Cohen–Macaulay rings

Cohen–Macaulay rings
Name	Cohen–Macaulay rings
Field	Commutative algebra
Introduced	1940s
Notable contributors	Irving S. Cohen, Francis S. Macaulay, Jean-Pierre Serre, Oscar Zariski, Alexander Grothendieck, Masayoshi Nagata, David Eisenbud

Cohen–Macaulay rings Cohen–Macaulay rings are a class of commutative rings with favorable homological properties that generalize regular rings and play a central role in Algebraic geometry, Singularity theory, and Commutative algebra. They were named after Irving S. Cohen and Francis S. Macaulay and have been developed further by researchers such as Jean-Pierre Serre, Alexander Grothendieck, and David Eisenbud. These rings provide a bridge between algebraic invariants like depth and Krull dimension and geometric notions such as smoothness and normality on schemes like those studied at Institute for Advanced Study and in works associated with Harvard University and University of Cambridge.

Definition and basic properties

A commutative Noetherian local ring (R, m) is defined to be Cohen–Macaulay if the depth of R equals its Krull dimension; this equality was emphasized in foundational results by Jean-Pierre Serre and contextualized by research at Princeton University and Massachusetts Institute of Technology. For non-local Noetherian rings, the condition is imposed on all localizations at prime ideals, a perspective used in monographs from University of California, Berkeley and Columbia University. Basic properties include stability under completion (as in results linked to Nagata's Compactification Theorem contexts) and behavior under flat local homomorphisms, topics explored by researchers at Stanford University and in seminars at University of Chicago.

Examples and non-examples

Standard examples include regular local rings such as power series rings kx1,...,xn studied in seminars at University of Oxford and polynomial rings k[x1,...,xn] over a field k, which appear in texts from Princeton University Press and Cambridge University Press. Cohen–Macaulayness holds for coordinate rings of nonsingular affine varieties as in work related to Hermann Weyl and Oscar Zariski. Hypersurface rings and determinantal rings provide rich families of Cohen–Macaulay examples treated in lecture series at Institute for Advanced Study and École Normale Supérieure. Non-examples include certain curve singularities and rings with embedded associated primes, instances analyzed in case studies at Massachusetts Institute of Technology and University of Chicago.

Homological characterizations

Homological criteria for Cohen–Macaulay rings connect Ext and Tor vanishing patterns first illuminated by Jean-Pierre Serre and systematically treated in texts by David Eisenbud and William Fulton. The Auslander–Buchsbaum formula, developed by Maurice Auslander and David Buchsbaum, relates projective dimension to depth, yielding characterizations for finite projective dimension modules over regular rings encountered at Columbia University and Rutgers University. Local duality theorems, influenced by Alexander Grothendieck and formalized in works from IHÉS, link canonical modules and Matlis duality, tools used to detect Cohen–Macaulayness in research from University of California, Los Angeles.

Depth and dimension theories

Depth and Krull dimension are central invariants: depth measures length of maximal regular sequences while Krull dimension counts chains of prime ideals, concepts rooted in the classical studies at Cambridge University and University of Göttingen. The relationship depth ≤ dimension and equality defining Cohen–Macaulay rings appears in expositions from Princeton University and lectures influenced by Emmy Noether's structural insights. Multiplicity theory, Hilbert–Samuel polynomials, and reduction numbers—subjects of seminars at Cornell University and University of Michigan—interact with depth/dimension considerations to study leading coefficients and invariants tied to Cohen–Macaulay properties.

Cohen–Macaulay modules and rings variants

The notion extends to Cohen–Macaulay modules over a ring R, studied in representation theory contexts at Université Paris-Sud and Technische Universität München. Variants include Gorenstein rings, Buchsbaum rings, and quasi-Gorenstein rings, topics treated in courses at University of Cambridge and University of Bonn. Maximal Cohen–Macaulay modules and their categories play a key role in connections to Representation theory of orders and cluster categories investigated in collaborations involving University of Oxford and Imperial College London. The theory also interacts with canonical modules and dualizing complexes used in seminars at Harvard University.

Local and graded Cohen–Macaulay rings

Local Cohen–Macaulay rings appear naturally in completion and localization procedures studied at Stanford University and in treatments by Nagata. Graded Cohen–Macaulay rings, including standard graded algebras like coordinate rings of projective varieties, are central in projective geometry contexts associated with Princeton University and ETH Zurich. Castelnuovo–Mumford regularity, studied in collaborative workshops at Max Planck Institute for Mathematics and Institut Fourier, often interacts with graded Cohen–Macaulay conditions in the study of syzygies and free resolutions.

Applications and connections in algebraic geometry and commutative algebra

Cohen–Macaulay rings underpin many structural results in algebraic geometry, such as canonical and duality theories developed by Alexander Grothendieck and applied in studies at IHÉS and Institute for Advanced Study. They appear in the analysis of singularities, resolution problems, and moduli questions addressed in conferences at Clay Mathematics Institute and Mathematical Sciences Research Institute. Intersections with invariant theory, as in classical work by David Hilbert and Emmy Noether, and with deformation theory, as studied at Courant Institute and University of California, Berkeley, make Cohen–Macaulay properties a pervasive tool across modern research programs in Algebraic geometry and Commutative algebra.

Category:Commutative algebra