nLab
ringed space

Context

Topos Theory

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Definition

A ringed space is a pair (X,O X)(X,O_X) where XX is a topological space and O XO_X is a sheaf of unital rings. The sheaf O XO_X is called the structure sheaf of the ringed space (X,O X)(X,O_X).

If all stalks of the structure sheaf are local rings, it is called a locally ringed space.

A morphism of ringed spaces (f,f ):(X,O X)(Y,O Y)(f,f^\sharp):(X,O_X)\to (Y,O_Y) is a pair where f:XYf:X\to Y is a continuous map and the comorphism f :O Yf *O Xf^\sharp : O_Y\to f_* O_X is a morphism of sheaves of rings over YY. Here f *f_* denotes the direct image functor for sheaves. Any sheaf of abelian modules \mathcal{M} equipped with actions O X(U)×(U)(U)O_X(U)\times\mathcal{M}(U)\to\mathcal{M}(U) making (U)\mathcal{M}(U) left O XO_X-modules, and such that the actions strictly commute with the restrictions, is called a sheaf of left O XO_X-modules.

Remarks

  • Every ringed space induces a ringed site: To a ringed space (X,O X)(X,O_X) assign the ringed site (Op X,O X)(Op_X,O_X) where Op XOp_X is the category of open sets and inclusions equipped with the pretopology of open covers and O XO_X is just viewed as a sheaf of rings on Op XOp_X.

  • In toric geometry and sometimes in relation to the “absolute” algebraic geometry over F 1F_1, one talks about monoided or monoidal space (Kato; Deitmar); which is a topological space together with a sheaf of monoids. N. Durov on the other hand develops a generalized algebraic geometry based on a notion of generalized ringed space, which is a space equipped with a sheaf of (commutative) generalized rings, which are finitary (= algebraic) monads in Set\mathrm{Set} with a commutativity condition (which are related to higher analogues of Eckmann-Hilton argument).

References

Section 6.25 of

Revised on July 4, 2011 12:16:06 by Urs Schreiber (82.113.99.41)