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Theorem 7.25. Suppose \( 0 < p < 1 \) and \( \alpha = n\left( {{p}^{-1} - 1}\right) \) . Then the dual space of \( {H}^{p} \) can be identified with \( {\Lambda }_{\alpha } \) under the integral pairing \[ \langle f, g\rangle = \mathop{\lim }\limits_{{r \rightarrow {1}^{ - }}}{\int }_{{\mathbb{S}}_{n}}{f}_{r}\bar{g}{d\... | Proof. Let \( \beta = \left( {n/p}\right) - \left( {n + 1}\right) \) . Then a computation using Lemma 1.11 shows that \[ {\int }_{{\mathbb{S}}_{n}}f\left( \zeta \right) \overline{g\left( \zeta \right) }{d\sigma }\left( \zeta \right) = {\int }_{{\mathbb{B}}_{n}}f\left( z\right) \overline{{R}^{-1,\alpha }g\left( z\right)... | Yes |
Theorem 7.26. Suppose \( \alpha > 0,\beta > - 1 \), and \( s = \beta - \alpha \) . If \( s > - 1 \), then \( {\left( {\Lambda }_{\alpha ,0}\right) }^{ * } = \) \( {A}_{\beta }^{1} \) (with equivalent norms) under the integral pairing\n\n\[ \langle f, g{\rangle }_{s} = \mathop{\lim }\limits_{{r \rightarrow {1}^{ - }}}{\... | Proof. Let \( t \) be any real parameter such that the fractional differential operator \( {R}^{t,\alpha } \) is a bounded invertible operator from \( {\Lambda }_{\alpha ,0} \) onto \( {\mathcal{B}}_{0} \) ; see Theorem 7.20. Then \( F \) is a bounded linear functional on \( {\Lambda }_{\alpha ,0} \) if and only if \( ... | Yes |
Lemma 1.2 Every monomial ideal has a unique minimal set of monomial generators, and this set is finite. | Proof. The Hilbert Basis Theorem says that every ideal in \( S \) is finitely generated. It implies that if \( I \) is a monomial ideal, then \( I = \left\langle {{\mathbf{x}}^{{\mathbf{a}}_{1}},\ldots ,{\mathbf{x}}^{{\mathbf{a}}_{r}}}\right\rangle \) . The direct sum condition means that a polynomial \( f \) lies insi... | Yes |
Theorem 1.7 The correspondence \( \Delta \rightsquigarrow {I}_{\Delta } \) constitutes a bijection from simplicial complexes on vertices \( \{ 1,\ldots, n\} \) to squarefree monomial ideals inside \( S = \mathbb{k}\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) . Furthermore, \[ {I}_{\Delta } = \mathop{\bigcap }... | Proof. By definition, the set of squarefree monomials that have nonzero images in the Stanley-Reisner ring \( S/{I}_{\Delta } \) is precisely \( \left\{ {{\mathbf{x}}^{\sigma } \mid \sigma \in \Delta }\right\} \) . This shows that the map \( \Delta \rightsquigarrow {I}_{\Delta } \) is bijective. In order for \( {\mathb... | Yes |
The Hilbert series of \( S \) itself is the rational function\n\n\[ H\left( {S;\mathbf{x}}\right) = \mathop{\prod }\limits_{{i = 1}}^{n}\frac{1}{1 - {x}_{i}} \] | \[ = \text{sum of all monomials in}S\text{.} \] | No |
Theorem 1.13 The Stanley-Reisner ring \( S/{I}_{\Delta } \) has the \( K \) -polynomial\n\n\[ \mathcal{K}\left( {S/{I}_{\Delta };\mathbf{x}}\right) = \mathop{\sum }\limits_{{\sigma \in \Delta }}\left( {\mathop{\prod }\limits_{{i \in \sigma }}{x}_{i} \cdot \mathop{\prod }\limits_{{j \notin \sigma }}\left( {1 - {x}_{j}}\... | Proof. The definition of \( {I}_{\Delta } \) says which squarefree monomials are not in \( {I}_{\Delta } \) . However, because the generators of \( {I}_{\Delta } \) are themselves squarefree, a monomial \( {\mathbf{x}}^{\mathbf{a}} \) lies outside \( {I}_{\Delta } \) precisely when the squarefree monomial \( {\mathbf{x... | Yes |
Example 1.14 Consider the simplicial complex \( \Gamma \) depicted in Fig. 1.1. (The reason for not calling it \( \Delta \) is because we will compare \( \Gamma \) in Example 1.36 with the simplicial complex \( \Delta \) of Examples 1.5 and 1.8.) The Stanley-Reisner ideal of \( \Gamma \) is\n\n\[ \n{I}_{\Gamma } = \lan... | \[ \n= \langle a, d\rangle \cap \langle a, e\rangle \cap \langle b, c, d\rangle \cap \langle b, e\rangle \cap \langle c, e\rangle \cap \langle d, e\rangle \n\] | No |
Corollary 1.15 Letting \( {f}_{i} \) be the number of \( i \) -faces of \( \Delta \), we get\n\n\[ H\left( {S/{I}_{\Delta };t,\ldots, t}\right) = \frac{1}{{\left( 1 - t\right) }^{n}}\mathop{\sum }\limits_{{i = 0}}^{d}{f}_{i - 1}{t}^{i}{\left( 1 - t\right) }^{n - i}, \]\n\nwhere \( d = \dim \left( \Delta \right) + 1 \) ... | Canceling \( {\left( 1 - t\right) }^{n - d} \) from the sum and the denominator \( {\left( 1 - t\right) }^{n} \) in Corollary 1.15, the numerator polynomial \( h\left( t\right) \) on the right-hand side of\n\n\[ \frac{1}{{\left( 1 - t\right) }^{d}}\mathop{\sum }\limits_{{i = 0}}^{d}{f}_{i - 1}{t}^{i}{\left( 1 - t\right... | No |
Example 1.25 Let \( \Gamma \) be the simplicial complex from Example 1.14. The Stanley-Reisner ring \( S/{I}_{\Gamma } \) has minimal free resolution  | in which the maps are denoted by monomial matrices. We have used the more succinct monomial labels \( {\mathbf{x}}^{{\mathbf{a}}_{p}} \) and \( {\mathbf{x}}^{{\mathbf{a}}_{q}} \) instead of the vector labels \( {\mathbf{a}}_{p} \) and \( {\mathbf{a}}_{q} \) . Below each free module is a list of the degrees in \( {\math... | Yes |
Proposition 1.28 The Koszul complex \( \mathbb{K} \) . is a minimal free resolution of \( \mathbb{k} = S/\mathfrak{m} \) for the maximal ideal \( \mathfrak{m} = \left\langle {{x}_{1},\ldots ,{x}_{n}}\right\rangle \) . | Proof. The essential observation is that a free module generated by \( {1}_{\tau } \) in squarefree degree \( \tau \) is nonzero in squarefree degree \( \sigma \) precisely when \( \tau \subseteq \sigma \) (equivalently, when \( {\mathbf{x}}^{\tau } \) divides \( {\mathbf{x}}^{\sigma } \) ). The only contribution to th... | Yes |
Tensoring the minimal free resolution in Example 1.25 with \( \mathbb{k} = S/\mathfrak{m} \) yields a complex of \( S \) -modules, each of which is a direct sum of translates of \( \mathbb{k} \), and where all the maps are zero. The translation vectors, which are listed below each direct sum, are identified with the ro... | The modules \( {\operatorname{Tor}}_{i}^{S}\left( {M, N}\right) \) are by definition calculated by applying \( \_ \otimes N \) to a free resolution of \( M \) and taking homology [Wei94, Definition 2.6.4]. However, it is a general theorem from homological algebra (see [Wei94, Application 5.6.3] or do Exercise 1.12) tha... | No |
Lemma 1.32 The \( {i}^{\text{th }} \) Betti number of an \( {\mathbb{N}}^{n} \) -graded module \( M \) in degree \( \mathbf{a} \) equals the vector space dimension \( {\dim }_{\mathbb{k}}{\operatorname{Tor}}_{i}^{S}{\left( \mathbb{k}, M\right) }_{\mathbf{a}} \) . | Proof. Tensoring a minimal free resolution of \( M \) with \( \mathbb{k} = S/\mathfrak{m} \) turns all of the differentials \( {\phi }_{i} \) into zero maps. | No |
Theorem 1.34 Given a vector \( \mathbf{b} \in {\mathbb{N}}^{n} \), the Betti numbers of \( I \) and \( S/I \) in degree \( \mathbf{b} \) can be expressed as\n\n\[{\beta }_{i,\mathbf{b}}\left( I\right) = {\beta }_{i + 1,\mathbf{b}}\left( {S/I}\right) = {\dim }_{\mathbb{k}}{\widetilde{H}}_{i - 1}\left( {{K}^{\mathbf{b}}\... | Proof. For the first equality, use a minimal free resolution of \( I \) achieved by snipping off the copy of \( S \) occurring in homological degree 0 of a minimal free resolution of \( S/I \) . To equate \( {\beta }_{i,\mathbf{b}}\left( I\right) \) with the dimension of the indicated homology, use Lemma 1.32 and Propo... | Yes |
