What are some good examples of (non-quasicoherent) sheaves not satisfying the conclusion of Hartshorne Lemma II.5.3?

Question

Hartshorne, Algebraic Geometry, Lemma II.5.3 reads (roughly):

Let $X = \operatorname{Spec} A$, let $f \in A$, and let $\mathscr{F}$ be a quasicoherent sheaf on $X$.

(a) If $s \in \Gamma(X, \mathscr{F})$ with $s|_{D(f)} = 0$ then $f^n s = 0$ for $n \gg 0$.

(b) If $s \in \Gamma(D(f), \mathscr{F})$ then $f^n s$ extends to all of $X$ for $n \gg 0$.

The proof essentially amounts to clearing denominators.

I'm trying to get a good picture of exactly what this is ruling out, but I don't have a great mental picture of a (obviously non-quasicoherent) sheaf where (a) and (b) don't hold. Are there any good examples to think of?

Answer 1

I never liked how Hartshorne presented this — my understanding is that the lemma says $\Gamma(D(f), \mathscr{F}) = \Gamma(X, \mathscr{F})_f$. There's always a map going left, and he shows (a) injectivity (b) surjectivity. This characterizes quasicoherence, so any non-quasicoherent $\mathscr{O}_X$-module should do.

Let's try a standard example of a non-quasicoherent ideal sheaf: $A = k[x]_{(x)}$ and $\mathscr{F}$ assigns $k(x)$ to the generic point and $0$ to the whole space. Then (b) fails with $f = x$.

I wish I had more intuition to offer you but non-quasicoherent sheaves seem pretty exotic to me since they appear to break out of the usual correspondence between geometry and algebra.

Answer 2

The standard example I usually think of (for (b)) is the following: let $X = \mathbb{A}^1$ and let $\mathscr{F}$ be the sheaf of holomorphic functions. (In other words, the "analytic structure sheaf".)

So $\Gamma(X,\mathscr{F})$ is the set of entire holomorphic functions on $\mathbb{C}$, and $\Gamma(X,\mathscr{F})_x$ is the set of meromorphic functions with (finite-order) poles at 0 (since we can invert $x$).

But this is not the same as $\Gamma(\mathbb{C}-\{0\},\mathscr{F})$: this contains holomorphic functions with essential singularities at $0$, like $e^{1/x}$.

So you're right that the quasicoherent case is about "clearing denominators" -- in this case, we can't do that since no power of $x$ can make $e^{1/x}$ extend to $x=0$. (As Hoot points out, this is one of the ways the correspondence between geometry and algebra can break down. Things like this are why the GAGA theorems only hold for projective varieties!)

Answer 3

Since Hoot's and Jake's excellent answers don't address point a), here is a counterexample to a) in the non quasi-coherent case.

Let $A$ be a discrete valuation ring with uniformizing parameter $t$, so that $X=\operatorname {Spec}(A)=\{\eta,m\}$ with $\eta=(0)$ the generic point and $m=(t)$ the closed point.
Define the sheaf $\mathcal F$ of $\mathcal O_X$-modules by $\Gamma(X, \mathcal{F})=A$ and $\Gamma(D(t), \mathcal{F})=0$
Then the section $s=1_A\in \Gamma(X, \mathcal{F})=A$ has the property that $s|_{D(t)} = 0\in \Gamma(D(t), \mathcal{F})$ but $t^n\cdot 1_A=t^n=0 \in \Gamma(X, \mathcal{F})=A$ is false for any $n\gt 0$ .