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:
“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)