Proposition 1.37 If \( \Delta \) is a simplicial complex, then its Alexander dual is \( {\Delta }^{ \star } = \{ \bar{\tau } \mid \tau \notin \Delta \} \), consisting of the complements of the nonfaces of \( \Delta \) . | Proof. By Definition 1.6, \( {I}_{\Delta } = \left\langle {{\mathbf{x}}^{\tau } \mid \tau \notin \Delta }\right\rangle \), so \( {I}_{{\Delta }^{ \star }} = \mathop{\bigcap }\limits_{{\tau \notin \Delta }}{\mathfrak{m}}^{\tau } \) by Definition 1.35. However, this intersection equals \( \mathop{\bigcap }\limits_{{\bar{... | Yes |
Corollary 1.40 (Hochster’s formula, dual version) All nonzero Betti numbers of \( {I}_{\Delta } \) and \( S/{I}_{\Delta } \) lie in squarefree degrees \( \sigma \), where\n\n\[ \n{\beta }_{i,\sigma }\left( {I}_{\Delta }\right) = {\beta }_{i + 1,\sigma }\left( {S/{I}_{\Delta }}\right) = {\dim }_{\mathbb{k}}{\widetilde{H... | Proof. For squarefree degrees, apply Theorem 1.34 by first checking that \( {K}^{\sigma }\left( {I}_{\Delta }\right) = {\operatorname{link}}_{{K}^{1}\left( {I}_{\Delta }\right) }\left( \bar{\sigma }\right) \) and then verifying that \( {K}^{1}\left( {I}_{\Delta }\right) = {\Delta }^{ \star } \) . Both of these claims a... | No |
Proposition 2.1 A nonzero ideal \( I \) inside \( S \) is fixed under the action of the torus \( {T}_{n}\left( \mathbb{k}\right) \) if and only if \( I \) is a monomial ideal. | Proof. Torus elements map each variable and hence each monomial-to a multiple of itself, so monomial ideals are fixed by \( {T}_{n}\left( \mathbb{k}\right) \) . Conversely, let \( I \) be an arbitrary torus-fixed ideal, and suppose that \( p = \sum {c}_{\mathbf{a}}{\mathbf{x}}^{\mathbf{a}} \) is a polynomial in \( I \)... | Yes |
Corollary 2.2 A nonzero ideal \( I \) in \( S \) is fixed under the action of the general linear group \( G{L}_{n}\left( \mathbb{k}\right) \) if and only if \( I \) is a power \( {\mathfrak{m}}^{d} \) of the irrelevant maximal ideal \( \mathfrak{m} = \left\langle {{x}_{1},\ldots ,{x}_{n}}\right\rangle \), for some posi... | Proof. The vector space of homogeneous polynomials of degree \( d \) is fixed by \( G{L}_{n}\left( \mathbb{k}\right) \), and hence so is the ideal \( {\mathfrak{m}}^{d} \) it generates. Conversely, suppose \( I \) is a \( G{L}_{n}\left( \mathbb{k}\right) \) -fixed ideal and that \( p \) is a nonzero polynomial in \( I ... | Yes |
Proposition 2.3 The following are equivalent for a monomial ideal I.\n\n(i) I is Borel-fixed.\n\n(ii) If \( m \in I \) is any monomial divisible by \( {x}_{j} \), then \( m\frac{{x}_{i}}{{x}_{j}} \in I \) for \( i < j \) . | Proof. Suppose that \( I \) is a Borel-fixed ideal. Let \( m \in I \) be any monomial divisible by \( {x}_{j} \) and consider any index \( i < j \) . Let \( g \) be the elementary matrix in \( {B}_{n}\left( \mathbb{k}\right) \) that sends \( {x}_{j} \) to \( {x}_{j} + {x}_{i} \) and that fixes all other variables. The ... | Yes |
Example 2.4 Here is a typical Borel-fixed ideal in three variables:\n\n\[ I = \left\langle {{x}_{1}^{2},{x}_{1}{x}_{2},{x}_{2}^{3},{x}_{1}{x}_{3}^{3}}\right\rangle . \] | Each of the four generators satisfies condition (ii). The ideal \( I \) has the following unique irreducible decomposition (see Chapter 5.2 if these are unfamiliar), which is also a primary decomposition:\n\n\[ I = \left\langle {{x}_{1},{x}_{2}^{3}}\right\rangle \cap \left\langle {{x}_{1}^{2},{x}_{2},{x}_{3}^{3}}\right... | No |
Lemma 2.6 For a fixed ideal \( I \) and term order \( < \), the number of equivalence classes in \( G{L}_{n}\left( \mathbb{k}\right) \) is finite. One of these classes is a nonempty Zariski open subset \( U \) inside of \( G{L}_{n}\left( \mathbb{k}\right) \) . | Proof. Consider the polynomial ring \( S\left\lbrack {{g}_{11},\ldots ,{g}_{nn}}\right\rbrack = \mathbb{k}\left\lbrack {\mathbf{g},\mathbf{x}}\right\rbrack \) in \( {n}^{2} + n \) unknowns. Suppose that \( {p}_{1}\left( \mathbf{x}\right) ,\ldots ,{p}_{r}\left( \mathbf{x}\right) \) are generators of the given ideal \( I... | Yes |
Example 2.8 Let \( n = 2 \) and consider the ideal \( I = \left\langle {{x}_{1}^{2},{x}_{2}^{2}}\right\rangle \), where \( < \) is the lexicographic order with \( {x}_{1} > {x}_{2} \) . For this term order, the ideal \( J \) defined in the proof of Lemma 2.6 has the comprehensive Gröbner basis | \[ \mathcal{C} = \left\{ {{g}_{11}^{2}{x}_{1}^{2} + 2{g}_{11}{g}_{12}{x}_{1}{x}_{2} + {g}_{12}^{2}{x}_{2}^{2},{g}_{21}^{2}{x}_{1}^{2} + 2{g}_{21}{g}_{22}{x}_{1}{x}_{2} + {g}_{22}^{2}{x}_{2}^{2},}\right. \] \[ 2{g}_{21}{g}_{11}\left( {{g}_{22}{g}_{11} - {g}_{21}{g}_{12}}\right) {x}_{1}{x}_{2} + \left( {{g}_{22}{g}_{11} ... | Yes |
Theorem 2.9 The generic initial ideal \( {\sin }_{ < }\left( I\right) \) is Borel-fixed. | Proof. We refer to Eisenbud's commutative algebra textbook, where this result appears as [Eis95, Theorem 15.20]. A complete proof is given there. \( ▱ \) | No |
Example 2.10 Let \( f, g \in \mathbb{k}\left\lbrack {{x}_{1},{x}_{2},{x}_{3},{x}_{4}}\right\rbrack \) be generic forms of degrees \( d \) and \( e \), respectively. Considering the three smallest nontrivial cases, we list the generic initial ideal of \( I = \langle f, g\rangle \) for both the lexicographic order and th... | The ideals \( J = {\operatorname{gin}}_{\text{lex }}\left( I\right) \) are:\n\n\[ \left( {d, e}\right) = \left( {2,2}\right) \;J = \left\langle {{x}_{2}^{4},{x}_{1}{x}_{3}^{2},{x}_{1}{x}_{2},{x}_{1}^{2}}\right\rangle \]\n\n\[ = \left\langle {{x}_{1},{x}_{2}^{4}}\right\rangle \cap \left\langle {{x}_{1}^{2},{x}_{2},{x}_{... | Yes |
Lemma 2.11 Each monomial \( m \) in the Borel-fixed ideal \( I = \left\langle {{m}_{1},\ldots ,{m}_{r}}\right\rangle \) can be written uniquely as a product \( m = {m}_{i}{m}^{\prime } \) with \( \max \left( {m}_{i}\right) \leq \min \left( {m}^{\prime }\right) \) . | Proof. Uniqueness: Suppose \( m = {m}_{i}{m}_{i}^{\prime } = {m}_{j}{m}_{j}^{\prime } \) both satisfy the condition, with \( {u}_{i} \leq {u}_{j} \) . Then \( {m}_{i} \) and \( {m}_{j} \) agree in every variable with index \( < {u}_{i} \) . If \( {x}_{{u}_{i}} \) divides \( {m}_{j}^{\prime } \), then \( {u}_{i} = {u}_{... | No |
Proposition 2.12 For the Borel-fixed ideal \( I = \left\langle {{m}_{1},\ldots ,{m}_{r}}\right\rangle \), the quotient S/I has K-polynomial | Proof. By Lemma 2.11, the set of monomials in \( I \) is the disjoint union over \( i = 1,\ldots, r \) of the monomials in \( {m}_{i} \cdot \mathbb{k}\left\lbrack {{x}_{{u}_{i}},\ldots ,{x}_{n}}\right\rbrack \) . The sum of all monomials in such a translated subalgebra of \( S \) equals the series\n\n\[ \frac{{m}_{i}}{... | Yes |
