Definition
The cotangent space of a local ring R, with maximal ideal m is defined to be
- m/m2
It is a vector space over the residue field k := R/m. Its dual (as a k-vector space) is called tangent space of R.
This definition is a generalization of the above example to higher dimensions: suppose given an affine algebraic variety V and a point v of V. Morally, modding out m2 corresponds to dropping the non-linear terms from the equations defining V inside some affine space, therefore giving a system of linear equations that define the tangent space.
(One often defines the tangent and cotangent spaces for a manifold in the analogous manner.)
Read more about this topic: Zariski Tangent Space
Famous quotes containing the word definition:
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)