Is the sheafification of a presheaf of graded modules graded?

Question

Let $X$ be a scheme, and $F$ be a presheaf of graded $O_X$-modules (give $O_X(U)$ the trivial grading for every open set $U$). Does the sheafification of $F$ also have a graded structure such that the natural map $i : F(U) \rightarrow F^+(U)$ preserves grading?

Suppose $F(U) = \oplus_{n \in \mathbb{N}} F_n(U)$. Since colimits commute with other colimits, we have that for $p \in X$, $F_p = \varinjlim_{U \ni p} \oplus_{n \in \mathbb{N}} F_n(U) = \oplus_{n \in \mathbb{N}} \varinjlim_{U \ni p} F_n(U)= \oplus_{n \in \mathbb{N}} (F_n)_p$. Therefore , the stalks are graded $O_{X,p}$-modules.

My idea is that given a collection of compatible germs $(s_p)_{p \in U} \in F^+(U)$, for each $n \in \mathbb{N}$, take the degree $n$ components of each $s_p$, which are compatible germs in $F^+_n(U)$. This gives a morphism $F^+(U) \rightarrow \prod_{n \in \mathbb{N}} F^+_n(U)$. I'm stuck trying to show that only finitely many nonzero terms, so we can replcae $\prod$ with $\oplus$.

Edit: By "graded sheaf of modules", I mean that for every open set $U$, $F(U)$ is a graded $O_X(U)$-module, and restriction maps preserve degree.

Answer 1

I think this works:

Let $k$ be a field and $X = \coprod_{n \in \mathbb{N}} \operatorname{Spec}(k)$, the infinite disjoint union of $\operatorname{Spec}(k$). Denote the points in $X$ as $(p_0, p_1, ...)$.

Consider the presheaf $F$ that sends every open set $U$ to $T(O_X(U))$, where $T$ denotes the tensor algebra. This is a presheaf of graded modules. The stalk at $p_i$ is $T(O_X(p_i)) = T(k) = k \oplus k \oplus ...$ Note that none of the degree $n$ terms are zero. Let $e_i \in k \oplus k \oplus ...$ be the element with a 1 in the $i$th entry and 0 everywhere else.

Then, consider the collection of compatible germs that is $e_i$ on $p_i$. This is a collection of compatible germs so corresponds to an element $s \in F^+(X)$. Then, the stalk of $s$ at every $p_i$ is homogeneous of degree $i$. Therefore, if $F^+(X)$ was a graded $O_X(X)$-module, there must be an element whose $n$th homogeneous component is not zero for every $n$. This is a contradiction.

Therefore, the sheafification of a presheaf of graded $O_X$ modules does not have to be graded, at least not in the usual way.