Example 2.13 Let \( I \) be the ideal in Example 2.4. Its \( K \) -polynomial is\n\n\[ \mathcal{K}\left( {S/I;\mathbf{x}}\right) = 1 - {x}_{1}^{2} - {x}_{1}{x}_{2}\left( {1 - {x}_{1}}\right) - {x}_{2}^{3}\left( {1 - {x}_{1}}\right) - {x}_{1}{x}_{3}^{3}\left( {1 - {x}_{1}}\right) \left( {1 - {x}_{2}}\right) \] | \[ = 1 - {x}_{1}^{2} - {x}_{1}{x}_{2} - {x}_{2}^{3} - {x}_{1}{x}_{3}^{3} \] \n\n\[ + {x}_{1}^{2}{x}_{3}^{3} + {x}_{1}{x}_{2}{x}_{3}^{3} + {x}_{1}{x}_{2}^{3} + {x}_{1}^{2}{x}_{2} \] \n\n\[ \text{ - }{x}_{1}^{2}{x}_{2}{x}_{3}^{3}\text{. } \] \n\nThis expansion suggests that the minimal resolution of \( S/I \) has the for... | Yes |
Example 2.19 Let \( n = 4 \) and \( r = 7 \), and consider the following ideal:\n\n\[ \left\langle {{x}_{1}{x}_{2}{x}_{4}^{4},\;{x}_{1}{x}_{2}{x}_{3}{x}_{4}^{2},\;{x}_{1}{x}_{3}^{6},\;{x}_{1}{x}_{2}{x}_{3}^{2},\;{x}_{2}^{6},\;{x}_{1}{x}_{2}^{2},\;{x}_{1}^{2}}\right\rangle . \] | This monomial ideal is Borel-fixed. Beneath the seven generators, we wrote in 12 rows the 12 minimal first syzygies (2.1) on the generators. These form a Gröbner basis for the syzygy module \( M \), and the initial module is\n\n\[ \operatorname{in}\left( M\right) = \left\langle {{x}_{1}{\mathbf{e}}_{1},\;{x}_{2}{\mathb... | Yes |
Let \( n = d = 3 \), and use the variable set \( \{ x, y, z\} \) . The Betti numbers and Eliahou-Kervaire resolution of the Borel-fixed ideal \( I = \) \( \langle x, y, z{\rangle }^{3} \) can be visualized as follows: | The numbers in the left-hand diagram determine the binomial coefficients \( \left( \begin{matrix} \max \left( {m}_{j}\right) - 1 \\ i \end{matrix}\right) \) from Theorem 2.18, which are given in the triangles below. By adding these triangles, we get the Betti numbers of the minimal free resolution\n\n![9d852306-8a03-41... | Yes |
Lemma 2.26 If \( B \) is a Borel set in \( {S}_{d} \) then\n\n\[ \n{\beta }_{i}\left( B\right) = \left( \begin{matrix} n - 1 \\ i \end{matrix}\right) \cdot \left| B\right| - \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{\mu }_{ \leq j}\left( B\right) \left( \begin{matrix} j - 1 \\ i - 1 \end{matrix}\right) .\n\] | Proof. Rewrite (2.6) for \( W = B \) as follows:\n\n\[ \n{\beta }_{i}\left( B\right) = \mathop{\sum }\limits_{{j = 1}}^{n}{\mu }_{j}\left( B\right) \left( \begin{matrix} j - 1 \\ i \end{matrix}\right) \n\]\n\n\[ \n= \mathop{\sum }\limits_{{j = 1}}^{n}\left( {{\mu }_{ \leq j}\left( B\right) - {\mu }_{ \leq j - 1}\left( ... | Yes |
Proposition 3.1 The minimal free resolution of an ideal generated by \( r \) monomials in \( S = \mathbb{k}\left\lbrack {x, y}\right\rbrack \) has the format\n\n\[ 0 \leftarrow S \leftarrow {S}^{r} \leftarrow {S}^{r - 1} \leftarrow 0. \] | The minimal first syzygies are the vectors \( {y}^{{b}_{i + 1} - {b}_{i}}{\mathbf{e}}_{i} - {x}^{{a}_{i} - {a}_{i + 1}}{\mathbf{e}}_{i + 1} \) corresponding to adjacent pairs \( \left\{ {{x}^{{a}_{i}}{y}^{{b}_{i}},{x}^{{a}_{i + 1}}{y}^{{b}_{i + 1}}}\right\} \) of minimal generators of \( I \) .\n\nProof. The kernel of ... | Yes |
Proposition 3.2 \( I \subset \mathbb{k}\left\lbrack {x, y}\right\rbrack \) has the irredundant irreducible decomposition\n\n\[ I = \left\langle {y}^{{b}_{1}}\right\rangle \cap \left\langle {{x}^{{a}_{1}},{y}^{{b}_{2}}}\right\rangle \cap \left\langle {{x}^{{a}_{2}},{y}^{{b}_{3}}}\right\rangle \cap \cdots \cap \left\lang... | Proof. After removing common factors from the generators, we may assume that \( {b}_{1} = 0 \) and \( {a}_{r} = 0 \), so that \( I \) is artinian. The given ideals \( \left\langle {{x}^{{a}_{i}},{y}^{{b}_{i} + 1}}\right\rangle \) are irreducible and clearly contain \( I \) . Inspection of the staircase diagram shows th... | No |
Proposition 3.5 The syzygy module \( \operatorname{syz}\left( I\right) \) is generated by syzygies \( {\sigma }_{ij} \) corresponding to edges \( \left( {i, j}\right) \) in the Buchberger graph \( \operatorname{Buch}\left( I\right) \) . | Proof. The following identity holds for all \( i, j, k \in \{ 1,\ldots, r\} \) :\n\n\[ \frac{\operatorname{lcm}\left( {{m}_{i},{m}_{j},{m}_{k}}\right) }{\operatorname{lcm}\left( {{m}_{i},{m}_{j}}\right) }{\sigma }_{ij} + \frac{\operatorname{lcm}\left( {{m}_{i},{m}_{j},{m}_{k}}\right) }{\operatorname{lcm}\left( {{m}_{j}... | Yes |
Proposition 3.9 If \( I \) is a strongly generic monomial ideal in \( \mathbb{k}\left\lbrack {x, y, z}\right\rbrack \) , then the Buchberger graph \( \operatorname{Buch}\left( I\right) \) is planar and connected. If, in addition, \( I \) is artinian, then \( \operatorname{Buch}\left( I\right) \) consists of the edges i... | Sketch of proof. First observe that it suffices to consider artinian monomial ideals \( I \), meaning that the minimal generators of \( I \) include pure powers in each of the three variables, say \( {x}^{a},{y}^{b} \), and \( {z}^{c} \) . Indeed, erasing all edges and regions incident to one or more of \( \left\{ {{x}... | Yes |
Theorem 3.11 Given a strongly generic monomial ideal \( I \) in \( \mathbb{k}\left\lbrack {x, y, z}\right\rbrack \), the planar map \( \operatorname{Buch}\left( I\right) \) provides a minimal free resolution of \( I \) . | Sketch of proof. Begin by throwing high powers \( {x}^{a},{y}^{b} \), and \( {z}^{c} \) into \( I \) . What results is still strongly generic, but now artinian. If we are given a minimal free resolution of this new ideal by a planar map, then deleting all edges and regions incident to one or more of \( \left\{ {{x}^{a}... | No |
Proposition 3.14 Suppose \( I \) is a monomial ideal in \( \mathbb{k}\left\lbrack {x, y, z}\right\rbrack \) and \( {I}_{\epsilon } \) is a strong deformation resolved by a planar map \( {G}_{\epsilon } \) . Specializing the vertices (hence also the edges and regions) of \( {G}_{\epsilon } \) yields a planar map resolut... | Proof. Consider the minimal free resolution \( {\mathcal{F}}_{{G}_{\epsilon }} \) determined by the triangulation \( {G}_{\epsilon } \) as in (3.2). The specialization \( G \) of the labeled planar map \( {G}_{\epsilon } \) still gives a complex \( {\mathcal{F}}_{G} \) of free modules over \( \mathbb{k}\left\lbrack {x,... | Yes |
Corollary 3.15 An ideal \( I \) generated by \( r \geq 3 \) monomials in \( \mathbb{k}\left\lbrack {x, y, z}\right\rbrack \) has at most \( {3r} - 6 \) minimal first syzygies and \( {2r} - 5 \) minimal second syzygies. These Betti number bounds are attained if \( I \) is artinian, strongly generic, and \( {xyz} \) divi... | Proof. Choose a strong deformation \( {I}_{\epsilon } \) of \( I \) that is strongly generic. Proposition 3.14 implies that \( I \) has Betti numbers no larger than those of \( {I}_{\epsilon } \), so we need only prove the first sentence of the theorem for \( {I}_{\epsilon } \) . Theorem 3.11 implies that \( {I}_{\epsi... | Yes |
