Does the locally-ringed spaces viewpoint on topology actually do what we want?

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There's a post here about how we know that the morphisms of smooth manifolds as locally-ringed spaces are the same as the morphisms of smooth manifolds as charts and atlases. I didn't understand the argument, but it was mentioned that the argument breaks down for fields other than $mathbbR$. Does this mean that attempts to take a locally-ringed spaces perspective on various kinds of manifolds over anything other than $mathbbR$ will tend to either fail, or at least have some surprising and perhaps undesirable pathologies due to the morphisms being "too general"?




manifolds differential-topology riemann-surfaces complex-manifolds ringed-spaces




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    There's a post here about how we know that the morphisms of smooth manifolds as locally-ringed spaces are the same as the morphisms of smooth manifolds as charts and atlases. I didn't understand the argument, but it was mentioned that the argument breaks down for fields other than $mathbbR$. Does this mean that attempts to take a locally-ringed spaces perspective on various kinds of manifolds over anything other than $mathbbR$ will tend to either fail, or at least have some surprising and perhaps undesirable pathologies due to the morphisms being "too general"?




    manifolds differential-topology riemann-surfaces complex-manifolds ringed-spaces




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      There's a post here about how we know that the morphisms of smooth manifolds as locally-ringed spaces are the same as the morphisms of smooth manifolds as charts and atlases. I didn't understand the argument, but it was mentioned that the argument breaks down for fields other than $mathbbR$. Does this mean that attempts to take a locally-ringed spaces perspective on various kinds of manifolds over anything other than $mathbbR$ will tend to either fail, or at least have some surprising and perhaps undesirable pathologies due to the morphisms being "too general"?




      manifolds differential-topology riemann-surfaces complex-manifolds ringed-spaces




      share|cite|improve this question











      There's a post here about how we know that the morphisms of smooth manifolds as locally-ringed spaces are the same as the morphisms of smooth manifolds as charts and atlases. I didn't understand the argument, but it was mentioned that the argument breaks down for fields other than $mathbbR$. Does this mean that attempts to take a locally-ringed spaces perspective on various kinds of manifolds over anything other than $mathbbR$ will tend to either fail, or at least have some surprising and perhaps undesirable pathologies due to the morphisms being "too general"?





      manifolds differential-topology riemann-surfaces complex-manifolds ringed-spaces





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      asked Aug 6 at 8:25









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