Extension by zero operation

Question

Suppose you have a closed subset $Z$ of a topological space $X$, and $F$ is a sheaf on $Z$. Then one can consider the extension by zero sheaf $F^X$ on $X$.

What are some examples and situations which illustrate the utility of extension by zero sheaves? Alternatively, what are some situations in which extension by zero arises naturally?

Note I am leaving this question quite open. So feel free to restrict your example/situation to when the data of $(Z,X,F)$ is, for example, algebraic or analytic.

Answer 1

If you were inventing sheaves from scratch, I suspect you would first reinvent extension by zero when considering quotient sheaves.

Let $X$ be a topological space, and let $Y\subset X$ and $Z:=X\smallsetminus Y$ be complementary subspaces. For a field $k$, we have a natural inclusion of sheaves $k_Y \rightarrow k_X$ between the respective constant sheaves.

• On a connected open set $U\subset X$ which is disjoint from $Y$ (and therefore is contained in $Z$), the map $k_Y(U) \rightarrow k_X(U)$ is the zero map $0\rightarrow k$, so the quotient sheaf is $$ k_X(U) / k_Y(U) = k $$
• On a connected open set $U\subset X$ which intersects $Y$ (and therefore is not contained in $Z$), the map $k_Y(U) \rightarrow k_X(U)$ is the identity $k\rightarrow k$, so the quotient sheaf is $$ k_X(U) / k_Y(U) = 0 $$

We see that the quotient sheaf $k_X/k_Y$ is not the constant sheaf on the complement $Z$, as one might naively expect, but rather the extension by zero of the constant sheaf on $Z$ to all of $X$.

This argument extends as follows, with $X,Y,Z$ as above and $f:Y\rightarrow Z$, $g:Z\rightarrow X$ the respective inclusions. Then every sheaf $\mathcal{F}$ on $X$ fits into a short exact sequence of sheaves $$ 0 \rightarrow f_*f^*\mathcal{F} \rightarrow \mathcal{F} \rightarrow g_!g^*\mathcal{F} \rightarrow 0 $$ The `lower shriek' operation $g_!$ is the extension by zero of the restriction $g^*$ of $\mathcal{F}$ to $Z$.