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.
Publisher
University of Nairobi
Rights
Attribution-NonCommercial-NoDerivs 3.0 United StatesUsage Rights
http://creativecommons.org/licenses/by-nc-nd/3.0/us/Collections
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