Properties
- For a scheme locally of finite type over a field k, a point x is closed if and only if k(x) is a finite extension of the base field k. This is a geometric formulation of Hilbert's Nullstellensatz. In the above example, the points of the first kind are closed, having residue field k, whereas the second point is the generic point, having transcendence degree 1 over k.
- A morphism Spec(K) → X, K some field, is equivalent to giving a point x ∈ X and an extension K/k(x).
- The dimension of a scheme of finite type over a field is equal to the transcendence degree of the residue field of the generic point.
Read more about this topic: Residue Field
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)