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Lemma 9.2.8. Let \( G \) be a finite graph on \( \left\lbrack n\right\rbrack \) and \( \Gamma \) a simplicial complex on \( \left\lbrack n\right\rbrack \) such that \( G \) is the 1-skeleton of \( \Gamma \) . Then \( \Gamma = \Delta \left( G\right) \) if and only if \( \Gamma \) is a flag complex. | Proof. Let \( \left( \begin{matrix} \left\lbrack n\right\rbrack \\ 2 \end{matrix}\right) \) denote the set of 2-element subsets of \( \left\lbrack n\right\rbrack \) and \( \mathcal{N}\left( \Gamma \right) \) the set of minimal nonfaces of \( \Gamma \). If \( F \) is a face of \( \Gamma \), then \( F \) is a clique of \... | Yes |
Corollary 9.2.9. A finite graph \( G \) has a perfect elimination ordering if and only if \( G \) is the 1-skeleton of a quasi-forest. | Proof. Since \( G \) is the 1-skeleton of \( \Delta \left( G\right) \), it follows that \( G \) is the 1-skeleton of a quasi-forest if \( \Delta \left( G\right) \) is a quasi-forest. Conversely, if \( G \) is the 1-skeleton of a quasi-forest \( \Gamma \), then by Lemma 9.2.7 and Lemma 9.2.8 one has \( \Gamma = \Delta \... | Yes |
Lemma 9.2.10. A simplicial complex \( \Delta = \left\langle {{F}_{1},\ldots ,{F}_{q}}\right\rangle \) on \( \left\lbrack n\right\rbrack \) is a quasi-forest if and only if the matrix \( {M}_{\Delta } \) contains a Hilbert-Burch matrix for the ideal \( \left( {{x}_{\left\lbrack n\right\rbrack }/{x}_{{F}_{1}},\ldots ,{x}... | Proof. \ | No |
Theorem 9.2.12. Given a finite graph \( G \) on \( \left\lbrack n\right\rbrack \) with \( E\left( G\right) \neq \left( \begin{matrix} \left\lbrack n\right\rbrack \\ 2 \end{matrix}\right) \), the following conditions are equivalent:\n\n(i) \( G \) is chordal;\n\n(ii) \( {I}_{\Delta \left( G\right) } \) has a linear reso... | Proof. First, Theorem 9.2.3 says that \( G \) is chordal if and only if \( {I}_{\Delta \left( G\right) } \) has a linear resolution. Second, the ideal \( {I}_{\Delta \left( G\right) } = I\left( \bar{G}\right) \) is generated by quadratic monomials since \( \Delta \left( G\right) \) is flag. Thus \( {I}_{\Delta \left( G... | Yes |
Lemma 9.3.2. Let \( R \) be a Noetherian ring, \( S = R\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) the polynomial ring over \( R, k \) an integer with \( 0 \leq k < n \), and \( J \) the ideal\n\n\[ \left( {{I}_{1}{x}_{1},\ldots ,{I}_{k}{x}_{k},{\left\{ {x}_{i}{x}_{j}\right\} }_{1 \leq i < j \leq n}}\right) ... | Proof. For a subset \( T \subset \left\lbrack n\right\rbrack \) we let \( {L}_{T} \) be the ideal generated by all monomials \( {x}_{i}{x}_{j} \) with \( i, j \in T \) and \( i < j \), and we set \( {I}_{T} = \mathop{\sum }\limits_{{j \in T}}{I}_{j} \) and \( {X}_{T} = \left( {\left\{ {x}_{j}\right\} }_{j \in T}\right)... | Yes |
Corollary 9.3.3. Let \( G \) be a chordal graph, and let \( {F}_{1},\ldots ,{F}_{m} \) be the facets of \( \Delta \left( G\right) \) which have a free vertex. Let \( {i}_{j} \) be a free vertex of \( {F}_{j} \) for \( j = 1,\ldots, m \) , and let \( {G}^{\prime } \) be the induced subgraph of \( G \) on the vertex set ... | Proof. (a) Let \( F \subset \left\lbrack n\right\rbrack \) and \( S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) . If \( J \) is the ideal generated by the set of monomials \( \left\{ {{x}_{i}{x}_{j} : i, j \in F}\right. \) and \( \left. {i < j}\right\} \) and if \( x = \mathop{\sum }\limits_{{i \in F}}{x}... | Yes |
