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I'm fairly certain there is both a typo and an omission in this exercise. It reads

"Let $X$ be a scheme and $f \in \mathcal{O}_X(X)$. Show that $U \mapsto f|_U \mathcal{O}_X(U)$ for every affine open subset $U$ defines a sheaf of ideals on $X$. We denote this sheaf of ideals by $f \mathcal{O}_X$..."

Surely "affine" above should be omitted, but my real question is

are we missing any hypotheses on $f$?

It is clear that $f \mathcal{O}_X$ defines a presheaf satisfying the uniqueness condition, i.e., if an global section restricts to zero everywhere on an open over, then the element is identically $0$, as $\mathcal{O}_X$ is a sheaf to begin with. However, in proving that is satisfies the criterion regarding the glueing of local sections,

do we not need the hypothesis that $f|_U$ is not a zero-divisor for all $U$?

This is the only condition in which I'm able to get at the result. Am I missing anything?

EDIT: Please see Martin Brandenburg's answer. I was mistaken. (But it was not a regrettable mistake, as the point which I missed deserves an extra line or two of qualification.)

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2 Answers

up vote 8 down vote accepted

"Surely "affine" above should be omitted"

No! The presheaf $U \mapsto (f|_U)$ is not a sheaf in general. But the associated sheaf is given on affine opens by this formula.

This sheaf is best seen as a special case of general constructions (whereas doing the exercise directly is unnecessarily cumbersome). Let $M$ be a quasi-coherent sheaf on $X$ (in the exercise $M = \mathcal{O}_X$) and $f$ a global section of $\mathcal{O}_X$. Then $f$ induces a homomorphism $f : M \to M$. It is well-known that the category of quasi-coherent sheaves is abelian, in particular we can construct the image of $f : M \to M$, denoted by $fM$, which is a quasi-coherent subsheaf of $M$. Obviously this construction is local on $X$. If $X=\mathrm{Spec}(A)$ is affine, then there is an equivalence of categories between quasi-coherent sheaves on $X$ and $A$-modules, given by the global section functor. This equivalence preserves, in particular, images.

For general $X$ it follows that for every open affine $U \subseteq X$ the module $\Gamma(U,fM)$ is the image of $f|_U : \Gamma(U,M) \to \Gamma(U,M)$, i.e. $\Gamma(U,fM) = f|_U \cdot \Gamma(U,M)$. This does not hold for arbitrary opens $U$. In that case we only have $f|_U \cdot \Gamma(U,M) \subseteq \Gamma(U,fM)$. The converse holds when $f$ is regular (i.e. $f : M \to M$ is a monomorphism), and this is what you have proven:

If $s \in \Gamma(U,fM)$, there is an open covering $U = \cup_i U_i$ and sections $t_i \in \Gamma(U_i,M)$ satisfying $s|_{U_i} = f|_{U_i} \cdot t_i$. We have $f|_{U_i \cap U_j} t_i|_{U_i \cap U_j} = f|_{U_i \cap U_j} t_j|_{U_i \cap U_j}$. Since $f|_{U_i \cap U_j}$ is regular, this shows $t_i|_{U_i \cap U_j}=t_j|_{U_i \cap U_j}$, i.e. the $t_i$ glue to a section $t \in \Gamma(U,M)$ satisfying $s = f|_U \cdot t$.

Perhaps someone can add an example for $\Gamma(U,f \mathcal{O}) \neq f|_U \cdot \Gamma(U,\mathcal{O})$ in a separate answer?

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thanks for the quick response. How does one get a presheaf on $X$ by only defining it on the affine opens? Is it by the following? Let $\{ U_i = spec A_i\}$ be an affine open cover. We define, for $a_i \in A_i$, $\mathcal{O}_X(D(a_i)) = f|_{U_i} (A_i)_{a_i}$, - and then extend to all open sets of $X$ after defining it on the basis of the principal open sets of the $A_i$? –  Joshua Seaton Feb 19 '13 at 17:26
Yeah, this is what I meant by cumbersome. Lots of things to check, but in fact it is not necessary from the correct point of view. –  Martin Brandenburg Feb 19 '13 at 17:32
Ah. That's a bit much to be left tacit so early on in the book. Thanks again. –  Joshua Seaton Feb 19 '13 at 17:46
@MartinBrandenburg: at that point of the book, quasi-coherent sheaves are not yet introduced, so some dirty things (but I find instructive for beginners) have to be done. What is interesting is to find an example with $(fO_X)(U)\ne f O_X(U)$. –  user18119 Feb 19 '13 at 23:03
Yes, perhaps you can add such an example in a separate answer? –  Martin Brandenburg Feb 21 '13 at 10:43
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An example showing that "affine" should not be omitted.

Let $X$ be the union of an affine line $\Gamma_1$ and a projective line $\Gamma_2$ meeting at two points. Suppose the ground field $k$ has characteristic different from $2$. Let $t$ be the parameter of $\Gamma_1$ such that the intersection points are $t=1$ and $t=-1$. One can check that:

  1. $O_X(X)=k+(t^2-1)k[t]$;

  2. Let $f=t^2-1$. Then $(fO_X)(X)=(t^2-1)k[t]\supsetneq f.O_X(X)$.

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Can you explain a little bit about the construction of $X$, and the calculation of $O_X(X)$? I hope to understand the nice example, thanks! –  mqx Oct 2 '13 at 4:38
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