Constant sheaf

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

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

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

Basics

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

A detailed example

Constant presheaf on a two-point discrete space

Two-point discrete topological space

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

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

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

Intermediate step for the constant sheaf

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

Constant sheaf on a two-point topological space

Modifying further to satisfy the gluing axiom, let

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

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

See also

• Locally constant sheaf

References

1. "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