Proposition 10.1.1. The toric ideal \( {I}_{\mathcal{A}} \) of \( \mathcal{A} \) is spanned by those binomials \( u - v \) of \( R \) with \( \pi \left( u\right) = \pi \left( v\right) \) . In particular \( {I}_{\mathcal{A}} \) is a binomial ideal. | Proof. Let \( f = {c}_{1}{u}_{1} + \cdots + {c}_{r}{u}_{r} \) be a polynomial belonging to \( {I}_{\mathcal{A}} \), where \( {u}_{i} \) is a monomial of \( R \) with \( {c}_{i} \in K \) . Suppose that \( \pi \left( {u}_{1}\right) = \cdots = \pi \left( {u}_{k}\right) \) and \( \pi \left( {u}_{1}\right) \neq \pi \left( {... | Yes |
Proposition 10.1.2. A reduced Gröbner basis of \( {I}_{\mathcal{A}} \) consists of primitive binomials. | Proof. If \( f \) and \( g \) are binomials, then their \( S \) -polynomial \( S\left( {f, g}\right) \) is a binomial. If \( {f}_{1},\ldots ,{f}_{s} \) and \( g \) are binomials, then every remainder of \( g \) with respect to \( {f}_{1},\ldots ,{f}_{s} \) is a binomial. Since \( {I}_{\mathcal{A}} \) is generated by bi... | Yes |
Lemma 10.1.4. If \( f \in {I}_{{\mathcal{A}}_{G}} \) is a primitive binomial, then there is a primitive even closed walk \( W \) of \( G \) with \( f = {f}_{W} \) . | Proof. If the binomial \( {f}_{W} \) arising from an even closed walk \( W \) of \( G \) is primitive, then clearly \( W \) is a primitive even closed walk of \( G \) . Thus what we must prove is that, for every primitive binomial \( f \) of \( {I}_{{\mathcal{A}}_{G}} \), there is an even closed walk \( W \) of \( G \)... | Yes |
Proposition 10.1.6. Suppose that \( I \subset S \) is a graded ideal generated in degree \( d \) . Then\n\n\[ \operatorname{reg}\left( {I}^{k}\right) \leq {kd} + {\operatorname{reg}}_{x}\left( {\mathcal{R}\left( I\right) }\right) \]\n\nIn particular if \( {\operatorname{reg}}_{x}\left( {\mathcal{R}\left( I\right) }\rig... | Proof. The bigraded minimal free \( R \) -resolution \( \mathbb{F} \) of \( \mathcal{R}\left( I\right) \) gives the exact sequence\n\n\[ 0 \rightarrow {\left( {F}_{p}\right) }_{\left( *, k\right) } \rightarrow \cdots \rightarrow {\left( {F}_{1}\right) }_{\left( *, k\right) } \rightarrow {\left( {F}_{0}\right) }_{\left(... | Yes |
Corollary 10.1.7. Let \( I \subset S \) be a graded ideal generated in one degree and \( \mathcal{R}\left( I\right) = R/P \) . Suppose that there exists a monomial order \( < \) on \( R \) such that the defining ideal \( P \) of \( \mathcal{R}\left( I\right) \) has a Gröbner basis \( \mathcal{G} \) whose elements are a... | Proof. The initial ideal \( {\operatorname{in}}_{ < }\left( P\right) \) is generated by monomials \( {u}_{1},\ldots ,{u}_{m} \) with each \( {\deg }_{x}\left( {u}_{j}\right) \leq 1 \) . Let \( \mathbb{T} \) be the Taylor resolution of \( {\operatorname{in}}_{ < }\left( P\right) \) ; see Section 7.1. Recall that the mod... | Yes |
