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I want to know more classes of examples of locally ringed spaces. The reason is that when I want to prove/disprove something about locally ringed spaces, my examples are often not eclectic enough. Besides, it is interesting to what extent special theories can be generalized to locally ringed spaces.

Here are the classes of examples I know:

  • Local rings (underlying topological space is a point)
  • Schemes
  • Algebraic Varieties in the classical sense (just closed points)
  • Manifolds with smooth/analytic/holomorphic functions
  • Topological spaces with continuous functions
  • Various subcategories of the above examples
  • The category of locally ringed spaces is complete and cocomplete. Thus you can build new locally ringed spaces out of the above ones.

Which substantial different examples do you know? Please no fancy Grothendieck topologies ;).

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  • $\begingroup$ You can also view supermanifolds and superschemes as locally ringed spaces, but you have to have use supercommutative rings, such as exterior algebras, rather than commutative rings. $\endgroup$ Jun 1 '10 at 13:08
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    $\begingroup$ Complex/real-analytic spaces, formal schemes, topological space equipped with locally constant functions valued in a local ring. Strictly speaking, examples with exotic Grothendieck topologies yield a locally ringed topos but are not locally ringed spaces (e.g., etale or crystalline of fppf topos of a scheme equipped with "structure sheaf", rigid-analytic spaces, Berkovich spaces, adic spaces, simplicial objects of various sorts, etc.). A nifty related fact is that there are rigid-analytic spaces admitting abelian sheaves having a nonzero global section whose stalks at all points vanish. $\endgroup$
    – BCnrd
    Jun 1 '10 at 14:10
  • $\begingroup$ Please post this as answer. $\endgroup$ Jun 2 '10 at 6:55
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    $\begingroup$ Balmer gives a way to associate a locally ringed space to any tensor triangulated category. number 18 on this list: math.ucla.edu/~balmer/research/publications.html $\endgroup$ Jan 30 '11 at 22:35
  • $\begingroup$ @Dylan: Interesting! $\endgroup$ Jan 30 '11 at 23:52
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The following is BCnrd's answer, posted as requested by Martin: "Complex/real-analytic spaces, formal schemes, topological space equipped with locally constant functions valued in a local ring. Strictly speaking, examples with exotic Grothendieck topologies yield a locally ringed topos but are not locally ringed spaces (e.g., etale or crystalline of fppf topos of a scheme equipped with "structure sheaf", rigid-analytic spaces, Berkovich spaces, adic spaces, simplicial objects of various sorts, etc.). A nifty related fact is that there are rigid-analytic spaces admitting abelian sheaves having a nonzero global section whose stalks at all points vanish."

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