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Let $F$ be a sheaf of $k$-algebras on a topological space $X$. So by definition, $F$ is a contravariant functor from the category of open sets of $X$ to the category of $k$-algebras, satisfying a certain patching condition and a uniqueness condition.

If $U$ is open in $X$, then we can define a restricted sheaf $F_{|U}$ on $U$, where for $V \subseteq U$ open, we set $F_{|U}(V) = F(V)$.

Suppose now that for each $U$ open, $F(U)$ is actually a $k$-subalgebra of functions $U \rightarrow k$. Then for any subset $W$ of $X$, not necessarily open, we may define a restriction sheaf $F_{|W}$ on $W$, where for $U$ open in $W$, we say that a function $f:U \rightarrow k$ lies in $F_{|W}(U)$ if and only if there exist open sets $U_i$ of $X$, and functions $f_i \in F(U_i)$, such that $U = W \cap (\bigcup\limits_i U_i)$, and $f_{|W \cap U_i} = f_{i|W \cap U_i}$ for each $i$.

My question is, suppose we do not assume that the objects $F(U)$ (for $U$ open in $X$) are actually sets of functions from $U$ to $k$. Is there some way to make sense out of a restriction sheaf $F_{|W}$ for any subset $W$ of $X$? Or in general, will this only make sense for $W$ open in $X$?

D_S
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1 Answers1

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When $W$ is closed, there is the notion of the subsheaf with supports in $W$. This is mentioned in Hartshorne's Algebraic Geometry, exercise 1.20.

Let $F$ be a sheaf on $X$, and let $s\in F(U)$ be a section over an open set $U$. Recall the support of $s$ is defined to be the set $\{p\in U : s_p\neq 0\}$. One can show this is a closed subset.

Fix a closed subset $Z\subset X$. We can define a presheaf on $Z$ by assigning to an open set $U$ the set $\Gamma_{Z\cap U}(U, \mathcal F|_U)$, where $\Gamma_{Z\cap U}$ is the subgroup of $\Gamma(U,\mathcal F|_U)$ defined by taking all sections with support in $Z\cap U$. This turns out to be a sheaf and is called the subsheaf of $F$ with supports in $Z$.

More generally, you can take the inclusion $W\rightarrow X$ and consider the inverse image sheaf, as noted in the comments.

Potato
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