Theorem 3.17 Every monomial ideal \( I \) in \( \mathbb{k}\left\lbrack {x, y, z}\right\rbrack \) has a minimal free resolution by some planar map. If \( I \) is artinian then the graph \( G \) underlying any such planar map is almost 3-connected. | Proof of correctness. If \( I \) is generic then the algorithm terminates immediately and correctly by Theorem 3.11. By induction on the number of passes through the while-do loop, assume that \( {I}_{\epsilon } \) at the beginning of the loop is minimally resolved by the regions, edges, and vertices of \( G \) . Once ... | No |
Example 3.19 If \( I = \left\langle {{x}^{2},{xy},{xz},{y}^{2},{yz},{z}^{2}}\right\rangle \) is the square of the maximal ideal \( \langle x, y, z\rangle \), then \( {I}_{\epsilon } = \left\langle {{x}^{2}, x{y}^{1.1},{x}^{1.1}z,{y}^{2}, y{z}^{1.1},{x}^{2}}\right\rangle \) is a strongly generic deformation satisfying t... | Proof of Theorem 3.17. The argument beginning the proof of Theorem 3.11 also works here, reducing everything to the artinian case. Algorithm 3.18 produces a minimal planar map resolution. What remains is to show that the underlying graph \( G \) is almost 3-connected. It is enough to produce three independent paths, on... | No |
Proposition 4.5 The cellular free complex \( {\mathcal{F}}_{X} \) supported on \( X \) is a cellular resolution if and only if \( {X}_{ \preccurlyeq \mathbf{b}} \) is acyclic over \( \mathbb{k} \) for all \( \mathbf{b} \in {\mathbb{N}}^{n} \) . When \( {\mathcal{F}}_{X} \) is acyclic, it is a free resolution of \( S/I ... | Proof. The free modules contributing to the part of \( {\mathcal{F}}_{X} \) in degree \( \mathbf{b} \in {\mathbb{N}}^{n} \) are precisely those generated in degrees \( \preccurlyeq \mathbf{b} \) . This proves the criterion for acyclicity, noting that if this degree \( \mathbf{b} \) complex is acyclic, then its homology... | Yes |
Theorem 4.7 If \( {\mathcal{F}}_{X} \) is a cellular resolution of the monomial quotient \( S/I \) , then the Betti numbers of \( I \) can be calculated for \( i \geq 1 \) as\n\n\[ \n{\beta }_{i,\mathbf{b}}\left( I\right) = {\dim }_{\mathbb{k}}{\widetilde{H}}_{i - 1}\left( {{X}_{ \prec \mathbf{b}};\mathbb{k}}\right) .\... | Proof. When \( {\mathbf{x}}^{\mathbf{b}} \) does not lie in \( I \), the complex \( {X}_{ \prec \mathbf{b}} \) consists at most of the empty face \( \varnothing \in X \), which has no homology in homological degrees \( \geq 0 \) . This is good, because \( {\beta }_{i,\mathbf{b}}\left( I\right) \) is zero unless \( {\ma... | Yes |
Example 4.8 Consider the ideal \( I = \left\langle {{x}_{1}{x}_{2},{x}_{1}{x}_{3},{x}_{1}{x}_{4},{x}_{2}{x}_{3},{x}_{2}{x}_{4},{x}_{3}{x}_{4}}\right\rangle \) , and let \( X \) be the boundary complex of the (solid) octahedron. Label the six vertices of \( X \) with the six generators of \( I \) so that opposite vertic... | \[ 0 \leftarrow {S}^{1} \leftarrow {S}^{6} \leftarrow {S}^{12} \leftarrow {S}^{8} \leftarrow {S}^{1} \leftarrow 0. \] | Yes |
Theorem 4.11 If a labeled cell complex \( X \) supports a cellular free resolution of a monomial quotient \( S/I \), then the \( K \) -polynomial of \( S/I \) equals the \( {\mathbb{N}}^{n} \) -graded Euler characteristic of \( X \) :\n\n\[ \mathcal{K}\left( {S/I;{x}_{1},\ldots ,{x}_{n}}\right) = \chi \left( {X;{x}_{1}... | Proof. Dividing \( \chi \left( {X;\mathbf{x}}\right) \) by \( \left( {1 - {x}_{1}}\right) \cdots \left( {1 - {x}_{n}}\right) \) yields an alternating sum of power series that we wish to show is the Hilbert series of \( S/I \) . However, the number of times a monomial \( {\mathbf{x}}^{\mathbf{b}} \) appears in this alte... | No |
Example 4.12 When \( \mathcal{P} \) is two-dimensional, so \( \mathcal{P} \) is a polygon, these resolutions follow the pattern\n\n\n\nThe number of variables equals the number of facets of \( \mathcal{P} \) . | The ideal \( {I}_{\mathcal{P}} \) plays an important role in the study of toric varieties (cf. Chapter 10). Briefly, each smooth (or just simplicial) projective toric variety is specified by a simple polytope \( \mathcal{P} \) called its moment polytope. The facets of \( \mathcal{P} \) correspond to the torus invariant... | No |
Lemma 4.13 The set \( {\mathcal{P}}_{t} \) is a polyhedron in \( {\mathbb{R}}^{n} \) . More precisely, we have\n\n\[{\mathcal{P}}_{t} = {\mathbb{R}}_{ \geq 0}^{n} + \operatorname{conv}\left\{ {{t}^{\mathbf{a}} \mid {\mathbf{x}}^{\mathbf{a}} \in \min \left( I\right) }\right\}\] | Proof. First we prove the inclusion \( \subseteq \) . Let \( {\mathbf{x}}^{\mathbf{b}} \) be any monomial in \( I \) . Then there is a minimal generator \( {\mathbf{x}}^{\mathbf{a}} \in \min \left( I\right) \) dividing \( {\mathbf{x}}^{\mathbf{b}} \) . This implies \( {t}^{{a}_{i}} \leq {t}^{{b}_{i}} \) for all \( i \)... | Yes |
Proposition 4.14 The face poset (i.e., the set of faces partially ordered by inclusion) of the polyhedron \( {\mathcal{P}}_{t} \) is independent of \( t \in \mathbb{R} \) for \( t > \left( {n + 1}\right) ! \) . The same holds for the subposet consisting of all bounded faces of \( {\mathcal{P}}_{t} \) . | Proof. The face poset of \( {\mathcal{P}}_{t} \) can be computed as follows. Let \( {C}_{t} \subset {\mathbb{R}}^{n + 1} \) be the cone spanned by the vectors \( \left( {{t}^{\mathbf{a}},1}\right) \) for all minimal generators \( {\mathbf{x}}^{\mathbf{a}} \) of \( I \) together with the unit vectors \( \left( {{\mathbf... | Yes |
Lemma 4.15 Let \( {a}_{ij} \) be integers for \( 1 \leq i, j \leq r \) . Then the Laurent polynomial \( f\left( t\right) = \det \left( {\left\lbrack {t}^{{a}_{ij}}\right\rbrack }_{1 \leq i, j \leq r}\right) \) either vanishes identically or has no real roots for \( t > r \) !. | Proof. Suppose that \( f \) is not the zero polynomial and write \( f\left( t\right) = {c}_{\alpha }{t}^{\alpha } + \) \( \mathop{\sum }\limits_{\beta }{c}_{\beta }{t}^{\beta } \), where the first term has the highest degree in \( t \) . For \( t > r \) ! we have the chain of inequalities\n\n\[ \left| {\mathop{\sum }\l... | Yes |
Lemma 4.18 Let \( F \) be a face of a polytope \( \mathcal{Q} \) . If \( K \) is the subcomplex of \( \partial \mathcal{Q} \) consisting of all faces of \( \mathcal{Q} \) that are disjoint from \( F \), then \( K \) is contractible. | Proof. Consider the barycentric subdivision \( B\left( {\partial \mathcal{Q}}\right) \) of the boundary of \( \mathcal{Q} \) . This is a triangulation of \( \partial \mathcal{Q} \) whose simplices are in bijection with chains (flags) of faces of \( \mathcal{Q} \) . A geometric realization of \( B\left( {\partial \mathc... | Yes |
Corollary 4.20 Every monomial quotient \( S/I \) has a \( K \) -polynomial; it is given by the \( {\mathbb{N}}^{n} \) -graded Euler characteristic \( \chi \left( {\operatorname{hull}\left( I\right) ;\mathbf{x}}\right) \) of its hull complex. | Of course, we could have stated the same corollary using the full Taylor resolution (Section 4.3.2) instead of the hull complex; but the hull complex has length at most \( n \) (the number of variables), and it has far fewer free summands. All \( {\mathbb{N}}^{n} \) -graded degrees of cells in hull \( \left( I\right) \... | No |
