category of locally ringed spaces

 algebraic geometry
Notation LRSR\LRS_R Objects locally ringed spaces over R≠0R \neq 0, i.e. pairs (X,OX)(X,\O_X) consisting of a topological space and a sheaf of commutative RR-algebras on it whose stalks are local rings Morphisms A morphism (f,f♯):(X,OX)→(Y,OY)(f,f^\sharp) : (X,\O_X) \to (Y,\O_Y) consists of a continuous map f:X→Yf : X \to Y and a homomorphism f♯:OY→f∗OXf^\sharp : \O_Y \to f_* \O_X of sheaves of RR-algebras such that for every x∈Xx \in X the induced map on stalks fx:OY,f(x)→OX,xf_x : \O_{Y,f(x)} \to \O_{X,x} is local. Related CAlg(R)\CAlg(R), SchR\Sch_R, Top\Top External nLab Link

Here, RR is a fixed non-trivial commutative ring. The general properties of this category will not depend on the choice of RR. For R=ZR = \IZ we obtain the category LRS\LRS of locally ringed spaces.

Satisfied Properties

Assigned properties

Deduced properties

Unsatisfied Properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

There are 11 properties for which the database doesn't have an answer if they are satisfied or not. Please help to contribute the data!

Special objects

Special morphisms

  • isomorphisms: A morphism (f,f♯):(X,OX)→(Y,OY)(f,f^\sharp) : (X, \O_X) \to (Y,\O_Y) is an isomorphism iff f:X→Yf : X \to Y is a homeomorphism and f♯:OY→f∗OXf^\sharp : \O_Y \to f_* \O_X is an isomorphism of sheaves.
  • monomorphisms:
  • epimorphisms: A morphism (f,f♯):(X,OX)→(Y,OY)(f,f^\sharp) : (X, \O_X) \to (Y,\O_Y) is an epimorphism iff f:X→Yf : X \to Y is surjective and f♯:OY→f∗OXf^\sharp : \O_Y \to f_* \O_X is a monomorphism sheaves.
  • regular monomorphisms:
  • regular epimorphisms:

Comments

  • Monomorphisms X→YX \to Y can be formally described by the condition that the diagonal morphism X→X×YXX \to X \times_Y X is an isomorphism. Since the fiber product of locally ringed spaces has a concrete description (MSE/1033675), we thus have a description of the monomorphisms, albeit a very complicated one. At least, we can say that every monomorphism is injective.