Corollary 10.1.8. Let \( I \subset S \) be a graded ideal generated in one degree and \( \mathcal{R}\left( I\right) = R/P \) . Suppose that there exists a monomial order \( < \) on \( R \) such that the defining ideal \( P \) of \( \mathcal{R}\left( I\right) \) has a Gröbner basis \( \mathcal{G} \) consisting of polyno... | Proof. Let \( \mathcal{G} = \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \) be a Gröbner basis with \( \deg {g}_{i} = 2 \) for all \( i \) . Since the defining ideal of \( \mathcal{R}\left( I\right) \) is bihomogeneous, we may assume that each \( {g}_{i} \) is bihomogeneous. Suppose \( u \in \operatorname{supp}\left( {g}_... | Yes |
Lemma 10.2.1. Let \( I \subset S \) be a squarefree monomial ideal with 2-linear resolution. Then, after suitable renumbering of the variables, one has the following property: if \( {x}_{i}{x}_{j} \in I \) with \( i \neq j, k > i \) and \( k > j \), then either \( {x}_{i}{x}_{k} \) or \( {x}_{j}{x}_{k} \) belongs to \(... | Proof. Let \( G \) be the finite graph on \( \left\lbrack n\right\rbrack \) with \( I = I\left( G\right) \) . Since \( I \) has a linear resolution, the complementary graph \( \bar{G} \) is a chordal graph, see Theorem 9.2.3. Let \( \Delta \) be the quasi-forest on \( \left\lbrack n\right\rbrack \) whose 1-skeleton coi... | Yes |
Lemma 10.2.2. Let \( I \) be a monomial ideal generated in degree 2 and \( J \subset I \) the ideal generated by all squarefree monomials belonging to I. Suppose that I has a linear resolution. Then \( J \) has a linear resolution. | Proof. Let \( \left\{ {{x}_{{i}_{1}}^{2},\ldots ,{x}_{{i}_{k}}^{2}}\right\} = I \cap \left\{ {{x}_{1}^{2},\ldots ,{x}_{n}^{2}}\right\} \) . Then \( I = \left( {{x}_{{i}_{1}}^{2},\ldots ,{x}_{{i}_{k}}^{2}, J}\right) \) . Recall from Subsection 1.6 that the polarization of \( I \) is the squarefree ideal \( {I}^{ * } = \... | Yes |
Lemma 10.2.3. Work with the situation as in the proof of Lemma 10.2.2. Let \( \Delta \) be the quasi-forest whose 1-skeleton coincides with \( \bar{G} \) . Then\n\n(a) the vertex \( {i}_{j} \) is a free vertex of \( \Delta \) for \( j = 1,\ldots, k \) ;\n\n(b) no two of these vertices \( {i}_{1},\ldots ,{i}_{k} \) belo... | Proof. Let \( {\Delta }^{ * } \) be the quasi-forest whose 1-skeleton is \( \overline{{G}^{ * }} \) .\n\n(a) Suppose that \( {i}_{j} \) is not a free vertex of \( \Delta \) . Then there exist edges \( \left\{ {{i}_{j}, r}\right\} \) and \( \left\{ {{i}_{j}, s}\right\} \) of \( \bar{G} \) such that \( \{ r, s\} \) is no... | Yes |
Corollary 10.2.4. Let \( I \) be a monomial ideal of \( S \) generated in degree 2. Suppose that \( I \) has a linear resolution and that \( {x}_{i}^{2} \in I \) . Then with the numbering of the variables as given in Lemma 10.2.1 one has the following property: for all \( j > i \) for which there exists \( k \) such th... | Proof. Suppose \( {x}_{i}^{2} \in I \) and there exists \( j > i \) for which there exists \( k \) such that \( {x}_{k}{x}_{j} \in I \), but neither \( {x}_{i}{x}_{j} \) nor \( {x}_{i}{x}_{k} \) belongs to \( I \) . Since \( {x}_{i}^{2} \in I \), one has \( k \neq i \) .\n\nLet \( k \neq j \) . Then \( \{ k, j\} \) is ... | Yes |