The staircase diagram of \( I = \langle x, y, z{\rangle }^{5} \) is at left below. For the hull resolution, consider the convex hull of \( \left\{ {\left( {{t}^{i},{t}^{j},{t}^{k}}\right) \mid i + j + k = 5}\right\} \) , for \( t > 1 \), and look at this convex polyhedron from the point \( \left( {1,1,1}\right) \) : ![... | The hull resolution of \( \langle x, y, z{\rangle }^{5} \), depicted at right above, respects the \( {S}_{3} \) - symmetry. In general, the hull resolution of \( \langle x, y, z{\rangle }^{d} \) has two classes of second syzygies: the \ | No |
Example 4.23 Consider the ideal \( I = \left\langle {{x}^{2}z,{xyz},{y}^{2}z,{x}^{3}{y}^{5},{x}^{4}{y}^{4},{x}^{5}{y}^{3}}\right\rangle \) . In Chapter 3 we saw that the Buchberger graph of \( I \) contains the nonplanar graph \( {K}_{3,3} \) . The minimal free resolution of \( S/I \) has the format | \[ 0 \leftarrow {S}^{1} \leftarrow {S}^{6} \leftarrow {S}^{7} \leftarrow {S}^{2} \leftarrow 0, \] where \( S = \mathbb{k}\left\lbrack {x, y, z}\right\rbrack \) . The computer algebra system Macaulay 2 [GS04] produces the minimal cellular resolution on the left:  in Section 4.3.5 is not the hull resolution. It is a good exercise to check this directly by writing down the hull complex explicitly, but there is a much easier reason: the hull complex is independent of characteristic, whereas the minimal ... | No |
Lemma 4.27 The polytope \( {\mathcal{Q}}_{t} \) is \( n \) -dimensional, and it has the simplex \( \Delta \left( J\right) \) as a facet. | Proof. The \( n \) points \( {v}_{i} = \left( {1,\ldots ,1,{t}^{{d}_{i}},1,\ldots ,1}\right) \) and the additional point \( \mathbf{1} = \left( {1,\ldots ,1}\right) \) are affinely independent. Their convex hull is the translate by 1 of an \( n \) -simplex given by the origin and \( n \) points on the \( n \) positive ... | Yes |
Lemma 4.28 Every bounded face of \( {\mathcal{P}}_{t} \) is a face of \( {\mathcal{Q}}_{t} \) . | Proof. If \( F \) is a face of \( {\mathcal{P}}_{t} \) then \( F \cap {\mathcal{Q}}_{t} \) is a face of \( {\mathcal{Q}}_{t} \) because \( {\mathcal{Q}}_{t} \subseteq {\mathcal{P}}_{t} \) . Suppose that \( F \) is bounded. Then \( F \) is the convex hull of a subset of the vertices of \( {\mathcal{P}}_{t} \) . But sinc... | Yes |
Lemma 4.29 A face \( F \) of \( {\mathcal{Q}}_{t} \) is a face of \( {\mathcal{P}}_{t} \) if and only if \( F \) has a strictly positive inner normal vector (all coordinates positive). The collection of such faces \( F \) is the hull complex hull \( \left( J\right) \) . | Proof. Suppose \( F \) is the face of \( {\mathcal{Q}}_{t} \) at which a strictly positive vector \( w \) attains its minimum. Then the face of \( {\mathcal{P}}_{t} \) at which \( w \) attains its minimum is bounded, and it contains \( F \), so by the previous lemma it must equal \( F \) . Hence \( F \in \operatorname{... | Yes |
Theorem 4.31 The hull complex \( \operatorname{hull}\left( J\right) \) of an artinian monomial ideal \( J \) in \( n \) variables is a polyhedral subdivision of the \( \left( {n - 1}\right) \) -simplex \( \Delta \left( J\right) \) . A face \( G \) lies in the boundary of \( \operatorname{hull}\left( J\right) \) if and ... | Proof. Pick any point \( p \) in the simplex \( \Delta \left( J\right) \) and imagine walking from \( p \) toward the point 1 along a straight line segment \( \ell \) . Since \( {\mathcal{Q}}_{t} \) is a closed subset of \( {\mathbb{R}}^{n} \), there is a unique last point \( \ell \left( p\right) \) along \( \ell \) th... | Yes |
Example 4.33 Consider the following subideal of the one in Example 4.22:\n\n\[ J = \left\langle {{x}^{5},{y}^{5},{z}^{5},{x}^{3}{y}^{2},{x}^{2}{y}^{3},{x}^{3}{yz},{x}^{2}y{z}^{2}, x{y}^{3}z, x{y}^{2}{z}^{2}}\right\rangle . \] | The three-dimensional polytope \( {\mathcal{Q}}_{t} \) has two distinguished facets, namely the triangle \( \Delta \left( J\right) \) with vertex set \( \left\{ {\left( {{t}^{5},1,1}\right) ,\left( {1,{t}^{5},1}\right) ,\left( {1,1,{t}^{5}}\right) }\right\} \) and the hexagon with vertex set \( \left\{ {\left( {{t}^{3}... | Yes |
Proposition 5.1 If \( I \) is a squarefree monomial ideal, then \( {\left( {I}^{ \star }\right) }^{ \star } = I \) . Equivalently, \( {\left( {\Delta }^{ \star }\right) }^{ \star } = \Delta \) for any simplicial complex \( \Delta \) . | Proof. View Alexander duality as poset duality in the Boolean lattice \( {2}^{\left\lbrack n\right\rbrack } \) of subsets of \( \left\lbrack n\right\rbrack \mathrel{\text{:=}} \{ 1,\ldots, n\} \), as follows. Proposition 1.37 says that removing \( \Delta \) from \( {2}^{\left\lbrack n\right\rbrack } \) leaves a poset i... | Yes |
Fix a simple polytope \( \mathcal{P} \) with \( n \) facets \( {F}_{1},\ldots ,{F}_{n} \) and \( r \) vertices \( {v}_{1},\ldots ,{v}_{r} \) . If \( \Delta \) is the boundary of the simplicial \( d \) -polytope polar to \( \mathcal{P} \), so that the \( n \) vertices of \( \Delta \) are in bijection with the \( n \) fa... | For example, let \( \Delta \) be the octahedron\n\n\[ \n{I}_{\Delta } = \left\langle {{x}_{0},{y}_{0},{z}_{0}}\right\rangle \cap \left\langle {{x}_{0},{y}_{0},{z}_{1}}\right\rangle \cap \left\langle {{x}_{0},{y}_{1},{z}_{0}}\right\rangle \n\]\n\n\[ \n\cap \langle {x}_{0},{y}_{1},{z}_{1}\rangle \cap \langle {x}_{1},{y}_... | Yes |
Proposition 5.5 The coKoszul complex \( {\mathbb{K}}^{ \bullet } \) minimally resolves \( \mathbb{k} = S/\mathfrak{m} \) . | Proof. Suppose \( \mathbf{b} \in {\mathbb{N}}^{n} \) has support \( \sigma \) . The degree \( \mathbf{b} \) part \( {\left( {\mathbb{K}}^{ \bullet }\right) }_{\mathbf{b}} \) of the complex \( {\mathbb{K}}^{ \bullet } \) comes from those rows and columns labeled by faces \( \tau \subseteq \sigma \) . These rows and colu... | Yes |
Theorem 5.6 (Alexander duality) \( {\widetilde{H}}_{i - 1}\left( {{\Delta }^{ \star };\mathbb{k}}\right) \cong {\widetilde{H}}^{n - 2 - i}\left( {\Delta ;\mathbb{k}}\right) \) . | Proof. We have already calculated the left-hand side to be \( {\operatorname{Tor}}_{i + 1}^{S}{\left( \mathbb{k}, S/{I}_{\Delta }\right) }_{\mathbf{1}} \) for \( \mathbf{1} = \left( {1,\ldots ,1}\right) \) in the proof of Theorem 1.34, by tensoring the Koszul complex \( \mathbb{K} \) . with \( {I}_{\Delta } \) and taki... | Yes |
Theorem 5.11 Given a vector \( \mathbf{b} \in {\mathbb{N}}^{n} \) with support \( \sigma = \left\{ {i \mid {b}_{i} \neq 0}\right\} \), the Betti numbers of \( I \) and \( S/I \) in degree \( \mathbf{b} \) can be expressed as\n\n\[ \n{\beta }_{i - 1,\mathbf{b}}\left( I\right) = {\beta }_{i,\mathbf{b}}\left( {S/I}\right)... | Proof. Apply Theorem 5.6 to Theorem 1.34, using Lemma 5.10. | No |
