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Definition. Morphism of ringed spaces [00AJ]

A morphism of ringed spaces and is a continous map , together with a morphism of sheaves . Where is the pushforward given by .

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Definition. Morphism of locally ringed spaces [00AO]

A morphism of locally ringed spaces is a morphism of ringed spaces , such that the induces map on stalks is a local homomorphism of local rings. I.e the unique maximal ideal is sent to the unique maximal ideal.

Recall that is a continous map, and is a morphism of sheaves, where is a sheaf on by .