Ringed Spaces and Locally Ringed Spaces
A pair consisting of a topological space X and a sheaf of rings on X is called a ringed space. Many types of spaces can be defined as certain types of ringed spaces. The sheaf is called the structure sheaf of the space. A very common situation is when all the stalks of the structure sheaf are local rings, in which case the pair is called a locally ringed space. Here are examples of definitions made in this way:
- An n-dimensional Ck manifold M is a locally ringed space whose structure sheaf is an -algebra and is locally isomorphic to the sheaf of Ck real-valued functions on Rn.
- A complex analytic space is a locally ringed space whose structure sheaf is a -algebra and is locally isomorphic to the vanishing locus of a finite set of holomorphic functions together with the restriction (to the vanishing locus) of the sheaf of holomorphic functions on Cn for some n.
- A scheme is a locally ringed space that is locally isomorphic to the spectrum of a ring.
- A semialgebraic space is a locally ringed space that is locally isomorphic to a semialgebraic set in Euclidean space together with its sheaf of semialgebraic functions.
Read more about this topic: Sheaf (mathematics)
Famous quotes containing the words ringed, spaces and/or locally:
““But such as you and I do not seem old
Like men who live by habit. Every day
I ride with falcon to the river’s edge
Or carry the ringed mail upon my back,
Or court a woman; neither enemy,
Game-bird, nor woman does the same thing twice....””
—William Butler Yeats (1865–1939)
“Surely, we are provided with senses as well fitted to penetrate the spaces of the real, the substantial, the eternal, as these outward are to penetrate the material universe. Veias, Menu, Zoroaster, Socrates, Christ, Shakespeare, Swedenborg,—these are some of our astronomers.”
—Henry David Thoreau (1817–1862)
“To see ourselves as others see us can be eye-opening. To see others as sharing a nature with ourselves is the merest decency. But it is from the far more difficult achievement of seeing ourselves amongst others, as a local example of the forms human life has locally taken, a case among cases, a world among worlds, that the largeness of mind, without which objectivity is self- congratulation and tolerance a sham, comes.”
—Clifford Geertz (b. 1926)