Corollary 5.12 (Hochster’s formula) The nonzero Betti numbers of \( {I}_{\Delta } \) and \( S/{I}_{\Delta } \) lie only in squarefree degrees \( \sigma \), and we have\n\n\[ \n{\beta }_{i - 1,\sigma }\left( {I}_{\Delta }\right) = {\beta }_{i,\sigma }\left( {S/{I}_{\Delta }}\right) = {\dim }_{\mathbb{k}}{\widetilde{H}}^... | Proof. The nonzero Betti numbers lie in squarefree degrees by Corollary 1.40. Hence the result is obtained by applying Theorem 5.6 to Theorem 5.11, once we show that \( {K}_{\sigma }\left( {I}_{\Delta }\right) \) is the restriction \( {\left. \Delta \right| }_{\sigma } \) . This follows directly from the definitions of... | Yes |
Theorem 5.14 (Alexander inversion formula) If \( \Delta \) is any simplicial complex, then the \( K \) -polynomial of its Stanley-Reisner ring satisfies\n\n\[ \mathcal{K}\left( {S/{I}_{\Delta };\mathbf{x}}\right) = \mathcal{K}\left( {{I}_{{\Delta }^{ \star }};\mathbf{1} - \mathbf{x}}\right) \] | Proof. By Proposition 1.37, the Hilbert series of \( {I}_{{\Delta }^{ \star }} \) is the sum of all monomials \( {\mathbf{x}}^{\mathbf{b}} \) divisible by \( \mathop{\prod }\limits_{{j \notin \sigma }}{x}_{j} \) for some face \( \sigma \in \Delta \) :\n\n\[ H\left( {{I}_{{\Delta }^{ \star }};\mathbf{x}}\right) = \matho... | Yes |
Example 5.15 Let \( \Delta \) be the simplicial complex in Example 1.5, which is Alexander dual to \( \Gamma \) in Example 1.14. Starting with the \( K \) -polynomial \( 1 - {abcd} - {abe} - {ace} - {de} + {abce} + {abde} + {acde} \) in Example 1.14, we calculate to be the \( K \) -polynomial of the Stanley-Reisner ide... | \[ 1 - \left( {1 - a}\right) \left( {1 - b}\right) \left( {1 - c}\right) \left( {1 - d}\right) - \left( {1 - a}\right) \left( {1 - b}\right) \left( {1 - e}\right) - \left( {1 - a}\right) \left( {1 - c}\right) \left( {1 - e}\right) - \left( {1 - d}\right) \left( {1 - e}\right) \] \[ + \left( {1 - a}\right) \left( {1 - b... | Yes |
Lemma 5.18 Every monomial ideal has an irreducible decomposition. | Proof. If \( m \) is a minimal generator of \( I \) and \( m = {m}^{\prime }{m}^{\prime \prime } \) is a product of relatively prime monomials \( {m}^{\prime } \) and \( {m}^{\prime \prime } \), then \( I = \left( {I + \left\langle {m}^{\prime }\right\rangle }\right) \cap \left( {I + \left\langle {m}^{\prime \prime }\r... | Yes |
Example 5.21 Let \( \mathbf{a} = \left( {4,4,4}\right) \) . Then | \[ \begin{aligned} I & = \left\langle {{x}^{3},{xy}, y{z}^{2}}\right\rangle & \Rightarrow {I}^{\left\lbrack \mathbf{a}\right\rbrack } & = \left\langle {x}^{2}\right\rangle \cap \left\langle {{x}^{4},{y}^{4}}\right\rangle \cap \left\langle {{y}^{4},{z}^{3}}\right\rangle \\ & = \left\langle {{x}^{3}, y}\right\rangle \cap... | No |
Example 5.22 Let \( n = 3 \), so that \( S = \mathbb{k}\left\lbrack {x, y, z}\right\rbrack \) . Fig. 5.1 lists the minimal generators and irreducible components of an ideal \( I \subseteq S \) and its dual \( {I}^{\left\lbrack {455}\right\rbrack } \) with respect to \( \mathbf{a} = \left( {4,5,5}\right) \) . The (trunc... | Alexander duality in three dimensions comes down to the familiar optical illusion in which isometrically rendered cubes appear alternately to point \ | No |
Proposition 5.23 Suppose that all minimal generators of the ideal I divide \( {\mathbf{x}}^{\mathbf{a}} \) . If \( \mathbf{b} \preccurlyeq \mathbf{a} \), then \( {\mathbf{x}}^{\mathbf{b}} \) lies outside \( I \) if and only if \( {\mathbf{x}}^{\mathbf{a} - \mathbf{b}} \) lies inside \( {I}^{\left\lbrack \mathbf{a}\righ... | Proof. Suppose \( I = \left\langle {{\mathbf{x}}^{\mathbf{c}} \mid \mathbf{c} \in C}\right\rangle \) . Then \( {\mathbf{x}}^{\mathbf{b}} \notin I \) if and only if we have \( \mathbf{b} \nsucceq \mathbf{c} \) , or equivalently, \( \mathbf{a} - \mathbf{b} \npreceq \mathbf{a} - \mathbf{c} \), for all \( \mathbf{c} \in C ... | Yes |
Theorem 5.24 If all minimal generators of \( I \) divide \( {\mathbf{x}}^{\mathbf{a}} \), then all minimal generators of \( {I}^{\left\lbrack \mathbf{a}\right\rbrack } \) divide \( {\mathbf{x}}^{\mathbf{a}} \), and \( {\left( {I}^{\left\lbrack \mathbf{a}\right\rbrack }\right) }^{\left\lbrack \mathbf{a}\right\rbrack } =... | Proof. Suppose \( I = \left\langle {{\mathbf{x}}^{{\mathbf{b}}_{1}},\ldots ,{\mathbf{x}}^{{\mathbf{b}}_{r}}}\right\rangle \) . The powers of variables generating the irreducible components of \( {I}^{\left\lbrack \mathbf{a}\right\rbrack } \) all divide \( {\mathbf{x}}^{\mathbf{a}} \) by definition. Since every minimal ... | Yes |
Corollary 5.25 If all generators of \( I \) divide \( {\mathbf{x}}^{\mathbf{a}} \), then \( {I}^{\left\lbrack \mathbf{a}\right\rbrack } \) is the unique ideal with generators dividing \( {\mathbf{x}}^{\mathbf{a}} \) that satisfies \( \left( {{\mathfrak{m}}^{\mathbf{a} + \mathbf{1}} : I}\right) = {I}^{\left\lbrack \math... | Proof. Observe that \( {\mathbf{x}}^{\mathbf{b}} \notin I \) if and only if all monomials dividing \( {\mathbf{x}}^{\mathbf{b}} \) lie outside of \( I \) . If \( \mathbf{b} \preccurlyeq \mathbf{a} \), then this occurs precisely when all monomials dividing \( {\mathbf{x}}^{\mathbf{a}} \) lie outside of \( {\mathbf{x}}^{... | Yes |
Lemma 5.26 Suppose that \( \mathbf{b} \preccurlyeq \mathbf{a} \) and \( \mathbf{c} \preccurlyeq \mathbf{a} \) in \( {\mathbb{N}}^{n} \) . Then \( {\mathbf{x}}^{\mathbf{a} \smallsetminus \mathbf{b}} \) divides \( {\mathfrak{x}}^{\mathbf{a} \smallsetminus \mathbf{c}} \) if and only if \( {\mathfrak{m}}^{\mathbf{b}} \subs... | Proof. We have \( {\mathfrak{m}}^{\mathbf{b}} \subseteq {\mathfrak{m}}^{\mathbf{c}} \) if and only if \( {b}_{i} \geq {c}_{i} \) whenever \( {c}_{i} \geq 1 \) and also \( {b}_{i} = 0 \) whenever \( {c}_{i} = 0 \) . This occurs if and only if \( {a}_{i} - {b}_{i} \leq {a}_{i} - {c}_{i} \) whenever \( {c}_{i} \geq 1 \) a... | Yes |
Theorem 5.27 Assume that all minimal generators of \( I \) divide \( {\mathbf{x}}^{\mathbf{a}} \) . Then I has a unique irredundant irreducible decomposition, and it is given by\n\n\[ I = \bigcap \left\{ {{\mathfrak{m}}^{\mathbf{a} \smallsetminus \mathbf{b}} \mid {\mathbf{x}}^{\mathbf{b}}\text{ is a minimal generator o... | Proof. The given intersection is equal to \( I \) by Theorem 5.24. It is irredun-dant by Lemma 5.26 because the intersection is taken over minimal generators of \( {I}^{\left\lbrack \mathbf{a}\right\rbrack } \) . Now suppose that we are given any irredundant irreducible decomposition \( I = \mathop{\bigcap }\limits_{{\... | Yes |
Lemma 5.35 Fix an ideal I generated in degrees preceding a. If \( {\mathcal{F}}^{Y} \) is a cocellular resolution of \( {I}_{ \preccurlyeq \mathbf{a}} \), and \( {Y}_{ \preccurlyeq \mathbf{a}} \) is the set of faces of \( Y \) whose labels precede \( \mathbf{a} \), then \( {\mathcal{F}}^{{Y}_{ \preccurlyeq \mathbf{a}}}... | Proof. The faces \( G \) contributing a nonzero monomial to degree \( \mathbf{b} \) of \( {\mathcal{F}}^{Y \preccurlyeq \mathbf{a}} \) are precisely those faces \( G \in Y \) whose labels \( {\mathbf{a}}_{G} \) precede \( \mathbf{a} \). Therefore the complex of \( \mathbb{k} \) -vector spaces in degree \( \mathbf{b} \)... | Yes |
