}}_{x}} is the same as the inverse image sheaf i − 1 F {\displaystyle i^{-1}{\mathcal {F}}} . Notice that the only open sets of the one point space { x } {\displaystyle \{x\}} are { x } {\displaystyle \{x\}} and ∅ {\displaystyle \emptyset } , and there is no data over the empty set. Over { x } {\displaystyle \{x\}} , however, we get:
i − 1 F ( { x } ) = lim → U ⊇ { x } F ( U ) = lim → U ∋ x F ( U ) = F x . {\displaystyle i^{-1}{\mathcal {F}}(\{x\})=\varinjlim _{U\supseteq \{x\}}{\mathcal {F}}(U)=\varinjlim _{U\ni x}{\mathcal {F}}(U)={\mathcal {F}}_{x}.}
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