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Cohen–Macaulay.
Cohen–Macaulay denotes a central class of commutative rings and modules in algebra with deep connections to Mumford, Serre, Grothendieck, Zariski, and Nagata-style foundations in algebraic geometry and commutative algebra. It sits between regular and singular behavior encountered by authors such as Kaplansky, Hall, Bass, and Atiyah; Cohen–Macaulay objects frequently appear in the work of Hironaka, Eisenbud, Auslander, and Buchsbaum.
A local Noetherian ring R (or a finitely generated module M over R) is Cohen–Macaulay when the depth equals the Krull dimension; this criterion is used by Artin, Zariski, Serre, Grothendieck, and Nagata in structural results. Depth is computed via regular sequences or Ext functors familiar from the work of Loday, Cartan, Serre and Grothendieck. Essential properties include stability under completion and flat local homomorphisms commonly studied by Nagata, Tate, Grothendieck, Serre, and inheritance under localization influenced by methods of Zariski and Deligne.
Classical examples include regular local rings treated by Serre and Hironaka as Cohen–Macaulay, hypersurface rings studied by Milnor and Eisenbud, and complete intersections appearing in the studies of Auslander, Buchsbaum, and Stanley. Toric varieties examined by Cox, Sturmfels, Gelfand, and Gelfand give combinatorial Cohen–Macaulay rings linked to results of Stanley and Ziegler. Determinantal rings connected to Eisenbud, Fiedler, and Fulton provide families of Cohen–Macaulay examples; graded coordinate rings of projective varieties studied by Grothendieck, Serre, and Mumford often satisfy Cohen–Macaulay conditions seen in classical work of Zariski and Deligne.
The module-theoretic viewpoint originates in research by Kaplansky, Auslander, Bass, and Buchsbaum. A finitely generated R-module M is Cohen–Macaulay when depth_R(M)=dim_R(support M), a condition analyzed via Ext and Tor functors used by Serre and Grothendieck. Maximal Cohen–Macaulay modules studied by Auslander and Reiten play key roles in representation theory related to work of Keller, Rickard, and Broué and singularity categories investigated by Orlov and Balmer.
Homological criteria derive from the Auslander–Buchsbaum formula of Auslander and Bass with proofs in expositions by Eisenbud, Hartshorne, and Serre. The vanishing of certain Ext or Tor groups is used by Atiyah, Halperin, and Milnor in characterizations; projective dimension finiteness and depth interact as in results by Auslander and Buchsbaum and refinement in work of Hibi and Ginzburg.
Cohen–Macaulay schemes and varieties are treated by Grothendieck, Serre, Mumford, and Hartshorne in their studies of duality, Hilbert schemes, and moduli problems explored by Deligne, Grothendieck and Gieseker. Conditions on local rings of points on a scheme, studied in work by Zariski and Nagata, determine Cohen–Macaulayness for projective varieties treated by Deligne, Borel, and Weil. Toric and Schubert varieties investigated by Sturmfels, Fulton, Brion, and Drinfeld exhibit Cohen–Macaulay behavior important for intersection theory in the traditions of Serre and Grothendieck.
Canonical modules and dualizing complexes are central in the work of Grothendieck, Serre, Hartshorne, and Mayer, and are used by Eisenbud and H. Matsumura to formulate local duality results. Gorenstein rings characterized by self-dual canonical modules were analyzed by Bass and Auslander and extended in treatments by Stanley and Nagata. Local duality theorems connect to Grothendieck’s duality and residues developed by Grothendieck, Koszul, and Serre.
Cohen–Macaulay theory informs classification problems studied by Auslander, Eisenbud, Bass, and Serre; it relates to Gorenstein properties investigated by Bass, regularity criteria of Serre and Nagata, and Buchsbaum rings introduced by Buchsbaum. Applications include linkage theory of ideals pursued by Huneke and Huneke, deformation theory in the work of Artin and Mumford, and combinatorial commutative algebra advanced by Stanley, Sturmfels, and Kalai. Computational methods tied to Eisenbud, Grayson, Ramachandran, and software projects influenced by Gries aid in verifying Cohen–Macaulay properties in explicit examples.