Dimension theory (algebra)

In mathematics, dimension theory is a part of commutative algebra that helps us understand the idea of the dimension of an algebraic variety and, more generally, the dimension of a scheme. This might sound simple, but there are many different ways to define dimension, and they only match up in very special cases. A big part of dimension theory looks at when these different definitions agree with each other.

One important example is the idea of a regular ring, which is a type of commutative ring where two specific dimensions—the homological dimension and the Krull dimension—are the same. This helps mathematicians classify and understand these rings better.

The theory becomes easier to work with when we look at commutative rings that are built from polynomial rings in a limited number of variables. In these cases, most definitions of dimension line up nicely. However, for more general rings, especially those that are not Noetherian rings, the theory is much harder, and there is still a lot we don’t know.

Basic results

When we look at special kinds of number systems called Noetherian rings or valuation rings, we find something interesting. If we take such a ring and add a new variable, the "size" or dimension of this new system increases by exactly one. This idea helps us understand how these systems grow when we add new elements.

For simpler systems called Artinian rings, which have a dimension of zero, adding several new variables results in the dimension matching the number of variables added. This gives us a clear way to measure how these systems expand.

Local rings

The study of local rings in algebra looks at special types of number systems. A key idea is the fundamental theorem, which says that three different ways to measure the "size" of these systems actually give the same answer. This helps us understand when these systems behave nicely.

Some important results follow from this theorem. For example, if a special kind of number system called a "regular local ring" is used, its size can be measured in a straightforward way. Other results help us compare sizes of different systems and understand how adding new elements changes these sizes.

Homological methods

Dimension theory in algebra looks at how we can measure the "size" of algebraic structures using tools from commutative algebra. This helps us understand when different ways of measuring size give the same answer.

One important idea is the concept of a "regular ring," which is a special type of ring where two different measures of size are equal. This equality helps mathematicians classify and study these rings more easily.

The study also involves looking at how rings behave under certain operations and how these behaviors relate to their dimensions. This gives insight into the structure of algebraic varieties and schemes, which are geometric objects defined by algebraic equations.

Dimensions of non-commutative rings

In this section, we look at how to measure the "size" of certain mathematical structures called non-commutative rings. These rings are more complex than the usual numbers we use, and they have special properties.

One way to measure their size is using something called the Gelfand–Kirillov dimension. Imagine you have a box that can hold smaller boxes inside it, and each smaller box can hold even more boxes. The Gelfand–Kirillov dimension helps us understand how these boxes grow when we keep putting more and more boxes inside.

For example, if the ring is very simple and can be described with a fixed number of elements, its Gelfand–Kirillov dimension is zero. For more complicated rings, this dimension can tell us more about their structure. In some special cases, this dimension matches another way we measure size, called the Krull dimension.