Proposition 10.3.1. Let \( I \subset S \) be a graded ideal. Then \( \operatorname{depth}S/{I}^{k} \) is constant for all \( k \gg 0 \) . | Proof. We will show that depth \( {I}^{k} \) is constant for \( k \gg 0 \) . This will yield the desired conclusion. In order to show this we consider the Koszul homology \( H\left( {\mathbf{x};\mathcal{R}\left( I\right) }\right) \) of the Rees algebra \( \mathcal{R}\left( I\right) \) of \( I \) with respect to \( \mat... | Yes |
Proposition 10.3.2. Let \( I \subset S \) be a nonzero graded ideal. Then\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}\operatorname{depth}S/{I}^{k} \leq n - \ell \left( I\right) \]\n\nEquality holds if \( \mathcal{R}\left( I\right) \) is Cohen-Macaulay. | Proof. Let \( r > 0 \) be any integer. Then \( \mathop{\lim }\limits_{{k \rightarrow \infty }}S/{I}^{kr} = \mathop{\lim }\limits_{{k \rightarrow \infty }}S/{I}^{k} \) and \( \ell \left( {I}^{r}\right) = \ell \left( I\right) \) . Moreover, \( \mathcal{R}\left( {I}^{r}\right) = \mathcal{R}{\left( I\right) }^{\left( r\rig... | Yes |
Lemma 10.3.5. Let \( J \subset I \) be graded ideals, and let \( d \) be the initial degree of \( I \) . Then\n\n\[{\beta }_{i, i + d}\left( J\right) \leq {\beta }_{i, i + d}\left( I\right)\]\n\nfor all \( i \) . | Proof. The short exact sequence\n\n\[0 \rightarrow J \rightarrow I \rightarrow I/J \rightarrow 0\]\n\n yields the long exact sequence\n\n\[\cdots \rightarrow {\operatorname{Tor}}_{i + 1}{\left( K, I/J\right) }_{i + 1 + \left( {d - 1}\right) } \rightarrow {\operatorname{Tor}}_{i}{\left( K, J\right) }_{i + d} \rightarrow... | Yes |
Theorem 10.3.6. Let \( t = d + n - 1 - \mathop{\sum }\limits_{{i = 1}}^{n}{e}_{i} \) . Then \( \operatorname{depth}S/{I}_{\left( d;{e}_{1},\ldots ,{e}_{n}\right) } = t \) . | Proof. Let \( {u}_{0} = {x}_{1}^{{e}_{1} - 1}\cdots {x}_{n - 1}^{{e}_{n - 1} - 1}{x}_{n}^{{e}_{n}} \) and \( u = {x}_{n - t}{x}_{n - t + 1}\cdots {x}_{n - 1}{u}_{0} \in G\left( I\right) \) . Let \( J = \left( \left\{ {w \in \bar{G}\left( I\right) : u{ < }_{\text{rev }}w}\right\} \right) \) . For each \( 1 \leq i \leq n... | Yes |
Corollary 10.3.9. Let \( P \) be an arbitrary finite poset with \( \left| P\right| = n \) . Then \[ \operatorname{depth}S/{H}_{P}^{k} = {2n} - \delta \left( {P;k}\right) - 1 \] for all \( k \geq 1 \) . | Proof. We work with the same notation as in the proof of Theorem 10.3.8. Recall that, for a monomial \( w = {u}_{{\alpha }_{1}}{u}_{{\alpha }_{2}}\cdots {u}_{{\alpha }_{k}} \in G\left( {H}_{P}^{k}\right) \), the colon ideal \( \left( \left\{ {v \in G\left( {H}_{P}^{k}\right) : w{ < }_{lex}v}\right\} \right) : w \) is g... | Yes |
Corollary 10.3.11. Given an integer \( n > 0 \) and given a finite sequence \( \left( {{a}_{1},{a}_{2},\ldots ,{a}_{r}}\right) \) of positive integers with \( {a}_{1} \geq {a}_{2} \geq \cdots \geq {a}_{r} \) and with \( {a}_{1} + \cdots + {a}_{r} = n \), there exists a squarefree monomial ideal \( I \subset S = \) \( K... | Proof. Let \( A\left( {a}_{i}\right) \) denote the antichain with \( \left| {A\left( {a}_{i}\right) }\right| = {a}_{i} \) and \( P \) the ordinal sum of the antichains \( A\left( {a}_{1}\right), A\left( {a}_{2}\right) ,\ldots, A\left( {a}_{r}\right) \) . In other words, \( P \) is the poset whose underlying set is the ... | Yes |