Lemma 5.36 (Nerve lemma) If \( \mathcal{U} \) is an acyclic cover of a polyhedral cell complex \( Y \) by polyhedral subcomplexes, then \( {\widetilde{H}}^{i}\left( {Y;\mathbb{k}}\right) \cong {\widetilde{H}}^{i}\left( {\mathcal{N}\left( \mathcal{U}\right) ;\mathbb{k}}\right) \) . | Proof. By barycentrically subdividing every face of \( Y \), we may assume that \( Y \) and all of the subcomplexes in \( \mathcal{U} \) are simplicial. Now the result is [Rot88, Theorem 7.26], but for cohomology instead of homology. (The argument in [Rot88] works just as well for cohomology; alternatively, use that we... | Yes |
Theorem 5.37 Fix a monomial ideal \( I \) generated in degrees preceding a and a length \( n \) cellular resolution \( {\mathcal{F}}_{X} \) of \( S/\left( {I + {\mathfrak{m}}^{\mathbf{a} + \mathbf{1}}}\right) \) such that all face labels on \( X \) precede \( \mathbf{a} + \mathbf{1} \) . If \( Y = \mathbf{a} + \mathbf{... | Proof. By Lemma 5.35, it is enough to show that \( {\mathcal{F}}^{\mathbf{a} + \mathbf{1} - X} \) is a weakly cocellular resolution of \( {\left( {I}^{\left\lbrack \mathbf{a}\right\rbrack }\right) }_{ \preccurlyeq \mathbf{a}} \) . The faces of \( \underline{X} \) contributing monomials to degree \( \mathbf{b} \) in \( ... | Yes |
Corollary 5.39 If the labeled cell complex \( X \) supports a minimal resolution of an artinian monomial quotient of \( S \), then \( X \) is pure of dimension \( n - 1 \) . | Proof. \( {\mathcal{F}}_{X} \) resolves \( S/I \) for an ideal \( I \) containing \( {\mathfrak{m}}^{\mathbf{a} + \mathbf{1}} \) for some \( \mathbf{a} \) . If \( G \) is a facet of \( X \), then the differential of \( {\mathcal{F}}^{\mathbf{a} + \mathbf{1} - X} \) is zero on \( G \) . Minimality of \( {\mathcal{F}}^{\... | Yes |
Proposition 5.40 Suppose \( G \) is a face of a labeled cell complex \( X \) supporting a minimal cellular resolution of an artinian quotient of \( S \) . If the \( {i}^{\text{th }} \) coordinate \( {\left( {\mathbf{a}}_{G}\right) }_{i} \) of the face label \( {\mathbf{a}}_{G} \) is nonzero, then \( {\left( {\mathbf{a}... | Proof. Let \( {X}_{G} \) be the subcomplex of \( X \) on faces whose labels have support contained in \( \operatorname{supp}\left( {\mathbf{a}}_{G}\right) \) . Since \( {X}_{G} \) equals \( {X}_{ \preccurlyeq d \cdot \operatorname{supp}\left( {\mathbf{a}}_{G}\right) } \) for \( d \gg 0 \), it supports a minimal cellula... | Yes |
Theorem 5.42 Fix a monomial ideal I generated in degrees preceding a, and let \( {\mathcal{F}}_{X} \) be a minimal cellular resolution of \( S/\left( {I + {\mathfrak{m}}^{\mathbf{a} + \mathbf{1}}}\right) \) . Writing \( \widehat{\mathbf{b}} = \mathop{\sum }\limits_{{{b}_{i} \leq {a}_{i}}}{b}_{i}{\mathbf{e}}_{i} \) for ... | Proof. Corollary 5.39 says that every facet \( G \) has dimension \( n - 1 \), so Proposition 5.40 implies that \( {\mathbf{a}}_{G} \) has full support. Therefore we find that \( \mathbf{a} + \mathbf{1} - {\mathbf{a}}_{G} \preccurlyeq \mathbf{a} \) for all facets \( G \in X \) . But then \( {\mathbf{x}}^{\mathbf{a} + \... | Yes |
Example 5.44 The \( n = 3 \) example \( {I}^{ \star } \) in Section 4.3.4 is Alexander dual to the ideal \( I \) in Section 4.3.3 with respect to \( \mathbf{a} = \left( {3,3,3}\right) \) . | It so happens that the hull resolution of \( \mathbb{k}\left\lbrack {x, y, z}\right\rbrack /\left( {I + {\mathfrak{m}}^{\left( 4,4,4\right) }}\right) \) is minimal; see the middle of Fig. 5.3. Therefore Theorem 5.37 produces a minimal cocellular resolution of \( {I}^{ \star } \), supported on the interior faces of the ... | No |
Corollary 5.46 \( {\mathcal{F}}^{{\operatorname{cohull}}_{\mathbf{a}}\left( I\right) } \) is a weakly cocellular free resolution of \( I \) . | Proof. The complex \( {\mathcal{F}}^{{\operatorname{cohull}}_{\mathbf{a}}\left( I\right) } \) is Alexander dual to the hull resolution of \( S/\left( {{I}^{\left\lbrack \mathbf{a}\right\rbrack } + {\mathfrak{m}}^{\mathbf{a} + \mathbf{1}}}\right) \), which satisfies the hypotheses of Theorem 5.37. | Yes |
Not all cellular resolutions come directly from hull and cohull resolutions. All resolutions in this example can be construed as being cellular, supported on labeled cell complexes depicted in Fig. 5.4. Set \( I = \left\langle {{z}^{2},{x}^{3}z,{x}^{4},{y}^{3},{y}^{2}z,{xyz}}\right\rangle \) so that \( {I}^{\left\lbrac... | In fact, this cocellular resolution is cellular, supported on the labeled cell complex \( Y \) . | No |
Theorem 5.48 (Duality for Betti numbers) If \( I \) is generated in degrees preceding \( \mathbf{a} \) and \( \mathbf{1} \preccurlyeq \mathbf{b} \preccurlyeq \mathbf{a} \), then \( {\beta }_{n - i,\mathbf{b}}\left( {S/I}\right) = {\beta }_{i,\mathbf{a} + \mathbf{1} - \mathbf{b}}\left( {I}^{\left\lbrack \mathbf{a}\right... | Proof. Let \( X = \operatorname{hull}\left( {I + {\mathfrak{m}}^{\mathbf{a} + \mathbf{1}}}\right) \) and \( Y = {\operatorname{cohull}}_{\mathbf{a}}\left( {I}^{\left\lbrack \mathbf{a}\right\rbrack }\right) \) . By Theorem 4.7 applied to \( X \), we get the equality \( {\beta }_{i,\mathbf{b}}\left( {S/I}\right) = {\beta... | Yes |
The following table lists some instances where the Betti numbers are 1 for the permutohedron and tree ideals \( I \) and \( {I}^{ \star } = {I}^{\left\lbrack {333}\right\rbrack } \) of Sections 4.3.3 and 4.3.4: | \[ \begin{matrix} 3 - i & \mathbf{b} & i & \mathbf{a} + \mathbf{1} - \mathbf{b} \\ 0 & \left( {1,2,3}\right) & 2 & \left( {3,2,1}\right) \\ 1 & \left( {1,3,3}\right) & 1 & \left( {3,1,1}\right) \\ 2 & \left( {3,3,3}\right) & 0 & \left( {1,1,1}\right) \end{matrix} \] \[ {\beta }_{3 - i,\mathbf{b}}\left( I\right) = {\bet... | Yes |
Theorem 5.56 (Eagon-Reiner Theorem) \( S/{I}_{\Delta } \) is Cohen-Macaulay if and only if \( {I}_{\Delta }^{ \star } \) has linear free resolution. | Proof. Suppose that the ideal \( {I}_{\Delta }^{ \star } \) is generated in degree \( d \) . Then \( {I}_{\Delta }^{ \star } \) has linear free resolution if and only if \( {\beta }_{i,\sigma }\left( {I}_{\Delta }^{ \star }\right) \) is zero whenever \( \left| \sigma \right| \neq d + i \) . The dual version of Hochster... | Yes |
The face ideal of a simplicial sphere \( \Delta \) is Cohen-Macaulay. In particular, if \( \Delta \) is the boundary of a simplicial polytope as in Example 5.3, then \( {I}_{\Delta } \) is Cohen-Macaulay. | By Theorem 5.56, \( {I}_{\Delta }^{ \star } \) has a linear resolution. Of course, we already know from Section 4.3 (and Exercise 4.5) that this linear resolution is cellular, supported on the polar polytope \( \mathcal{P} \). See Example 5.3 for an illustration of this linear resolution. | No |
