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:
“Not ringed but rare, not gilled but polyp-like, having sprung up
overnight—
These mushrooms of the gods, resembling human organs uprooted,
rooted only on the air,”
—William Jay Smith (b. 1918)
“Though there were numerous vessels at this great distance in the horizon on every side, yet the vast spaces between them, like the spaces between the stars,—far as they were distant from us, so were they from one another,—nay, some were twice as far from each other as from us,—impressed us with a sense of the immensity of the ocean, the “unfruitful ocean,” as it has been called, and we could see what proportion man and his works bear to the globe.”
—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)