Appendices a Algebraic Geometry
Appendices A Algebraic Geometry Affine varieties are ubiquitous in Differential Galois Theory. For many results (e.g., the definition of the differential Galois group and some of its basic properties) it is enough to assume that the varieties are defined over algebraically closed fields and study their properties over these fields. Yet, to understand the finer structure of Picard-Vessiot extensions it is necessary to understand how varieties behave over fields that are not necessarily algebraically closed. In this section we shall develop basic material concerning algebraic varieties taking these needs into account, while at the same time restricting ourselves only to the topics we will use. Classically, algebraic geometry is the study of solutions of systems of equations { fα(X1,... ,Xn) = 0}, fα ∈ C[X1,... ,Xn], where C is the field of complex numbers. To give the reader a taste of the contents of this appendix, we give a brief description of the algebraic geometry of Cn. Proofs of these results will be given in this appendix in a more general context. One says that a set S ⊂ Cn is an affine variety if it is precisely the set of zeros of such a system of polynomial equations. For n = 1, the affine varieties are fi- nite or all of C and for n = 2, they are the whole space or unions of points and curves (i.e., zeros of a polynomial f(X1, X2)). The collection of affine varieties is closed under finite intersection and arbitrary unions and so forms the closed sets of a topology, called the Zariski topology.
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