In Part 1, we introduce algebraic varieties and develop the related topics using an elementary approach. In the first chapter we define affine varieties as subsets of an affine space given by a finite set of polynomial equations. We see that affine varieties have a natural topology called Zariski topology. We introduce the concept of abstract affine variety to illustrate that giving an affine variety is equivalent to giving the ring of regular functions on each open set. We then study projective varieties and see how functions defined on a projective variety can be recovered by means of the open covering of the projective space by affine spaces. In the second chapter we study algebraic varieties, which include affine and projective ones. We define morphism of algebraic varieties, the dimen- sion of an algebraic variety, and the tangent space at a point. We analyze the dimension of the tangent space and define simple and singular points of a variety. We establish Chevalley’s theorem and Zariski’s main theorem which will be used in the construction of the quotient of an algebraic group by a closed subgroup. For more details on algebraic geometry see [Hu], [Kle], and [Sp]. For the results of commutative algebra see [A-M], [L], and [Ma]. Unless otherwise specified, C will denote an algebraically closed field of characteristic 0.
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