13.1 Monomial Ideals and Stanley-Reisner Rings

Monomial ideals and Stanley-Reisner rings bridge algebra and combinatorics. They let us study polynomial rings using simplicial complexes, connecting algebraic structures to geometric shapes. It's like translating between two languages!

These tools reveal hidden patterns in polynomials. By linking algebraic properties to combinatorial ones, we can use visual intuition to solve tricky math problems. It's a powerful way to unite different areas of math.

Monomial ideals and properties

Definition and structure of monomial ideals

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• A monomial in a polynomial ring $R = k[x_1, ..., x_n]$ is a product of the form $x_1^{a_1} * ... * x_n^{a_n}$, where each $a_i$ is a non-negative integer
• A monomial ideal $I$ in $R$ is an ideal generated by monomials, meaning that $I$ is the smallest ideal containing a given set of monomials
  • Every element of $I$ is a finite sum of the form $\Sigma c_im_i$, where each $c_i$ is a coefficient in $k$ and each $m_i$ is a monomial in the generating set of $I$
• Monomial ideals have a unique minimal generating set consisting of monomials, known as the minimal monomial generators of $I$

Properties and invariants of monomial ideals

• The monomials not belonging to a monomial ideal $I$ form a $k$-basis for the quotient ring $R/I$
• The lcm-lattice of a monomial ideal $I$ is the lattice of all least common multiples of subsets of the minimal monomial generators of $I$, ordered by divisibility
• Dickson's Lemma states that every monomial ideal in a polynomial ring over a field is finitely generated
  • This result ensures that monomial ideals have a finite minimal generating set
  • Example: The monomial ideal $I = \langle x^2, xy, y^3 \rangle$ in $k[x, y]$ is finitely generated by the monomials $x^2$, $xy$, and $y^3$

Stanley-Reisner rings from complexes

Simplicial complexes and their properties

• A simplicial complex $\Delta$ on a vertex set $[n] = \{1, ..., n\}$ is a collection of subsets of $[n]$, called faces, closed under taking subsets
  • The dimension of a face $\sigma \in \Delta$ is $\dim(\sigma) = |\sigma| - 1$, and the dimension of $\Delta$ is the maximum dimension of its faces
  • A facet of $\Delta$ is a maximal face under inclusion
  • Example: The simplicial complex $\Delta = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}\}$ has facets $\{1, 2\}$ and $\{1, 3\}$ and dimension 1

Construction and properties of Stanley-Reisner rings

• The Stanley-Reisner ring (or face ring) of a simplicial complex $\Delta$ over a field $k$ is the quotient ring $k[\Delta] = k[x_1, ..., x_n] / I_\Delta$, where $I_\Delta$ is the Stanley-Reisner ideal generated by the monomials corresponding to non-faces of $\Delta$
• The Krull dimension of $k[\Delta]$ equals the dimension of $\Delta$ plus 1
• The Hilbert series of $k[\Delta]$ encodes the dimensions of the graded components of $k[\Delta]$ and can be expressed in terms of the $f$-vector of $\Delta$
• The Stanley-Reisner ring $k[\Delta]$ is Cohen-Macaulay if and only if $\Delta$ is a Cohen-Macaulay complex, meaning that for every face $\sigma \in \Delta$, the subcomplex $\text{link}(\sigma)$ has the homology of a sphere of dimension $\dim(\text{link}(\sigma))$
  • Example: The Stanley-Reisner ring of the boundary complex of a simplex is always Cohen-Macaulay

Combinatorial vs algebraic properties

Combinatorial invariants of simplicial complexes

• The $f$-vector of a simplicial complex $\Delta$, denoted by $f(\Delta) = (f_{-1}, f_0, ..., f_{d-1})$, counts the number of faces of each dimension, where $f_i$ is the number of $i$-dimensional faces and $f_{-1} = 1$ (corresponding to the empty face)
• The $h$-vector of $\Delta$, denoted by $h(\Delta) = (h_0, h_1, ..., h_d)$, is a transformation of the $f$-vector that often has a more direct interpretation in terms of the algebraic properties of $k[\Delta]$
  • The entries of the $h$-vector are given by the relation: $\Sigma_i h_i t^i = \Sigma_i f_{i-1} (t-1)^{d-i}$, where $d = \dim(\Delta) + 1$
  • Example: For the boundary complex of a simplex, the $h$-vector is always $(1, 1, ..., 1)$

Connections between combinatorial and algebraic properties

• The Hilbert series of $k[\Delta]$ can be expressed as $\text{Hilb}(k[\Delta], t) = (\Sigma_i f_{i-1} t^i) / (1-t)^d$, connecting the $f$-vector of $\Delta$ to the Hilbert series of its Stanley-Reisner ring
• The Krull dimension of $k[\Delta]$ equals the dimension of $\Delta$ plus 1, linking the combinatorial dimension of the simplicial complex to the algebraic dimension of its Stanley-Reisner ring
• The Cohen-Macaulay property of $k[\Delta]$ is equivalent to the Cohen-Macaulay property of $\Delta$, relating the algebraic structure of the Stanley-Reisner ring to the topological structure of the simplicial complex
• The Gorenstein property of $k[\Delta]$ is equivalent to $\Delta$ being a Gorenstein complex, which is a strengthening of the Cohen-Macaulay property with additional symmetry conditions on the $h$-vector
  • Example: The Stanley-Reisner ring of the boundary complex of a simplex is always Gorenstein

Hilbert series and Betti numbers

Hilbert series of Stanley-Reisner rings

• The Hilbert series of a graded $k$-algebra $A = \oplus_{i\geq0} A_i$ is the generating function $\text{Hilb}(A, t) = \Sigma_{i\geq0} (\dim_k A_i) t^i$, where $\dim_k A_i$ is the dimension of the $i$-th graded component of $A$ as a $k$-vector space
• For a Stanley-Reisner ring $k[\Delta]$, the Hilbert series can be computed from the $f$-vector of $\Delta$ using the formula $\text{Hilb}(k[\Delta], t) = (\Sigma_i f_{i-1} t^i) / (1-t)^d$, where $d = \dim(\Delta) + 1$
  • Example: For the boundary complex of an $n$-simplex, the Hilbert series is $(1 + nt) / (1-t)^n$

Betti numbers and free resolutions

• The Betti numbers of a graded $k$-algebra $A$ are the ranks of the free modules in a minimal free resolution of $A$ over a polynomial ring
  • A free resolution of $A$ is an exact sequence of the form $0 \to F_n \to ... \to F_1 \to F_0 \to A \to 0$, where each $F_i$ is a free module over the polynomial ring
  • The $i$-th Betti number $\beta_i$ is the rank of the free module $F_i$ in a minimal free resolution
• The Betti numbers of a Stanley-Reisner ring $k[\Delta]$ can be computed using Hochster's formula, which expresses $\beta_i$ in terms of the reduced homology of certain subcomplexes of $\Delta$
• The Betti numbers of $k[\Delta]$ are related to the $h$-vector of $\Delta$ by the formula $\beta_i = \Sigma_{j\geq i} h_j \binom{j-1}{i-1}$, providing a connection between the algebraic invariants of the Stanley-Reisner ring and the combinatorial invariants of the simplicial complex
• The regularity of $k[\Delta]$ is the smallest integer $r$ such that $\beta_{i,j} = 0$ for all $j > i + r$, where $\beta_{i,j}$ are the graded Betti numbers. The regularity measures the complexity of the minimal free resolution of $k[\Delta]$
  • Example: The regularity of the Stanley-Reisner ring of the boundary complex of an $n$-simplex is always 1