🧮Commutative Algebra Unit 9 – Dimension Theory and Krull Dimension

Key Concepts and Definitions

• Krull dimension measures the size of a commutative ring by the supremum of the lengths of all chains of prime ideals
• A chain of prime ideals is a sequence of prime ideals $P_0 \subset P_1 \subset \cdots \subset P_n$ where each ideal is properly contained in the next
• The length of a chain is the number of strict inclusions in the chain
• A ring has Krull dimension $n$ if the supremum of the lengths of all chains of prime ideals is $n$
• A ring is called Noetherian if it satisfies the ascending chain condition on ideals, meaning every ascending chain of ideals stabilizes
  • In a Noetherian ring, every ideal is finitely generated
• The height of a prime ideal $P$ is the supremum of the lengths of all chains of prime ideals descending from $P$
• A local ring is a ring with a unique maximal ideal
  • The Krull dimension of a local ring is equal to the height of its maximal ideal

Historical Context and Development

• The concept of dimension in commutative algebra was introduced by Wolfgang Krull in the 1930s
• Krull's work built upon earlier ideas in algebraic geometry, such as the dimension of algebraic varieties
• The development of dimension theory was motivated by the need to understand the structure of commutative rings and their prime ideals
• Krull's definition of dimension using chains of prime ideals provided a way to measure the complexity of a ring
• The notion of Krull dimension was later extended to non-Noetherian rings by Oscar Zariski and Pierre Samuel
• Dimension theory has played a crucial role in the development of modern commutative algebra and algebraic geometry
  • It has led to important results in the classification of commutative rings and the study of their properties

Fundamental Theorems and Proofs

• Krull's Principal Ideal Theorem states that if $R$ is a Noetherian ring and $P$ is a prime ideal generated by $n$ elements, then the height of $P$ is at most $n$
  • This theorem provides an upper bound on the Krull dimension of a Noetherian ring
• Krull's Altitude Theorem states that if $R$ is a Noetherian local ring with maximal ideal $M$, then the Krull dimension of $R$ is equal to the minimum number of generators of $M$
• The Dimension Inequality states that if $R$ is a Noetherian ring and $P \subset Q$ are prime ideals, then $\text{height}(P) + \dim(R/P) \leq \text{height}(Q) + \dim(R/Q)$
  • This inequality relates the heights of prime ideals and the dimensions of their quotient rings
• The Hilbert-Samuel Polynomial is a polynomial associated to a finitely generated module over a local ring that encodes information about its dimension and multiplicity
• Serre's Intersection Theorem states that if $R$ is a regular local ring and $M, N$ are finitely generated $R$-modules, then $\dim(M \otimes_R N) \geq \dim(M) + \dim(N) - \dim(R)$
  • This theorem relates the dimensions of modules and their tensor products

Applications in Commutative Algebra

• Krull dimension is used to classify commutative rings and study their properties
  • For example, a Noetherian local ring is regular if and only if its Krull dimension equals the minimum number of generators of its maximal ideal
• Dimension theory is essential in the study of singularities and the resolution of singularities in algebraic geometry
• The concept of dimension is used to define the notion of codimension, which measures the difference in dimensions between a ring and a quotient ring
• Dimension theory plays a role in the study of homological properties of rings, such as the depth and the Cohen-Macaulay property
  • A local ring is Cohen-Macaulay if its depth equals its Krull dimension
• Krull dimension is used in the classification of commutative rings, such as the characterization of Artinian rings as Noetherian rings with Krull dimension zero
• Dimension theory is applied in the study of integral extensions and the going-up and going-down theorems, which relate dimensions of rings in tower of integral extensions

Examples and Problem-Solving Techniques

• To find the Krull dimension of a ring, one can construct chains of prime ideals and determine the supremum of their lengths
  • For example, in the ring $\mathbb{Z}[x]$, the chain $(0) \subset (x)$ shows that the Krull dimension is at least 1
• In a Noetherian local ring, the Krull dimension can be determined by finding the minimum number of generators of the maximal ideal
  • For instance, in the local ring $k[[x,y]]/(xy)$, the maximal ideal is generated by the images of $x$ and $y$, so the Krull dimension is 1
• To prove that a ring has a certain Krull dimension, one can use the properties of Noetherian rings and the dimension inequalities
  • For example, to show that the Krull dimension of $\mathbb{Z}[x]$ is 2, one can use the chain $(0) \subset (x) \subset (x,p)$ for any prime number $p$
• When working with finitely generated modules over a local ring, the Hilbert-Samuel polynomial can be used to determine the dimension and multiplicity of the module
• In problems involving integral extensions, the going-up and going-down theorems can be applied to relate dimensions of rings in the extension

Connections to Other Mathematical Areas

• Dimension theory in commutative algebra is closely related to the concept of dimension in algebraic geometry
  • The Krull dimension of a commutative ring corresponds to the dimension of the associated affine scheme
• Krull dimension is connected to the study of topological spaces, particularly in the context of the Zariski topology on the spectrum of a ring
• Dimension theory is related to the study of homological algebra, as the dimension of a ring can be characterized using homological invariants such as the depth and the projective dimension
• The concept of dimension is used in the theory of algebraic varieties and their singularities
  • The dimension of a variety is defined as the Krull dimension of its coordinate ring
• Krull dimension has applications in number theory, particularly in the study of rings of integers in algebraic number fields
• Dimension theory is connected to the study of invariant theory and the theory of group actions on rings and varieties

Advanced Topics and Current Research

• The study of non-Noetherian rings and their dimensions is an active area of research in commutative algebra
  • Techniques such as the use of filtrations and the theory of valuations are employed to extend dimension theory to non-Noetherian settings
• The connection between Krull dimension and other notions of dimension, such as the Gelfand-Kirillov dimension and the Hilbert-Samuel multiplicity, is a topic of ongoing investigation
• Researchers are exploring the relationship between dimension theory and the theory of tight closure, which provides a powerful tool for studying singularities in characteristic $p$
• The study of dimensions of modules and complexes is an active area of research, with applications to the theory of Cohen-Macaulay rings and the homological conjectures
• Dimension theory is being applied to the study of non-commutative rings and their representations, leading to the development of non-commutative dimension concepts
• The interplay between dimension theory and the theory of $D$-modules, which are modules equipped with a differential structure, is a current topic of research in algebraic geometry and representation theory

Study Tips and Common Pitfalls

• When studying dimension theory, it is essential to have a solid understanding of the basic concepts such as prime ideals, chains, and Noetherian rings
• Practice computing Krull dimensions of various rings and constructing examples of rings with specific dimensions to develop intuition
• Pay attention to the differences between Noetherian and non-Noetherian rings, as the properties and techniques used in dimension theory can vary significantly between the two cases
• Be careful when applying theorems and results, as they often have specific assumptions about the rings or modules involved
  • Always check the hypotheses before using a theorem or result
• When working with local rings, keep in mind the relationship between the Krull dimension and the number of generators of the maximal ideal
• Remember that the Krull dimension is a global property of a ring, while the height of a prime ideal is a local property
  • Be cautious when relating the two concepts
• When dealing with chain conditions, be precise about the type of chain being considered (ascending or descending) and the objects involved (ideals, prime ideals, or submodules)
• Take advantage of the connections between dimension theory and other areas of mathematics, such as algebraic geometry and homological algebra, to gain a deeper understanding of the concepts and their applications