11.3 Depth, regular sequences, and Cohen-Macaulay rings

Depth, regular sequences, and Cohen-Macaulay rings are key concepts in commutative algebra that bridge algebraic and geometric properties. They measure how "nice" a ring or module is, with implications for smoothness, singularities, and cohomology of algebraic varieties.

These ideas connect algebra to geometry by relating the length of regular sequences to the codimension of supports. They're crucial for understanding singularities, computing cohomology, and studying properties of algebraic varieties like normality and rationality.

Module Depth and Geometry

Definition and Geometric Interpretation

• Define the depth of a module M over a local ring (R,m) as the length of a maximal M-regular sequence in the maximal ideal m
• Interpret the depth of a module geometrically as the codimension of the support of the module in the spectrum of the ring (affine algebraic variety, projective variety)
• Understand that for a finite module M over a local ring (R,m), the depth of M is always less than or equal to the dimension of M
• Recognize that if M is a finite module over a local ring (R,m), then depth(M) = dim(M) if and only if M is a Cohen-Macaulay module

Measuring Distance from Cohen-Macaulay Property

• Interpret the depth of a module as a measure of how far the module is from being Cohen-Macaulay
• Understand that the closer the depth of a module is to its dimension, the closer it is to being Cohen-Macaulay
• Recognize that if the depth of a module is strictly less than its dimension, it fails to be Cohen-Macaulay
• Use the difference between depth and dimension to quantify the deviation from the Cohen-Macaulay property (local rings, graded rings)

Regular Sequences in Commutative Algebra

Definition and Properties

• Define an M-regular sequence as a sequence of elements x_1, ..., x_n in a ring R such that x_i is a non-zero-divisor on M/(x_1, ..., x_{i-1})M for all i = 1, ..., n
• Recognize the importance of regular sequences in the study of depth and Cohen-Macaulay rings and modules
• Understand that the length of a maximal regular sequence in an ideal I of a local ring (R,m) is equal to the grade of the ideal I
• Apply the property that if x_1, ..., x_n is an M-regular sequence, then M is a free module over R/(x_1, ..., x_n) if and only if M/(x_1, ..., x_n)M is a free R/(x_1, ..., x_n)-module

Applications in Homological Algebra

• Understand the role of regular sequences in defining the Koszul complex, a fundamental tool in homological algebra
• Use regular sequences to compute the depth of a module via the Koszul complex
• Apply the Auslander-Buchsbaum formula, which relates the projective dimension of a module to its depth using regular sequences
• Recognize the connection between regular sequences and the Tor functor in homological algebra (resolutions, spectral sequences)

Cohen-Macaulay Rings and Modules

Definition and Characterizations

• Define a local ring (R,m) as Cohen-Macaulay if depth(R) = dim(R)
• Define a finite module M over a local ring (R,m) as Cohen-Macaulay if depth(M) = dim(M)
• Understand equivalent characterizations of Cohen-Macaulay rings and modules, such as the existence of a maximal regular sequence generating the maximal ideal or the vanishing of certain local cohomology modules
• Recognize that regular local rings are always Cohen-Macaulay, but the converse is not true in general (singular Cohen-Macaulay rings)

Closure Properties

• Apply the property that if R is a Cohen-Macaulay local ring and x is a non-zero-divisor in R, then R/xR is also Cohen-Macaulay
• Understand that the Cohen-Macaulay property is preserved under localization and completion
• Recognize that the Cohen-Macaulay property is not necessarily preserved under taking quotients or extensions (non-flat homomorphisms, non-Cohen-Macaulay rings)
• Apply the result that if R is a Cohen-Macaulay ring and I is an ideal generated by a regular sequence, then R/I is also Cohen-Macaulay

Cohen-Macaulay Rings for Algebraic Geometry

Affine Algebraic Varieties

• Understand that an affine algebraic variety X over an algebraically closed field k is Cohen-Macaulay if and only if its coordinate ring k[X] is a Cohen-Macaulay ring
• Apply the property that the Cohen-Macaulay property of an algebraic variety is preserved under taking open subsets and irreducible components
• Use the Cohen-Macaulay property of an algebraic variety to study its singularities and compute its cohomology groups (singular cohomology, étale cohomology)

Connections to Other Geometric Properties

• Relate the Cohen-Macaulay property to Serre's criterion for normality and Hartshorne's connectedness theorem
• Understand the connection between the Cohen-Macaulay property and the canonical module of an algebraic variety
• Apply the Hochster-Roberts theorem, which states that rings of invariants of linearly reductive groups acting on regular rings are Cohen-Macaulay
• Recognize the role of Cohen-Macaulay rings in the study of singularities, such as rational singularities and Du Bois singularities

Cohen-Macaulay Property vs Geometry

Relation to Smoothness

• Understand that a smooth algebraic variety over a field k is always Cohen-Macaulay
• Recognize that the converse is not true in general: there exist singular Cohen-Macaulay varieties (cones, monomial curves)
• Apply the property that if X is an equidimensional algebraic variety over a field k, then X is Cohen-Macaulay if and only if the local ring O_{X,x} is Cohen-Macaulay for all x in X

Relation to Complete Intersections and Gorenstein Property

• Understand that the Cohen-Macaulay property is a necessary condition for an algebraic variety to be a complete intersection, but not sufficient
• Recognize that complete intersections are always Cohen-Macaulay, but not all Cohen-Macaulay varieties are complete intersections (determinantal varieties)
• Relate the Cohen-Macaulay property to the Gorenstein property, which is a stronger condition on the local rings of an algebraic variety
• Apply the result that Gorenstein rings are always Cohen-Macaulay, but the converse is not true in general (hypersurfaces, canonical singularities)