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functions on open subsets. Among ringed spaces, especially important and prominent is a locally ringed space: a ringed space in which the analogy between...

analytic space is a generalization of a complex manifold that allows the presence of singularities. Complex analytic varieties are locally ringed spaces that...

immersions of locally ringed spaces—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-08-05. "Section 17.8 (01B1): Modules locally generated by...

notion of spaces in algebraic geometry, are locally ringed spaces that are locally isomorphic to the spectrum of a ring. Given a ringed space, a sheaf...

particular kind of a locally ringed space. When this definition is used, a morphism of varieties is just a morphism of locally ringed spaces. The composition...

coordinate ring of D(f); that is, "regular-ness" can be patched together. Hence, ( X , O X ) {\displaystyle (X,{\mathcal {O}}_{X})} is a locally ringed space. A...

Weil reciprocity. Zariski–Riemann space A Zariski–Riemann space is a locally ringed space whose points are valuation rings. Proof: Let D be a Weil divisor...

mathematics, a ringed topos is a generalization of a ringed space; that is, the notion is obtained by replacing a "topological space" by a "topos". The...

variety, one requires the notion of a locally ringed space. A complex algebraic/analytic variety is a locally ringed space ( X , O X ) {\displaystyle (X,{\mathcal...

the possible ways they can differ outside of Z. An analytic space is a locally ringed space ( X , O X ) {\displaystyle (X,{\mathcal {O}}_{X})} such that...

{p}}} , and this is a local ring. Consequently, Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} is a locally ringed space. If R {\displaystyle R} is...

a Zariski–Riemann space or Zariski space of a subring k of a field K is a locally ringed space whose points are valuation rings containing k and contained...

Like all manifolds, it is locally homeomorphic to Euclidean space and thus locally metrizable (but not metrizable) and locally Hausdorff (but not Hausdorff)...

scheme is a locally ringed space isomorphic to the spectrum Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} of a commutative ring R {\displaystyle...

sections functor in the category of locally ringed spaces. If A {\displaystyle A} is a ring, we define projective n-space over A {\displaystyle A} to be the...

In mathematics, an invertible sheaf is a sheaf on a ringed space that has an inverse with respect to tensor product of sheaves of modules. It is the equivalent...

a wider class of rings. A scheme is a locally ringed space such that every point has a neighbourhood that, as a locally ringed space, is isomorphic to...

O) is an example of a locally ringed space: it is a topological space equipped with a sheaf whose stalks are each local rings. A differentiable manifold...

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional...

idea of a sheaf of local rings is central. This, however, is not quite the same thing; one speaks instead of a locally ringed space, because it is not true...