Corollary 10.3.12. Given a nonincreasing function \( f : \mathbb{N} \rightarrow \mathbb{N} \) with\n\n\[ f\left( 0\right) = 2\mathop{\lim }\limits_{{k \rightarrow \infty }}f\left( k\right) + 1 \]\n\nfor which \( {\Delta f} \) is nonincreasing, there exists a monomial ideal \( I \subset S \) such that depth \( S/{I}^{k}... | Proof. Let \( \mathop{\lim }\limits_{{k \rightarrow \infty }}f\left( k\right) = n - 1 \) and \( f\left( 0\right) = {2n} - 1 \) . Let \( {a}_{k} = \left( {\Delta f}\right) \left( {k - 1}\right) \) for all \( k \geq 1 \) . Thus \( f\left( k\right) = {2n} - \left( {{a}_{1} + \cdots {a}_{k}}\right) - 1 \) for all \( k \geq... | Yes |
Proposition 10.3.14. Let \( \mathbf{a} \in {\mathbb{Z}}_{ + }^{n} \) . Then\n\n(a) \( o\left( \mathbf{a}\right) = \min \left\{ {\langle \mathbf{c},\mathbf{a}\rangle : \mathbf{c} \in {\mathbb{Z}}_{ + }^{n}, M \cdot \mathbf{c} \geq \mathbf{1}}\right\} \) ;\n\n(b) \( \sigma \left( \mathbf{a}\right) = \max \left\{ {\langle... | Proof. (a) To say that \( {\mathbf{x}}^{\mathbf{a}} \in I{\left( \Delta \right) }^{\left( k\right) } = \mathop{\bigcap }\limits_{{C \in \mathcal{C}\left( \Delta \right) }}{P}_{C}^{k} \) is equivalent to saying that \( \mathop{\sum }\limits_{{i \in C}}{a}_{i} \geq k \) for all \( C \in \mathcal{C}\left( \Delta \right) \... | Yes |
Theorem 10.3.16. Let \( \Delta \) be a simplicial complex which has no special odd cycles. Then \( I\left( \Delta \right) \) is normally torsionfree. | Since the bipartite graphs are exactly those which have no odd cycles, we obtain | No |
Corollary 10.3.18. Let \( G \) be a bipartite graph with \( c \) connected components. Then \( \mathop{\lim }\limits_{{k \rightarrow \infty }}\operatorname{depth}S/I{\left( G\right) }^{k} = c \) . | Proof. Since \( I\left( G\right) \) is normally torsionfree, it follows from Corollary 10.3.15 that \( \mathcal{R}\left( {I\left( \Delta \right) }\right) \) is Cohen-Macaulay. Thus Proposition 10.3.2 implies that\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}\operatorname{depth}S/I{\left( G\right) }^{k} = n - \e... | Yes |
Lemma 10.3.19. Let \( I \subset S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) be a monomial ideal generated in a single degree with \( G\left( I\right) = \left\{ {{\mathbf{x}}^{{\mathbf{a}}_{1}},\ldots ,{\mathbf{x}}^{{\mathbf{a}}_{m}}}\right\} \), and let \( A \) be the \( m \times n \) matrix whose colum... | Proof. Since by assumption all generators of \( I \) have the same degree, it follows that \( \mathcal{R}\left( I\right) /\mathfrak{m}\mathcal{R}\left( I\right) \cong R \) where \( R = K\left\lbrack {{\mathbf{x}}^{{\mathbf{a}}_{1}},\ldots ,{\mathbf{x}}^{{\mathbf{a}}_{m}}}\right\rbrack \) . Therefore, \( \ell \left( I\r... | Yes |
