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locally ringed topological space

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topos theory

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Definition

A locally ringed space is a ringed space (X,𝒪)(X,\mathcal{O}) such that the stalks of the structure sheaf 𝒪\mathcal{O} are local rings.

A morphism of locally ringed space is a morphism of ringed spaces (f,f ):(X,𝒪 X)(Y,𝒪 Y)(f,f^\sharp):(X,\mathcal{O}_X)\to (Y,\mathcal{O}_Y), where f:XYf:X\to Y, such that the comorphism f :𝒪 Yf *𝒪 Xf^\sharp:\mathcal{O}_Y\to f_*\mathcal{O}_X is a morphism of local rings (that is, a map of rings which respects the maximal ideal (reflects invertibility)).

Examples

  • The category of smooth manifolds has a canonical inclusion into the category of locally ringed spaces. Indeed, any smooth manifold MM comes equipped with a sheaf of smooth real valued functions 𝒞 M \mathcal{C}^{\infty}_M, and the maximal ideal in a given stalk consists precisely of germs of smooth functions which vanish at that point. Moreover, a smooth map f:MNf: M \to N determines a comorphism f :𝒞 N f *𝒞 M f^\sharp: \mathcal{C}^{\infty}_N \to f_* \mathcal{C}^{\infty}_M by the usual precomposition. In fact, the inclusion of smooth manifolds into locally ringed spaces is fully faithful. For a proof, see Lucas Braune’s nice answer on math stackexchange.
  • Schemes are usually thought of as locally ringed spaces.

On the locality condition

The condition that all stalks 𝒪 X,x\mathcal{O}_{X,x} are local rings can be reformulated without referring to the points of XX:

  • The only open subset UU such that 𝒪 X(U)\mathcal{O}_X(U) is the zero ring is U=U = \emptyset.
  • Let f,g𝒪 X(U)f, g \in \mathcal{O}_X(U) be local sections such that f+gf + g is invertible in 𝒪 X(U)\mathcal{O}_X(U). Then there is an open covering U=VWU = V \cup W such that f| V𝒪 X(V)f|_V \in \mathcal{O}_X(V) and g| W𝒪 X(W)g|_W \in \mathcal{O}_X(W) are invertible. (It’s allowed that VV or WW are empty.)

References

Last revised on March 28, 2017 at 17:13:47. See the history of this page for a list of all contributions to it.