Constant sheaf

In mathematics, the constant sheaf on a topological space $X$ associated to a set $A$ is a sheaf of sets on $X$ whose stalks are all equal to $A$ . It is denoted by ${\underline {A}}$ or $A_{X}$ . The constant presheaf with value $A$ is the presheaf that assigns to each open subset of $X$ the value $A$ , and all of whose restriction maps are the identity map $A\to A$ . The constant sheaf associated to $A$ is the sheafification of the constant presheaf associated to $A$ . This sheaf identifies with the sheaf of locally constant $A$ -valued functions on $X$ .^[1]

In certain cases, the set $A$ may be replaced with an object $A$ in some category ${\textbf {C}}$ (e.g. when ${\textbf {C}}$ is the category of abelian groups, or commutative rings).

Constant sheaves of abelian groups appear in particular as coefficients in sheaf cohomology.

Basics

Let $X$ be a topological space, and $A$ a set. The sections of the constant sheaf ${\underline {A}}$ over an open set $U$ may be interpreted as the continuous functions $U\to A$ , where $A$ is given the discrete topology. If $U$ is connected, then these locally constant functions are constant. If $f:X\to \{{\text{pt}}\}$ is the unique map to the one-point space and $A$ is considered as a sheaf on $\{{\text{pt}}\}$ , then the inverse image $f^{-1}A$ is the constant sheaf ${\underline {A}}$ on $X$ . The sheaf space of ${\underline {A}}$ is the projection map $A$ (where $X\times A\to X$ is given the discrete topology).

A detailed example

Constant presheaf on a two-point discrete space

Let $X$ be the topological space consisting of two points $p$ and $q$ with the discrete topology. $X$ has four open sets: $\varnothing ,\{p\},\{q\},\{p,q\}$ . The five non-trivial inclusions of the open sets of $X$ are shown in the chart.

A presheaf on $X$ chooses a set for each of the four open sets of $X$ and a restriction map for each of the inclusions (with identity map for $U\subset U$ ). The constant presheaf with value ${\textbf {Z}}$ , denoted $F$ , is the presheaf where all four sets are ${\textbf {Z}}$ , the integers, and all restriction maps are the identity. $F$ 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, $\varnothing =\bigcup \nolimits _{U\in \{\}}U$ , and vacuously, any two sections in $F(\varnothing )$ are equal when restricted to any set in the empty family $\{\}$ . The local identity axiom would therefore imply that any two sections in $F(\varnothing )$ are equal, which is false.

To modify this into a presheaf $G$ that satisfies the local identity axiom, let $G(\varnothing )=0$ , a one-element set, and give $G$ the value ${\textbf {Z}}$ 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 $G(\varnothing )=0$ is forced by the local identity axiom.

Now $G$ is a separated presheaf (satisfies local identity), but unlike $F$ it fails the gluing axiom. Indeed, $\{p,q\}$ is disconnected, covered by non-intersecting open sets $\{p\}$ and $\{q\}$ . Choose distinct sections $m\neq n$ in $\mathbf {Z}$ over $\{p\}$ and $\{q\}$ respectively. Because $m$ and $n$ restrict to the same element 0 over $\varnothing$ , the gluing axiom would guarantee the existence of a unique section $s$ on $G(\{p,q\})$ that restricts to $m$ on $\{p\}$ and $n$ on $\{q\}$ ; but the restriction maps are the identity, giving $m=s=n$ , which is false. Intuitively, $G(\{p,q\})$ is too small to carry information about both connected components $\{p\}$ and $\{q\}$ .

Modifying further to satisfy the gluing axiom, let

$H(\{p,q\})=\mathrm {Fun} (\{p,q\},\mathbf {Z} )\cong \mathbb {Z} \times \mathbb {Z}$ ,

the $\mathbf {Z}$ -valued functions on $\{p,q\}$ , and define the restriction maps of $H$ to be natural restriction of functions to $\{p\}$ and $\{q\}$ , with the zero map restricting to $\varnothing$ . Then $H$ is a sheaf, called the constant sheaf on $X$ with value ${\textbf {Z}}$ . Since all restriction maps are ring homomorphisms, $H$ is a sheaf of commutative rings.

References

^ "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, Cambridge University Press, ISBN 978-0-521-20784-3

[1] "Does the extension by zero sheaf of the constant sheaf have some nice description?". Mathematics Stack Exchange. Retrieved 2022-07-08.

[1]

Basics

A detailed example

See also

References