Tittel: Introduction to commutative ring theory, from localization to complete intersections

Veileder: Peder Thompson

Sammendrag: This thesis will be an introduction to commutative ring theory, with an end goal of introducing complete intersection rings and reviewing some results about them. It will be written with the assumption that the reader is familiar with some basic algebraic concepts, such as groups, rings, and modules.

The first part is localisation of rings. It is important to have tools at hand to construct local rings in order to have a wider array of “nice” rings to work with. It is also important to know what properties such a construction will have. After that we will look at primary decomposition of ideals. This part consists of results about primary ideals, and how an intersection of them can be a way of representing an ideal, and that representation’s properties. The theory of primary ideals also comes up when working with dimension theory as we will work with systems of parameters of local rings.

The next part will be about the $\mathfrak{a}$-adic completions of rings and modules, and the Artin-Rees lemma. This construction is complicated and is based on taking the inverse limit of an inverse system constructed from the ring and an ideal $\mathfrak{a}$. The last part of what we might call the preliminaries of this thesis is dimension theory. In this part we introduce the concept of graded rings and modules, and Hilbert functions, as well as proving some properties about dimensions specific for Noetherian local rings.

The last part will be about complete intersection rings, and some results regarding them. For example, that any C.I ring is of the form a regular local ring quotient with an ideal generated by a regular sequence. Here we will need all the previous parts to describe them sufficiently. We also need to introduce some new theory to be able to define them.

There is included an appendix on Category Theory and Homological Algebra as some of theory included relies on knowing some basic definitions from the fields.