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 SchR\Sch_R, Top\Top, CAlg(R)\CAlg(R) 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 8 properties for which the database has no information on whether they are satisfied. Please help us fill in the gaps by contributing to this project.

Special objects

  • terminal object: Spec⁡(R)\Spec(R)
  • initial object: empty space
  • products: See Gillam. See also MSE/1033675.
  • coproducts: disjoint union with the product sheaf

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 of sheaves.
  • regular monomorphisms: A morphism i:Z→Yi : Z \to Y is a regular monomorphism iff its underlying continuous map is an embedding and its stalk maps OY,i(z)→OZ,z\O_{Y,i(z)} \to \O_{Z,z} are surjective for all z∈Zz \in Z.
  • 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.