13.4 Cohen-Macaulay Rings and Shellability

Cohen-Macaulay rings are a special type of commutative rings with cool properties. They connect algebra and combinatorics, making them super useful in math. These rings pop up in lots of places, from geometry to number theory.

Shellability is a neat feature of some geometric shapes called simplicial complexes. It's like building a complex piece by piece in a specific order. When a complex is shellable, its algebraic counterpart (the Stanley-Reisner ring) is Cohen-Macaulay.

Cohen-Macaulay Rings

Definition and Significance

• Cohen-Macaulay rings are a class of commutative Noetherian rings with desirable homological and combinatorial properties
• The definition of Cohen-Macaulay rings involves the notion of depth and dimension which measure the "size" and "complexity" of the ring
• Cohen-Macaulay rings play a central role in combinatorial commutative algebra as they provide a link between algebraic and combinatorial properties of rings and simplicial complexes
• Many important classes of rings are Cohen-Macaulay
  • Polynomial rings
  • Regular local rings
  • Determinantal rings
• The study of Cohen-Macaulay rings has applications in various fields
  • Algebraic geometry
  • Singularity theory
  • Combinatorics

Characterizing Cohen-Macaulay Rings

Depth and Dimension

• A local ring $(R, m)$ is Cohen-Macaulay if its depth equals its Krull dimension, i.e., $depth(R) = dim(R)$
• For a general Noetherian ring $R$, it is Cohen-Macaulay if the localization at every maximal ideal is Cohen-Macaulay
• The depth of a ring $R$ can be defined in two ways
  • The length of a maximal regular sequence in $R$
  • The minimal $i$ such that $Ext^i_R(k, R) ≠ 0$, where $k$ is the residue field
• The Krull dimension of a ring $R$ is the supremum of the lengths of all chains of prime ideals in $R$

Homological Properties

• Cohen-Macaulay rings have a simple characterization in terms of their local cohomology modules: $R$ is Cohen-Macaulay if and only if $H^i_m(R) = 0$ for all $i ≠ dim(R)$, where $m$ is the maximal ideal
• Cohen-Macaulay rings satisfy the Unmixedness Theorem: if $P$ is a minimal prime over an ideal generated by a system of parameters, then the height of $P$ equals the dimension of the ring
• Example: Regular local rings (rings with maximal ideal generated by a regular sequence) are always Cohen-Macaulay
• Example: The ring $k[x,y,z]/(xy, xz, yz)$ is not Cohen-Macaulay, as its depth (1) is less than its dimension (2)

Shellability of Simplicial Complexes

Simplicial Complexes and Shellability

• A simplicial complex $Δ$ is a collection of subsets (called faces) of a finite set $V$ that is closed under taking subsets, i.e., if $σ ∈ Δ$ and $τ ⊆ σ$, then $τ ∈ Δ$
• A simplicial complex $Δ$ is pure if all its maximal faces (called facets) have the same dimension
• A pure simplicial complex $Δ$ is shellable if its facets can be ordered $F_1, ..., F_s$ such that for all $1 ≤ i < j ≤ s$, there exists some $v ∈ F_j \ F_i$ and some $k < j$ with $F_i ∩ F_j ⊆ F_k ∩ F_j = F_j \ \{v\}$
• Shellability is a combinatorial property that implies many nice topological and algebraic properties
  • Being homotopy equivalent to a wedge of spheres
  • Having a Cohen-Macaulay Stanley-Reisner ring

Examples of Shellable Complexes

• Simplicial trees
• Boundary complexes of simplicial polytopes
• Independence complexes of matroids
• Example: The boundary complex of a simplex is shellable (any ordering of the facets gives a shelling)
• Example: The complex consisting of all subsets of $\{1,2,3,4\}$ except $\{1,2\}$ and $\{3,4\}$ is not shellable, as it is not homotopy equivalent to a wedge of spheres

Shellable Complexes vs Cohen-Macaulay Rings

Stanley-Reisner Rings

• The Stanley-Reisner ring of a simplicial complex $Δ$ on vertex set $V = \{v_1, ..., v_n\}$ over a field $k$ is the quotient ring $k[Δ] = k[x_1, ..., x_n] / I_Δ$, where $I_Δ$ is the ideal generated by the monomials corresponding to non-faces of $Δ$
• Example: If $Δ$ is the boundary complex of a triangle, then $k[Δ] ≅ k[x,y,z]/(xy, xz, yz)$
• Example: If $Δ$ is the complex of all subsets of $\{1,2,3\}$ except $\{1,3\}$, then $k[Δ] ≅ k[x,y,z]/(xz)$

Proof: Shellable Complexes have Cohen-Macaulay Stanley-Reisner Rings

• To prove that shellable complexes have Cohen-Macaulay Stanley-Reisner rings, one can use induction on the number of facets and the recursive structure of the shelling
  • Base case: If $Δ$ has only one facet, then $k[Δ]$ is a polynomial ring, which is Cohen-Macaulay
  • Inductive step: Suppose $Δ$ is a shellable complex with shelling order $F_1, ..., F_s$. Let $Δ'$ be the subcomplex generated by $F_1, ..., F_{s-1}$. By the induction hypothesis, $k[Δ']$ is Cohen-Macaulay
  • One can show that there is a short exact sequence $0 → k[Δ'] → k[Δ] → k[Δ] / (x_v) → 0$, where $v$ is the vertex in $F_s \ F_i$ for some $i < s$, as guaranteed by the shelling property
  • Using the depth lemma and the induction hypothesis, one can conclude that $depth(k[Δ]) = depth(k[Δ']) + 1 = dim(Δ') + 1 = dim(Δ)$, proving that $k[Δ]$ is Cohen-Macaulay
• Alternatively, one can prove the result using the characterization of Cohen-Macaulay rings in terms of local cohomology and the interpretation of local cohomology modules as reduced cohomology of certain subcomplexes