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?
1 Answers
This is an example showing that the kernel of a homomorphism between quasicoherent sheaves on a locally ringed space is not necessarily quasicoherent. 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 quasicoherence for $M$ is not satisfied at $s$.
Note that the kernel of $[f]$ in the category of quasicoherent sheaves on $S$ exists and is zero, so, stricto sensu, the above example is not an example where the category of quasicoherent sheaves is not abelian.
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1Just a comment. This answer may be a little misleading to some people. Actually, the category of quasicoherent sheaves over any scheme is abelian (even Grothendieck abelian, but the canonical inclusion into modules does not preserve limits, .e.g, kernel). – user40276 Apr 07 '19 at 12:41