Question about cokernel of a morphism of sheaves

Question

Let $X$ be a scheme and let $Q$ be an $O_X$-ideal ($Q(U)$ is an ideal of $O_X(U)$ for all open $U$). Let $L$ be the cokernel of $0 \to Q \to O_X$. Let $U = \operatorname{Spec}R \subseteq X$ be an affine open and $A = Q(U)$.

1. I want to prove that $L(U) = R/A$. $L$ is the sheafification of $L'$ where $L'(V) = O_X(V)/Q(V)$, but I wasn't sure if sheafifying changes this..

2. How can I show that $(Y \cap U, O_Y) \cong \operatorname{Spec} (R/A)$? where $Y = \{ y \in X : 1 \notin Q_y \}$ and $O_Y$ is the sheaf on $Y$ given by restricting $L$ to $Y$.

Any clarification is appreciated. Thank you!

Answer 1

First, note that we don't ever use anything outside $U$, so we might as well assume that $X=U$ here. To handle 2), we refer to a result in the section on "closed subpreschemes" (my copy has this as chapter II, section 5, theorem 3 on page 106, though my version is the "second, expanded edition" which may not have the same page numbers as the original):

Let $X=\operatorname{Spec} R$ and let $Y\subset X$ be a closed subprescheme with inclusion $f:Y\to X$. Let $Q$ be that $O_X$-ideal defining $Y$, $A=\Gamma(X,Q)$. Then $Y$ is canonically isomorphic to $\operatorname{Spec} R/A$.

The proof of that statement may/should also give enough ammunition to resolve 1) via some of the statements about stalks, but the route using quasicoherent sheaves is preferable (in my view). The big idea is that quasicoherent sheaves on an affine scheme $\operatorname{Spec} R$ are equivalent to $R$-modules (via taking global sections, with inverse functor the "associated sheaf" construction), and their sections on standard affine opens as well as stalks have particularly nice descriptions just in terms of the ring/module data. Now, as ideal sheaves of closed subschemes are always quasicoherent and the structure sheaf is quasicoherent, we see that $L$ must be quasicoherent as well - in a short exact sequence of sheaves, if two of the three are quasicoherent, the third must be as well. So $L$ is quasicoherent sheaf, and in fact it must be the quasicoherent sheaf associated to $R/A$, which has global sections exactly $R/A$.

(In general, on a non-affine scheme, you will need to sheafify which will alter global sections of $L$. The easiest example is the ideal sheaf of two distinct points in $\Bbb P^1_k$.)