Preliminary Study on the Dimension and Depth of Local Rings

Introduction

Local rings are fundamental objects in commutative algebra, characterized by having a unique maximal ideal. Their study provides insights into the structure of rings and modules, particularly in relation to their dimension and depth. The dimension of a local ring, often referred to as the Krull dimension, measures the length of the longest chain of prime ideals. Conversely, the depth of a local ring is a measure of the maximal length of a regular sequence in the ring. This preliminary study explores the definitions, properties, and interrelations of these invariants, focusing on their implications in algebraic geometry and homological algebra.

Definitions and Basic Properties

A local ring is a ring with a unique maximal ideal. For a Noetherian local ring $ R $, the dimension $ \dim R $ is the length of the longest chain of prime ideals. This concept is closely tied to the Krull dimension, which is defined for any Noetherian ring. The depth of a local ring $ R $, denoted $ \depth R $, is the maximum length of a regular sequence in $ R $. A regular sequence is a sequence of elements $ a_1, a_2, \ldots, a_n $ such that each element is not a zero divisor in the quotient $ R/(a_1, \ldots, a_{i-1})R $ for each $ i $.

In the case of a local ring, the depth is closely related to the ring's regularity. A ring is regular if its local ring is a regular local ring, meaning it has a finite number of prime ideals and its depth equals its dimension. The Auslander-Buchsbaum formula states that for a finitely generated module $ M $ over a Noetherian local ring $ R $, the depth $ \depth R(M) $ satisfies:
$$ \depth R(M) + \rank_R(M) = \dim R $$
This relationship underscores the interplay between the ring's dimension and the depth of its modules.

Dimension of Local Rings

The dimension of a local ring is a critical invariant, often determined by the length of the longest chain of prime ideals. For a Noetherian local ring $ R $, the dimension is equal to the length of the longest chain of prime ideals, which can be computed using the Krull dimension. In the case of a regular local ring, the dimension coincides with the depth, as regular rings are also Cohen-Macaulay.

For non-regular local rings, the dimension and depth may differ. For example, consider the ring $ R = k[x, y]/(x^2, y^2) $, a local ring of dimension 2. The depth of $ R $ is 1, as the maximal regular sequence has length 1. This illustrates that the depth can be less than the dimension in non-regular rings.

The Auslander-Buchsbaum theorem further clarifies this relationship. For a finitely generated module $ M $ over a Noetherian local ring $ R $, the depth $ \depth R(M) $ satisfies:
$$ \depth R(M) + \rank_R(M) = \dim R $$
This theorem highlights the dependency between the ring's dimension and the depth of its modules, emphasizing the role of the ring's regularity.

Depth of Local Rings

The depth of a local ring is a measure of the ring's "regularity" and is closely tied to the structure of its modules. For a local ring $ R $, the depth is the maximum length of a regular sequence in $ R $. In the case of a regular local ring, the depth equals the dimension, as regular rings are Cohen-Macaulay.

For non-regular local rings, the depth can vary. For instance, consider the ring $ R = k[x, y]/(x^2, xy, y^2) $, a local ring of dimension 2. The depth of $ R $ is 1, as the maximal regular sequence has length 1. This demonstrates that the depth can be less than the dimension in non-regular rings.

The Cohen-Macaulay property is crucial in determining the depth of a local ring. A ring is Cohen-Macaulay if its depth equals its dimension. For a Noetherian local ring, the Cohen-Macaulay property is equivalent to being regular. Thus, the depth of a local ring is directly tied to its regularity.

Relationship Between Dimension and Depth

The relationship between the dimension and depth of a local ring is governed by the Auslander-Buchsbaum theorem, which states that for a finitely generated module $ M $ over a Noetherian local ring $ R $, the depth $ \depth R(M) $ satisfies:
$$ \depth R(M) + \rank_R(M) = \dim R $$
This theorem highlights the interdependence between the ring's dimension and the depth of its modules. In regular local rings, the depth equals the dimension, as regular rings are Cohen-Macaulay.

For non-regular local rings, the depth can differ from the dimension. For example, in the ring $ R = k[x, y]/(x^2, xy, y^2) $, the depth is 1, while the dimension is 2. This illustrates that the depth can be less than the dimension in non-regular rings.

Examples and Applications

To illustrate these concepts, consider the ring $ R = k[x, y]/(x^2, y^2) $, a local ring of dimension 2. The depth of $ R $ is 1, as the maximal