I recently bought a copy of S. Ramanan's(adsbygoogle = window.adsbygoogle || []).push({}); Global Calculus. I skimmed around a bit. Naturally, I was confused when it defined a differentiable function ##f:M\to N## between differentiable manifolds as a continuous map such that, for each ##x\in M## and for each ##\phi\in\mathcal{O}_N(V)##, where ##V\ni f(x)## is some neighborhood of ##f(x)## in ##N## and ##\mathcal{O}_N## is the structure sheaf of ##N##, the composition ##\phi\circ f## is in ##f_*\mathcal{O}_M(V)##.

I'd think the more appropriate definition would be that ##f:M\to N## is differentiable if (and only if) it is a homomorphism of locally ringed spaces. I believe these two definitions are equivalent, but I haven't checked yet, due to a more general question to muse over: if two locally ringed spaces have structure sheaves that would allow for the composition in the first definition to make sense, would an analogue of the first definition define homomorphisms of locally ringed spaces between two such structures?

I do not think this holds for ringed spaces of the same nature, since definition #1 is essentially a "local" condition, but I don't know how to show this. Could someone please explain where this goes wrong for ringed spaces? Thank you.

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# Maps of Ringed Spaces

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