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)
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)
“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)