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Suppose $(X,\mathcal{O}_X)$ and $(Y,\mathcal{O}_Y)$ are locally ringed spaces. Then morphisms glue, that is, if $\{U_i\}_i$ is an open cover of $X$, then "to give a morphism $X\to Y$ is the same as giving morphisms $U_i\to Y$ that agree on overlaps".

This sentence is formally similar to the sentence that expresses the condition a presheaf needs to satisfy in order to be a sheaf. So one can try to say that the above sentence can be expressed as: there is an exact sequence

$0\to Mor(X,Y)\to \prod_i Mor(U_i, Y)\to \prod_{i,j} Mor(U_i\cap U_j,Y)$

and I have just seen this in a couple of solutions I found online to an exercise in Hungerford.

However I can't make sense of this. The first map would be $f\mapsto (f|_{U_i})$, but what would be the second map? The expression $(f_i)\mapsto ((f_i-f_j)|_{U_i\cap U_j})$ doesn't make sense, does it? What does it mean to substract morphisms of ringed spaces? And more importantly, what does exactness of the final arrow mean?

Am I missing something obvious? Is there any way around this?

Bruno Stonek
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    Perhaps I should just ditch solution guides to Hartshorne, as suggested in a comment to my previous post... – Bruno Stonek Nov 03 '13 at 00:17
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    This is really a statement saying the image of the first is the equalizer of the second set of arrows (which ones?), and the first map is an injection. It's technically incorrect, but intuitively useful notation. – Alex Youcis Nov 03 '13 at 00:23
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    The point being, of course, that one can just as easily define sheaves of sets, and the presheaf $U \mapsto \mathrm{Mor}(U, Y)$ is a sheaf. – Zhen Lin Nov 03 '13 at 01:18

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