Foundations to the Theory of Schemes

Abstract

Grothendieck’s magnificent theory of schemes pervades the spectrum of modern algebraic geometry and underpins its wide applications in the field of Number theory,Medicine, Physics , Applied Mathematics,image encryption and finger printing. This report which is a simple account of the foundations to the theory of schemes underscores and demonstrates the common geometric concepts that form the basis of the definitions. The report begins these foundations with Some local algebra where we make a mention of Noether’s Normalization Lemma, Going-up theorem of Cohen-Seidenberg and the Weak Nullstellensatz result before giving some properties of Cohen-Macaulay rings. The report then introduces the language of categories and functors which then leads to a discussion on the sheaf theory. We then introduce the spectrum of rings and the Zariski topology before defining an afine scheme and scheme in general. This is then followed by a number of examples of schemes and some of the properties of afine schemes. The report discusses dimension of a scheme and ends by exhibiting on the concept of gluing construction. In this dissertation all the results are well-known and therefore our contribution is only at the level of presentation.