Posts tagged ‘prime spectrum’

Schemes in algebraic geometry 2 : prime spectra and generic points

I just explained how the affine plane could be described by the ring \mathbb Z[x,y]. A point M of the affine plane whose coordinate ring is R is a morphism \mathbb Z[x,y] \to R defined by the assignment P \mapsto P(a,b) \in R, where (a,b) are the coordinates of M. In the case of points corresponding to morphisms \mathbb Z[x,y] \to \mathbb Z, there is a natural way of recovering the point from the ring morphism by looking at his equations, which are elements of the kernel of the morphism. If M satisfies the equations x=a and y=b, then M has the form (a,b). This motivates the abstract definition of point of the affine plane as a morphism \mathbb Z[x,y] \to R to some ring.

Conversely, the set of equations of M defines a canonically associated point p_M, which is the morphism \mathbb Z[x,y] \to \mathbb Z[x,y]/I, where I is the ideal generated by the equations. But this morphism has no reason to totally recover M if it wasn’t a point with integral coordinates. For example, the point (2,3) is a special point, satisfying a lot of equations, which characterize it. But (\log 2, \pi) do not satisfy any polynomial equations with integral coefficients, so the set of its equations is empty, and cannot be used to recover it. Moreover, the point (e, \log 3) does not satisfy equations either: their algebraic properties are exactly the same. These points are called generic.

The prime spectrum of a ring is a convenient way of describing equivalence classes of points of a given ring.
Definition. The prime spectrum of A = \mathbb Z[x,y] is the set of points p_I: A \to A/I for prime ideals I. If M: A \to R is any point of the affine plane with coordinates in an integral domain R, then Mis canonically associated to some p_M := p_I, where I is the kernel of the map A \to R.

10 March 2009 at 10:07 pm 2 comments