Category of quasicoherent sheaves not abelian

Question

Wikipedia mentions that the category of quasicoherent sheaves need not form an abelian category on general ringed spaces. Is there a `naturally occurring' example of this failing, even for locally ringed spaces?

Answer 1

This is an example showing that the kernel of a homomorphism between quasi-coherent sheaves on a locally ringed space is not necessarily quasi-coherent. This example is similar to the one given in EGA O$_\text{I}$ 5.1.1 (first edition).

Take for space $S$ the spectrum of a discrete valuation ring, with its Zariski topology. It has only two points, the closed point $s$, and the generic point $t$. The only neighborhood of $s$ is $S$. A sheaf of sets $F$ on $S$ is the datum of two sets: $F_s$ ($= \Gamma(S,F)$, the stalk of $F$ at $s$), and $F_t$, its stalk at $t$, together with a restriction (or "specialization") map $F_s \to F_t$. A sheaf of local rings $R$ on $S$ is a triple$$(R_s, R_t, u : R_s \to R_t),$$where $R_s$, $R_t$ are local rings, and $u : R_s \to R_t$ is a homomorphism of rings (not necessarily local). Choose $R$ such that there exists $f \in R_s$ which is a nonzero divisor, and the kernel of multiplication by $u(f)$ on $R_t$ is nonzero. One can take, for example, $R_s = Z_p$, $R_t = F_p$, $u$ the canonical surjection, and $f = p$. Let us denote by $\text{Hom}$ the set of homomorphisms of sheaves of $R$-modules on $S$. We have$$\text{Hom}(R,R) = \Gamma(S,R) = R_s.$$So the element $f$ defines a homomorphism $[f] : R \to R$, whose stalk at $s$ (resp. $t$) is multiplication by $f$ (resp. $u(f)$) on $R_s$ (resp. $R_t$). Consider the sheaf of $R$-modules $M = \text{Ker}([f])$. By construction, $M_t \ne 0$, so $M \ne 0$, but $$M_s = \Gamma(S,M) = \text{Ker}(f_s) = 0.$$Therefore any homomorphism $R^{(I)} \to M$ is automatically zero, and the axiom of quasi-coherence for $M$ is not satisfied at $s$.

Note that the kernel of $[f]$ in the category of quasi-coherent sheaves on $S$ exists and is zero, so, stricto sensu, the above example is not an example where the category of quasi-coherent sheaves is not abelian.