Posts filed under ‘schemes in algebraic geometry’

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.
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10 March 2009 at 10:07 pm 2 comments

Schemes in algebraic geometry I : the affine plane

I think most people blogging around algebraic geometry eventually write about schemes, (as in Rigorous trivialities or algebraic stacks (in the Secret Blogging Seminar), which are traditionnally seen as the main reason (not) to study algebraic geometry today. My turn now. I recommend Igor Dolgacev’s lectures which is one of my favorite ways of speaking of schemes.

Interesting mathematics come up when algebraic varieties (things defined by several polynomial equations) are no longer defined as mere sets (sets of tuples of numbers satisfying the equation) but mope complex mathematical objects. Differential geometry, for example, gives the structure of a complex manifold to algebraic varieties in \mathbb C^n, which is still an efficient way of proving theorems. However, during the 20th century, a lot of mathematicians tried to develop a new structure which would avoid the use of analysis to concentrate on the algebraic aspects (I don’t know exaclty who, but expect Hilbert, Zariski, Chevalley, Grothendieck to have played a role). Grothendieck approach using category theory and functors of points is now widely used and is a very impressive way of tying together intuition, commutative algebra, and geometry.

There are many ways of reverse engineering Grothendieck’s definition of a scheme (see EGA1 if you want to know how this is related to Chevalley’s definition of a scheme). The first thing to say is probably what properties and notions are needed for schemes:
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8 March 2009 at 12:49 pm 2 comments


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