🧮Commutative Algebra Unit 13 – Cohen–Macaulay Rings

Key Concepts and Definitions

• Cohen–Macaulay rings defined as Noetherian local rings with depth equal to dimension
• Regular local rings considered special cases of Cohen–Macaulay rings
  • Regular local rings have maximal ideal generated by a regular sequence
• Depth of a module $M$ over a local ring $(R,m)$ defined as the length of a maximal $M$-regular sequence in $m$
• Dimension of a ring refers to the Krull dimension, the supremum of lengths of chains of prime ideals
• Cohen–Macaulay modules generalize the concept to modules over a Noetherian local ring
  • A finitely generated module $M$ over a Noetherian local ring $R$ is Cohen–Macaulay if $depth(M) = dim(M)$
• Gorenstein rings form a subclass of Cohen–Macaulay rings with additional symmetry properties
• Cohen–Macaulay complexes in combinatorics are simplicial complexes with certain homological properties

Historical Context and Development

• Cohen–Macaulay rings introduced by Irvin Cohen and Francis Macaulay in the early 20th century
• Developed as a generalization of regular local rings in commutative algebra
• Macaulay's work on polynomial rings and ideals laid the foundation for the theory
• Gorenstein rings, named after Daniel Gorenstein, were introduced as a subclass of Cohen–Macaulay rings in the 1950s
• Melvin Hochster and John A. Eagon made significant contributions to the theory in the 1960s and 1970s
  • Hochster's work on the homological properties of Cohen–Macaulay rings was particularly influential
• In the 1980s, Stanley and Reisner introduced Cohen–Macaulay complexes in combinatorics
• Recent research has focused on applications in algebraic geometry, combinatorics, and computational algebra

Properties of Cohen–Macaulay Rings

• Local cohomology modules of a Cohen–Macaulay ring vanish except in the top dimension
• Cohen–Macaulay rings satisfy the unmixedness theorem, which relates the heights of primary components of an ideal to its codimension
• In a Cohen–Macaulay ring, every system of parameters is a regular sequence
  • A system of parameters is a sequence of elements generating an $m$-primary ideal, where $m$ is the maximal ideal
• Cohen–Macaulay rings are universally catenary, meaning all saturated chains of prime ideals between two fixed primes have the same length
• The canonical module of a Cohen–Macaulay ring is a maximal Cohen–Macaulay module and plays a crucial role in duality theory
• The Hilbert-Samuel multiplicity of a Cohen–Macaulay local ring coincides with its degree as a projective variety
• Cohen–Macaulay rings are closely related to the study of singularities in algebraic geometry
  • Rational singularities and log-terminal singularities are characterized by their Cohen–Macaulay property

Examples and Non-Examples

• Regular local rings, such as the ring of convergent power series over a field, are Cohen–Macaulay
• Polynomial rings over a field are examples of Cohen–Macaulay rings
  • More generally, all affine and projective varieties over a field are Cohen–Macaulay
• The coordinate ring of a smooth algebraic curve is always Cohen–Macaulay
• Gorenstein rings, such as complete intersections and determinantal rings, are Cohen–Macaulay
• Rings of invariants of finite groups acting on regular local rings are Cohen–Macaulay
• The cubical complex associated with a partially ordered set is Cohen–Macaulay if and only if the poset is a lattice
• The ring $k[[x,y,z]]/(x^2, y^2, z^2, xy, yz, zx)$ is an example of a non-Cohen–Macaulay ring
  • This ring has depth 1 but dimension 2

Connections to Other Algebraic Structures

• Cohen–Macaulay rings play a central role in the study of singularities in algebraic geometry
  • Rational singularities and log-terminal singularities are characterized by their Cohen–Macaulay property
• The canonical module of a Cohen–Macaulay ring is closely related to the dualizing complex in duality theory
• Cohen–Macaulay rings are connected to the study of Stanley-Reisner rings and simplicial complexes in combinatorial commutative algebra
• In the theory of integral closure of ideals, the associated graded ring of an ideal in a Cohen–Macaulay ring is also Cohen–Macaulay
• The Cohen–Macaulay property is preserved under certain ring constructions, such as tensor products and localizations
• Cohen–Macaulay rings have applications in the study of Hochster's formula for local cohomology and the Hilbert-Kunz multiplicity
• The study of Cohen–Macaulay rings is closely related to the theory of Gorenstein rings and complete intersections

Applications in Mathematics

• Cohen–Macaulay rings are fundamental in the study of singularities in algebraic geometry
  • They are used to characterize rational singularities and log-terminal singularities
• In combinatorics, Cohen–Macaulay complexes have applications in the study of face rings and Stanley-Reisner rings
  • The Stanley-Reisner ring of a simplicial complex is Cohen–Macaulay if and only if the complex is Cohen–Macaulay
• Cohen–Macaulay rings appear in the study of invariant theory and the rings of invariants of finite groups
• The Cohen–Macaulay property is important in the theory of integral closure of ideals and the associated graded rings
• Cohen–Macaulay rings have applications in the study of Hilbert functions and Hilbert polynomials
  • The Hilbert function of a Cohen–Macaulay graded ring has a simple form determined by its Hilbert polynomial
• In the theory of moduli spaces, Cohen–Macaulay rings are used to study the local structure of moduli spaces of algebraic varieties
• Cohen–Macaulay rings have connections to the study of Hochster's formula for local cohomology and the Hilbert-Kunz multiplicity

Computational Techniques and Methods

• Gröbner basis methods are used to compute the depth and dimension of a module over a polynomial ring
  • These methods can be used to check the Cohen–Macaulay property of a ring or module
• Syzygies and free resolutions are computational tools used to study the homological properties of Cohen–Macaulay rings
  • The Auslander-Buchsbaum formula relates the depth of a module to the projective dimension of its syzygy modules
• Hilbert series and Hilbert polynomials can be computed using Gröbner basis methods and are useful in the study of Cohen–Macaulay graded rings
• The Eisenbud-Goto conjecture, which bounds the regularity of a finitely generated graded module over a polynomial ring, has computational implications for Cohen–Macaulay rings
• Computational methods for determining the Cohen–Macaulay property of simplicial complexes have been developed in combinatorial commutative algebra
• The Hochster-Roberts theorem, which states that rings of invariants of linearly reductive groups acting on regular rings are Cohen–Macaulay, has computational applications
• Computational tools from toric geometry, such as toric varieties and polyhedra, are used to study Cohen–Macaulay semigroup rings

Advanced Topics and Current Research

• The study of Cohen–Macaulay rings in mixed characteristic, where the ring contains elements of positive characteristic and characteristic zero, is an active area of research
• Researchers are investigating the connections between Cohen–Macaulay rings and singularity theory, particularly in the context of minimal model program and birational geometry
• The theory of Cohen–Macaulay modules over non-commutative rings, such as finite-dimensional algebras, is a growing area of interest
• Generalizations of Cohen–Macaulay rings, such as sequentially Cohen–Macaulay rings and generalized Cohen–Macaulay rings, are being studied for their homological and geometric properties
• The study of Cohen–Macaulay rings in positive characteristic has led to the development of tight closure theory and F-singularity theory
  • These theories have applications in the study of singularities and the homological properties of rings
• Researchers are exploring the connections between Cohen–Macaulay rings and the theory of cluster algebras, which have applications in representation theory and algebraic geometry
• The study of Cohen–Macaulay rings in the context of derived categories and triangulated categories is an emerging area of research, with potential applications in homological algebra and algebraic geometry