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. Given a subset S ⊂ Cn, one can define an n ideal I(S) ={f ∈ C[X1,... ,Xn]| f(c1,... ,cn) = 0 for all (c1,... ,cn) ∈ C } ⊂ C[X1,... ,Xn]. A fundamental result (the Hilbert Basissatz) states that any ideal of C[X1,... ,Xn] is finitely generated and so any affine variety is determined by a finite set of polynomials. One can show that I(S) is a radial ideal, that is, if m f ∈ I(S) for some m > 0, then f ∈ I(S). Given an ideal I ⊂ C[X1,... ,Xn] one n can define a variety Z(I) ={(c1,... ,cn) ∈ C | f(c1,... ,cn) = 0 for all f ∈ I} ⊂ Cn. Another result of Hilbert (the Hilbert Nullstellensatz) states for any proper ideal I ⊂ C[X1,... ,Xn], the set Z(I) is not empty. This allows one to show that maps V → I(S) and I → Z(I) define a bijective correspondence between the col- n lection of affine varieties in C and the collection of radical ideals in C[X1,... ,Xn]. Given a variety V, one can consider a polynomial f in C[X1,... ,Xn] as a function f : V → C. The process of restricting such polynomials to V yields a homo- morphism from C[X1,... ,Xn] to C[X1,... ,Xn]/I(V) and allows one to identify 340 A Algebraic Geometry C[X1,... ,Xn]/I(V) with the collection of polynomial functions on V. This lat- ter ring is called the coordinate ring of V and denoted by C[V]. The ring C[V] is a finitely generated C-algebra and any finitely generated C-algebra R may be written as R = C[X1,... ,Xn]/I for some ideal I. I will be the ideal of an affine variety if it is a radical ideal or, equivalently, when R has no nilpotent elements. Therefore, there is a correspondence between affine varieties and finitely generated C-algebras without nilpotents. More generally, if V ⊂ Cn and W ⊂ Cm are affine varieties, a map φ : V → W is n m said to be a regular map if it is the restriction of a Φ = (Φ1,... ,Φm) : C → C , where each Φi is a polynomial in n variables. Given an element f ∈ C[W], one sees that f ◦ φ is an element of C[V]. In this way, the regular map φ induces a C-algebra homomorphism from C[W] to C[V]. Conversely, any such C-algebra homomorphism arises in this way. Two affine varieties V and W are said to be isomorphic if there are regular maps φ : V → W and ψ : W → V such that ψ ◦ φ = idV and φ ◦ ψ = idW . Two affine varieties are isomorphic if and only if their coordinate rings are isomorphic as C-algebras. We say that an affine variety is irreducible if it is not the union of two proper affine varieties and irreducible if this is not the case. One sees that an affine variety V is irreducible if and only if I(V) is a prime ideal or, equivalently, if and only if its coordinate ring is an integral domain. The Basissatz can be, furthermore, used to show that any affine variety can be written as the finite union of irreducible affine varieties. If one has such a decomposition where no irreducible affine variety is contained in the union of the others, then this decomposition is unique and we refer to the irreducible affine varieties appearing as the components of V. This allows us to frequently restrict our attention to irreducible affine varieties. All of the above concepts are put in a more general context in Sect. A.1.1. One peculiarity of the Zariski topology is that the Zariski topology of C2 = C × C is ( 2 + 2) not the product topology. For example, V X1 X2 is not a finite union of sets of the form {pt}×{pt}, {pt}×C, C ×{pt}, or C × C. We shall have occasion to deal with products of affine varieties. For example, the Galois theory of differential equations leads one to consider the affine groups G and these are defined as affine varieties where the group law is a regular map from G × G → G (as well as insisting that the map taking an element to its inverse is a regular map G → G). To do this efficiently we wish to give an intrinsic definition of the product of two varieties. In Sect. A.1.2, [ ]⊗ [ ] [ ] we show that for affine varieties V and W the tensor product C V CC W of C V and C[W] is a C-algebra that has no nilpotent elements. We define the product of V [ ]⊗ [ ] ⊂ n and W to be the affine variety associated with the ring C V C C W . If V C and W ⊂ Cm then we can identify V × W with point set V × W ⊂ Cn+m. This set is Zariski closed and has the above coordinate ring. The Basissatz implies that any decreasing chain of affine varieties V V1 ··· Vt ... must be finite. One can show that the length of such a chain A Algebraic Geometry 341 is uniformly bounded and one can define the dimension of an affine variety V to be the largest number d for which there is a chain of nonempty affine varieties V V1 ··· Vd. The dimension of an affine variety is the largest dimension of its irreducible components. For an irreducible affine variety V this coincides with the transcendence degree of C(V) over C, where C(V) is called the function field of V and is the quotient field of C[V]. These concepts are further discussed in Sect. A.1.3. Let V be an irreducible variety of dimension d and let p ∈ V. We may write the coordinate ring C[V] as C[X1,... ,Xn]/( f1,... , ft ). One can show that the matrix ∂ ( fi (p)) has rank, at most, n−d. We say that p is a nonsingular point of V if the rank ∂X j is exactly n − d. This will happen at a Zariski-open set of points on V. The Implicit Function Theorem implies that in a (Euclidean) neighborhood of a nonsingular point, V will be a complex manifold of dimension d. One can define the tangent space of V at a nonsingular point p = (p1,... ,pn) to be the zero set of the linear equations n ∂ f j (p)(X − p ) = 0 for j = 1,... ,t . ∂X i i i=1 i This formulation of the notions of nonsingular point and tangent space appear to depend on the choice of the fi and are not intrinsic. Furthermore, one would like to define the tangent space at nonsingular points as well. In Sect. A.1.4, we give an intrinsic definition of nonsingularity and tangent space at an arbitrary point of a (not necessarily irreducible) affine variety and show that these concepts are equivalent to the above in the classical case. A major use of the algebraic geometry that we develop will be to describe linear algebraic groups and sets on which they act. The prototypical example of a linear algebraic group is the group GLn(C) of invertible n × n matrices with entries in C. 2 We can identify this group with an affine variety in Cn +1 via the map sending −1 A ∈ GLn(C) to (A,(det(A)) ). The ideal in C[X1,1,... ,Xn,n, Z] defining this set is generated by Z det(Xi, j )−1. The entries of a product of two matrices A and B are clearly polynomials in the entries of A and B. Cramer’s rule implies that the entries of the inverse of a matrix A can be expressed as polynomials in the entries of A and (det(A))−1. In general, a linear algebraic group is defined to be an affine variety G such that the multiplication is a regular map from G × G to G and the inverse is a regular map from G to G.