Cohen-Macaulay and Gorenstein rings are special types of commutative rings with nice algebraic properties. They're important because they balance complexity and structure, making them useful in many areas of algebra and geometry.
These rings have specific and relationships, and their properties are often described using homological tools. Understanding them helps us tackle problems in algebraic geometry, commutative algebra, and even some areas of physics.
Cohen-Macaulay Rings
Definition and Properties
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is a commutative Noetherian local ring R whose depth equals its Krull dimension, depth(R)=dim(R)
is a Cohen-Macaulay ring with the additional property that its maximal ideal can be generated by a regular sequence (sequence of elements x1,…,xn such that xi is a non-zero divisor in R/(x1,…,xi−1) for all i)
Cohen-Macaulay rings have a ωR, a finitely generated module with special homological properties (e.g., ExtRi(ωR,R)=0 for i=dim(R) and ExtRdim(R)(ωR,R)≅R)
Type and Canonical Module
Type of a Cohen-Macaulay ring R is the dimension of the socle of the ring R/m as a vector space over the residue field k=R/m, where m is the maximal ideal
Canonical module ωR plays a crucial role in the study of Cohen-Macaulay rings
For a , the canonical module is isomorphic to the ring itself, ωR≅R
For a , the canonical module is isomorphic to the top exterior power of the cotangent space, ωR≅∧dim(R)m/m2
Gorenstein Rings
Definition and Properties
Gorenstein ring is a Cohen-Macaulay ring of type 1, meaning the socle of R/m is a one-dimensional vector space over the residue field k=R/m
Gorenstein rings have a , a complex of modules that generalizes the notion of a canonical module
characterization states that a local ring R is Gorenstein if and only if Hmi(R)=0 for all i=dim(R) and Hmdim(R)(R)≅ER(k), the of the residue field k
Dualizing Complex
is a complex of modules D∙ over a local ring R with special homological properties (e.g., the cohomology of D∙ is finitely generated and the natural map R→RHomR(D∙,D∙) is a quasi-isomorphism)
For a Gorenstein ring, the dualizing complex is isomorphic to a shift of the ring itself, D∙≅R[−dim(R)]
Existence of a dualizing complex is a key property that distinguishes Gorenstein rings from general Cohen-Macaulay rings
Properties and Characterizations
Serre's Conditions and Regular Local Rings
"" (Sn) and (Rn) are properties of a local ring R related to the depth of prime ideals and the regularity of the ring
(Sn): depth(Rp)≥min(n,dim(Rp)) for all prime ideals p of R
(Rn): Rp is regular for all prime ideals p with dim(Rp)≤n
Regular local ring satisfies both (Sn) and (Rn) for all n, making it a Cohen-Macaulay ring with additional regularity properties
Homological Characterizations
Canonical module and local cohomology provide homological characterizations of Cohen-Macaulay and Gorenstein rings
For a Cohen-Macaulay ring R, the canonical module ωR satisfies ExtRi(ωR,R)=0 for i=dim(R) and ExtRdim(R)(ωR,R)≅R
Local cohomology characterization of Gorenstein rings states that R is Gorenstein if and only if Hmi(R)=0 for all i=dim(R) and Hmdim(R)(R)≅ER(k), the injective hull of the residue field k
These characterizations highlight the deep connections between the homological properties of a ring and its Cohen-Macaulay or Gorenstein nature