Properties
If R is a Noetherian local ring, the dimension of the tangent space is at least the dimension of R:
- dim m/m2 ≧ dim R
R is called regular if equality holds. In a more geometric parlance, when R is the local ring of a variety V in v, one also says that v is a regular point. Otherwise it is called a singular point.
The tangent space has an interpretation in terms of homomorphisms to the dual numbers for K,
- K/:
in the parlance of schemes, morphisms Spec K/ to a scheme X over K correspond to a choice of a rational point x ∈ X(k) and an element of the tangent space. Therefore, one also talks about tangent vectors.
Read more about this topic: Zariski Tangent Space
Famous quotes containing the word properties:
“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)
“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)