What do the local sections of an nilpotent ideal sheaf look like? Let $(X, O_X)$ be a scheme. and $I$ an nilptoent ideal sheaf. i.e. $I^n=0$ for some $n$. Would this imply that each $I(U)$ is an nilpotent ideal of $O_X(U)$?
 A: I’m submitting this as an alternative to hm2020’s answer. It’s an elaboration of what I wrote in the comments.
Let $I\subseteq \mathcal{O}_X$ be an ideal sheaf, and let $\mathcal{F}$ be the presheaf assigning to each open subset $U$ of $X$ the ideal $I(U)^n\subseteq \mathcal{O}_X(U)$. You say that $I$ is nilpotent of degree $n$ if $\mathcal{F}^\#$ is zero, where $\#$ is used to denote sheafification. But, since $\mathcal{F}$ is a separated presheaf, being a subsheaf of the sheaf $\mathcal{O}_X$, one has that $\mathcal{F}=0$ if and only if $\mathcal{F}^\#=0$ (e.g. see [1, Tag00WB]). Thus, we deduce the following:

Fact: Let $X$ be a scheme and $I$ an ideal sheaf of $\mathcal{O}_X$. Then, the following are equivalent:

*

*For all open subsets $U$ the ideal $I(U)^n$ is zero.

*The sheafification of the presheaf $U\mapsto I(U)^n$ is zero.


[1] Various authors, 2020. Stacks project. https://stacks.math.columbia.edu/
A: Given a quasi coherent sheaf of ideals $\mathcal{I} \subseteq \mathcal{O}_X$ where $X$ is a scheme, we get a presheaf
$\mathcal{I}(n)(U):=\{ \sum_I s_Ix_{i_1}\cdots x_{i_n}: x_{i_j}\in \mathcal{I}(U)$ and $s_I\in \mathcal{O}_X(U)\} \subseteq \mathcal{I}(U)$
If $U_i:=Spec(A_i)$ and $\mathcal{I}(U_i):=I_i \subseteq A_i$ it follows
by definition
$\mathcal{I}(n)(U_i):=I_i^n.$
Definition 0. Let $\mathcal{I}^n:=\mathcal{I}(n)^{+}$ be the sheafification of $\mathcal{I}(n)$. It follows there is a canonical injective map of sheaves
$\mathcal{I}^n \rightarrow \mathcal{I}$.
It follows $\mathcal{I}^n(U_i)=I^n_i$.
Definition 1. Let us define a quasi coherent sheaf of ideals $\mathcal{I}$ to be "nilpotent" iff there is an integer $n\geq 1$ with $\mathcal{I}^n(U)=0$ for all open sets $U$.
If $I \subseteq A$ is an ideal with $I^n=0$ and $\mathcal{I}\subseteq \mathcal{O}_X$ with $X:=Spec(A)$ is the corresponding sheaf of ideals, it follows $\mathcal{I}^n(X)\cong I^n=0$. You may moreover prove that there is a canonical isomorphism
I1. $\mathcal{I}(U)\cong \mathcal{O}_X(U)\otimes_A I \rightarrow^j  \mathcal{O}_X(U)$
where the map $j$ is the canonical map
$j(u\otimes x):= ux$.
Here $u\in \mathcal{O}_X(U)$ and $x\in I$. The map $j$ is an injective map since the ring extension $A \rightarrow \mathcal{O}_X(U)$ is flat for any open set $U$. Hence the injective map $j$ is obtained from the inclusion $I \subseteq A$ via tensor product.
Note: To prove that the canonical map $A\rightarrow \mathcal{O}_X(U)$ is flat
you should use an exercise in Atiyah-Macdonald (Ex 4.23) where you give an alternative construction of the structure sheaf. You define
$\mathcal{O}_X(U) :=lim_{D(f) \subseteq U}\mathcal{O}_X(D(f)):= lim_{D(f) \subseteq U}A_f$.
Since $A\rightarrow A_f$ is flat, it follows $A\rightarrow \mathcal{O}_X(U)$ is flat since it is a direct limit of flat maps.
Note: I want to recommend the exercise in AM since it gives an alternative construction of the structure sheaf of a scheme.
If you choose arbitrary elements
$w_1:=u_1\otimes x_1,...,w_n:=u_n\otimes x_n$ in $\mathcal{I}(U)$ it follows
$w_1\cdots w_n = u_1\cdots u_nx_1\cdots x_n=0$ since $x_1\cdots x_n=0$ in $I^n=0$.
and from I1 it follows $\mathcal{I}^n(U)=0$.
Question: "Would this imply that each $I(U)$ is an nilpotent ideal of $O_X(U)$?"
Answer: It seems like the answer is "yes" by the above argument, but there may of course be errors.
Conversely if $\mathcal{I}^n(U)=0$ for all open sets $U$ it follows in particular that $\mathcal{I}^n(X)=I^n =0$.
It seems: In the affine situation it follows the quasi coherent ideal sheaf $\mathcal{I}$ is nilpotent (in the sense of Definition 1) iff the ideal $I$ is nilpotent.
Example: The general case. You must be more careful when $X$ is no longer an affine scheme. The correspondence between ideal sheaves $\mathcal{I}$ and ideals $I \subseteq S$ when $S$ is a graded ring and $X:=Proj(S)$ is more complicated (see Hartshorne Exercise II.5.9).
It seems we should give the following  general definition for a quasi coherent sheaf of ideals $\mathcal{I}$ to be nilpotent:
Definition 2. The quasi coherent ideal sheaf $\mathcal{I}$ is nilpotent iff there is an integer $n \geq 1$ with $\mathcal{I}^n_{\mathfrak{p}}=0$
in $\mathcal{O}_{X,\mathfrak{p}}$ for every point $\mathfrak{p}$.
It seems to me Definition 2 agrees with Definition 1 in the affine situation but you have to check this.