Example 6.2 Taking \( I = \left\langle {{x}^{2},{xy},{y}^{2}z,{z}^{2}}\right\rangle \), let \( \Delta \) be the simplicial complex consisting of the two triples \( \{ 1,2,4\} \) and \( \{ 2,3,4\} \) and their subsets. Here is a picture of \( \Delta \), with each face accompanied by its monomial label. | The Taylor complex \( {\mathcal{F}}_{\Delta } \) is given by the following monomial matrices: \n\nFor an example of the non-monomial matrix way to write this complex, note that the left column in the rightmost map co... | Yes |
Lemma 6.3 The Taylor complex \( {\mathcal{F}}_{\Delta } \) is acyclic if and only if for every monomial \( m \), the simplicial subcomplex \( {\Delta }_{ \preccurlyeq m} = \left\{ {\sigma \in \Delta \mid {m}_{\sigma }}\right. \) divides \( \left. m\right\} \) is acyclic over \( \mathbb{k} \) (homology only in degree 0)... | Proof. This is a special case of Proposition 4.5. | No |
Lemma 6.4 The Taylor complex \( {\mathcal{F}}_{\Delta } \) is minimal if and only if for all faces \( \sigma \in \Delta \) and all indices \( i \in \sigma \), the monomials \( {m}_{\sigma } \) and \( {m}_{\sigma \smallsetminus i} \) are different. | Proof. A complex of \( {\mathbb{N}}^{n} \) -graded free \( S \) -modules is minimal if in its representation by monomial matrices, every nonzero matrix entry has its column label different from its row label. Here, these labels are \( {m}_{\sigma } \) and \( {m}_{\sigma \smallsetminus i} \) . | Yes |
The tree ideal \( {I}^{ \star } \) in Section 4.3.4 is generated by the monomials \( {\omega }_{\sigma } = \mathop{\prod }\limits_{{s \in \sigma }}{x}_{s}^{n - \left| \sigma \right| + 1} \) for the nonempty subsets \( \sigma \subseteq \{ 1,\ldots, n\} \) . If \( \sigma \) and \( {\sigma }^{\prime } \) are distinct subs... | This is an instance of the Scarf complex construction in the next section. | No |
Lemma 6.8 The Scarf complex \( {\Delta }_{I} \) is a simplicial complex. Its dimension is at most \( n - 1 \) . | Proof. If \( \sigma \) is a face of the Scarf complex and \( i \) is an element of \( \sigma \), let \( \tau = \sigma \smallsetminus i \) . Suppose that \( {m}_{\tau } = {m}_{\rho } \) for some index set \( \rho \) . Then \( {m}_{\sigma } = {m}_{\rho \cup i} \) and consequently \( \rho \cup i = \sigma \), because \( \s... | Yes |
In all dimensions, every edge of the Scarf complex of a monomial ideal is an edge of the Buchberger graph. | \[ \operatorname{edges}\left( {\Delta }_{I}\right) \subseteq \operatorname{Buch}\left( I\right) \] | Yes |
Proposition 6.12 If \( I \) is a monomial ideal in \( S \), then every free resolution of \( S/I \) contains the algebraic Scarf complex \( {\mathcal{F}}_{{\Delta }_{I}} \) as a subcomplex. | Proof. Every free resolution contains a minimal free resolution (Exercise 1.11), so it is enough to show that \( {\mathcal{F}}_{{\Delta }_{I}} \) is contained in some minimal free resolution \( \mathcal{F} \) of \( S/I \) . In particular, we may choose \( \mathcal{F} \) to be a subcomplex of the full Taylor resolution,... | No |
Theorem 6.13 If \( I \) is a monomial ideal, then its Scarf complex \( {\Delta }_{I} \) is a subcomplex of the hull complex hull \( \left( I\right) \) . If \( I \) is generic then \( {\Delta }_{I} = \operatorname{hull}\left( I\right) \) , so its algebraic Scarf complex \( {\mathcal{F}}_{{\Delta }_{I}} \) minimally reso... | Proof. Let \( F = \left\{ {{\mathbf{x}}^{{\mathbf{a}}_{1}},\ldots ,{\mathbf{x}}^{{\mathbf{a}}_{p}}}\right\} \) be a face of the Scarf complex \( {\Delta }_{I} \) with \( {m}_{F} = {\mathbf{x}}^{\mathbf{u}} \) . For any index \( i \in \{ 1,\ldots, p\} \), the least common multiple \( {m}_{F \smallsetminus i} \) of \( F ... | Yes |
Lemma 6.14 Let \( I \) be a monomial ideal and \( F \) a face of \( \operatorname{hull}\left( I\right) \) . For each monomial \( m \in I \) there is a variable \( {x}_{j} \) such that \( {\deg }_{{x}_{j}}\left( m\right) \geq {\deg }_{{x}_{j}}\left( {m}_{F}\right) \) . | Proof. Suppose that \( m = {\mathbf{x}}^{\mathbf{u}} \) strictly divides \( {m}_{F} \) in each coordinate. Let \( {t}^{{\mathbf{a}}_{1}},\ldots ,{t}^{{\mathbf{a}}_{p}} \) be the vertices of the face \( F \) and consider their barycenter\n\n\[ \mathbf{v}\left( t\right) = \frac{1}{p} \cdot \left( {{t}^{{\mathbf{a}}_{1}} ... | Yes |
Corollary 6.16 The K-polynomial of \( S/I \) for a generic monomial ideal \( I \) equals the \( {\mathbb{N}}^{n} \) -graded Euler characteristic of the Scarf complex \( {\Delta }_{I} \) :\n\n\[ \mathcal{K}\left( {S/I;{x}_{1},\ldots ,{x}_{n}}\right) = \mathop{\sum }\limits_{{\sigma \in {\Delta }_{I}}}{\left( -1\right) }... | Proof. The Euler characteristic statement follows from Theorems 6.13 and 4.11. There can be no cancellation by definition of \( {\Delta }_{I} \) . | No |
Example 6.18 The generic ideal \( I = \left\langle {{x}^{2}{z}^{2},{xyz},{y}^{2}{z}^{4},{y}^{4}{z}^{3},{x}^{3}{y}^{5},{x}^{4}{y}^{3}}\right\rangle \) has staircase diagram and Scarf complex as follows: | \n\n## Observe that this Scarf complex is not pure, though it is still contractible. \( \diamond \) | Yes |
Corollary 6.19 If \( I = \left\langle {{m}_{1},\ldots ,{m}_{r}}\right\rangle \) is generic and \( S/I \) is artinian, with \( {m}_{i} = {x}_{i}^{{d}_{i}} \) for \( i = 1,\ldots, n \), then the Scarf complex \( {\Delta }_{I} \) is a regular triangulation (usually with additional vertices, some of which may lie on the bo... | Proof. This follows from Theorem 4.31 and Theorem 6.13. | No |
Corollary 6.20 Let \( I \) be a generic monomial ideal, and fix \( \mathbf{u} \in {\mathbb{N}}^{n} \) such that each minimial generator of \( I \) divides \( {\mathbf{x}}^{\mathbf{u}} \) . Set \( {I}^{ * } = I + {\mathfrak{m}}^{\mathbf{u} + \mathbf{1}} \), and for any \( \mathbf{b} \in {\mathbb{N}}^{n} \), abbreviate \... | Proof. Use Theorems 5.42 and 6.13, since \( {I}^{ * } \) is still generic (check this!). | No |
Example 6.21 Let \( I = \left\langle {{x}^{3}{y}^{2}z,{x}^{2}y{z}^{3}, x{y}^{3}{z}^{2}}\right\rangle \) be the ideal \( J \) from Section 3.2, but without any of the artinian generators \( \left\{ {{x}^{4},{y}^{4},{z}^{4}}\right\} \) . Here, we can take \( \mathbf{u} = \left( {3,3,3}\right) \) . The irreducible decompo... | \[ I = \langle z\rangle \cap \langle y\rangle \cap \langle x\rangle \cap \langle {y}^{2},{z}^{3}\rangle \cap \langle {x}^{3},{z}^{2}\rangle \cap \langle {x}^{2},{y}^{3}\rangle \cap \langle {x}^{3},{y}^{3},{z}^{3}\rangle \] | Yes |
Theorem 6.22 Let \( I \) be generic. The quotient \( S/I \) is Cohen-Macaulay if and only if all irreducible components of \( I \) have the same dimension. More generally, the projective dimension of \( S/I \) equals the maximum number of generators of an irreducible component of \( I \) . | Proof. By Theorem 6.13, \( S/I \) has projective dimension equal to the maximum cardinality \( \left| \sigma \right| \) of a facet \( \sigma \in {\Delta }_{I} \) . Suppose every generator of \( I \) divides \( {\mathbf{x}}^{\mathbf{a}} \), and set \( {I}^{ * } = I + {\mathfrak{m}}^{\mathbf{a} + \mathbf{1}} \) . By Coro... | Yes |
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