Constant sheaf
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In mathematics, the constant sheaf on a topological space associated to a set is a sheaf of sets on whose stalks are all equal to . It is denoted by or . The constant presheaf with value is the presheaf that assigns to each open subset of the value , and all of whose restriction maps are the identity map . The constant sheaf associated to is the sheafification of the constant presheaf associated to . This sheaf identifies with the sheaf of locally constant -valued functions on .[1]
In certain cases, the set may be replaced with an object in some category (e.g. when is the category of abelian groups, or commutative rings).
Constant sheaves of abelian groups appear in particular as coefficients in sheaf cohomology.
Basics
[edit]Let be a topological space, and a set. The sections of the constant sheaf over an open set may be interpreted as the continuous functions , where is given the discrete topology. If is connected, then these locally constant functions are constant. If is the unique map to the one-point space and is considered as a sheaf on , then the inverse image is the constant sheaf on . The sheaf space of is the projection map (where is given the discrete topology).
A detailed example
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Constantpresheaf.png/300px-Constantpresheaf.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/b5/2_point_discrete_space.png/100px-2_point_discrete_space.png)
Let be the topological space consisting of two points and with the discrete topology. has four open sets: . The five non-trivial inclusions of the open sets of are shown in the chart.
A presheaf on chooses a set for each of the four open sets of and a restriction map for each of the inclusions (with identity map for ). The constant presheaf with value , denoted , is the presheaf where all four sets are , the integers, and all restriction maps are the identity. is a functor on the diagram of inclusions (a presheaf), because it is constant. It satisfies the gluing axiom, but is not a sheaf because it fails the local identity axiom on the empty set. This is because the empty set is covered by the empty family of sets, , and vacuously, any two sections in are equal when restricted to any set in the empty family . The local identity axiom would therefore imply that any two sections in are equal, which is false.
To modify this into a presheaf that satisfies the local identity axiom, let , a one-element set, and give the value on all non-empty sets. For each inclusion of open sets, let the restriction be the unique map to 0 if the smaller set is empty, or the identity map otherwise. Note that is forced by the local identity axiom.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Constantsheaf_intermediate_step.png/300px-Constantsheaf_intermediate_step.png)
Now is a separated presheaf (satisfies local identity), but unlike it fails the gluing axiom. Indeed, is disconnected, covered by non-intersecting open sets and . Choose distinct sections in over and respectively. Because and restrict to the same element 0 over , the gluing axiom would guarantee the existence of a unique section on that restricts to on and on ; but the restriction maps are the identity, giving , which is false. Intuitively, is too small to carry information about both connected components and .
![](http://upload.wikimedia.org/wikipedia/commons/thumb/7/79/Constant_sheaf_with_categorical_product.png/300px-Constant_sheaf_with_categorical_product.png)
Modifying further to satisfy the gluing axiom, let
,
the -valued functions on , and define the restriction maps of to be natural restriction of functions to and , with the zero map restricting to . Then is a sheaf, called the constant sheaf on with value . Since all restriction maps are ring homomorphisms, is a sheaf of commutative rings.
See also
[edit]References
[edit]- ^ "Does the extension by zero sheaf of the constant sheaf have some nice description?". Mathematics Stack Exchange. Retrieved 2022-07-08.
- Section II.1 of Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
- Section 2.4.6 of Tennison, B.R. (1975), Sheaf theory, ISBN 978-0-521-20784-3