Proposition 10.3.21. Let \( \Delta \) be a forest. Then \( \Delta \) has no special cycles of length \( \geq 3 \) . In particular, \( I\left( \Delta \right) \) is normally torsionfree and \( \mathcal{R}\left( {I\left( \Delta \right) }\right) \) is Cohen-Macaulay. | Proof. Assume that \( \Delta \) has a special cycle \( {v}_{1},{F}_{1},\ldots ,{v}_{s},{F}_{s},{v}_{s + 1} = {v}_{1} \) with \( s \geq 3 \) . Let \( \Gamma \) be the subcomplex with the facets \( {F}_{1},\ldots ,{F}_{s} \) and \( {F}_{1} \) a leaf of \( \Gamma \) . Then there exists a facet \( {F}_{i} \neq {F}_{1} \) s... | Yes |
Corollary 10.3.22. Let \( \Delta \) be a forest with vertex set \( \left\lbrack n\right\rbrack \) . Assume that \( \Delta \) is pure and has \( m \) facets. Then \( \mathop{\lim }\limits_{{k \rightarrow \infty }}\operatorname{depth}S/I{\left( \Delta \right) }^{k} = n - m \) . | Proof. By Proposition 10.3.21, \( \mathcal{R}\left( {I\left( \Delta \right) }\right) \) is Cohen-Macaulay, so that we may apply Proposition 10.3.2 to conclude that \( \mathop{\lim }\limits_{{k \rightarrow \infty }}S/I{\left( \Delta \right) }^{k} = n - \ell \left( {I\left( \Delta \right) }\right) \) . Since all generato... | Yes |
Lemma 11.1.1. The operation \( \Delta \rightarrow {\operatorname{Shift}}_{ij}\left( \Delta \right) \) satisfies the conditions \( \left( {S}_{2}\right) \) , \( \left( {S}_{3}\right) \) and \( \left( {S}_{4}\right) \) . | Proof. Since \( \left| {{C}_{ij}\left( F\right) }\right| = \left| F\right| \) for all faces \( F \) of \( \Delta \) and since \( {C}_{ij}\left( F\right) \neq {C}_{ij}\left( G\right) \) if \( F \neq G \), it follows that \( \Delta \) and \( {\operatorname{Shift}}_{ij}\left( \Delta \right) \) have the same \( f \) -vecto... | Yes |
Lemma 11.1.2. There exists a finite sequence of pairs of integers\n\n\\[ \n\\left( {{i}_{1},{j}_{1}}\\right) ,\\left( {{i}_{2},{j}_{2}}\\right) ,\\ldots ,\\left( {{i}_{q},{j}_{q}}\\right) \n\\]\n\nwith each \\( 1 \\leq {i}_{k} < {j}_{k} \\leq n \\) such that\n\n\\[ \n{\\operatorname{Shift}}_{{i}_{q}{j}_{q}}\\left( {{\\... | Proof. For each face \\( F = \\left\{ {{j}_{1},\\ldots ,{j}_{d}}\\right\} \\) of \\( \\Delta \\), we set \\( c\\left( F\\right) = {j}_{1} + \\cdots + {j}_{d} \\) . Let \\( c\\left( \\Delta \\right) = \\mathop{\\sum }\\limits_{{F \\in \\Delta }}c\\left( F\\right) \\) . Obviously one has \\( c\\left( \\Delta \\right) \\l... | Yes |
Proposition 11.2.1. The operation \( \Delta \rightarrow {\Delta }^{e} \) is a shifting operation. | Proof. Since \( {\Delta }^{e} \) is shifted, the condition \( \left( {S}_{1}\right) \) is satisfied. Since \( {J}_{\Delta } \) is strongly stable, it follows that \( {\operatorname{gin}}_{{ < }_{\text{rev }}}\left( {J}_{\Delta }\right) = {J}_{\Delta } \), see Theorem 5.2.9. Thus \( \left( {S}_{2}\right) \) is satisfied... | Yes |
Lemma 11.2.2. Let \( I \subset S \) be a squarefree monomial ideal. Then \[ m\left( u\right) + \deg u \leq n + 1 \] for all monomial \( u \) belonging to \( G\left( {{\operatorname{gin}}_{{ < }_{\text{rev }}}\left( I\right) }\right) \) . | Proof. Since \( {\operatorname{gin}}_{{ < }_{\text{rev }}}\left( I\right) \) is strongly stable, it follows from Corollary 7.2.3 that \[ {\beta }_{{ii} + j}\left( I\right) = \mathop{\sum }\limits_{{u \in G{\left( {\operatorname{gin}}_{{ < }_{\text{rev }}}\left( I\right) \right) }_{j}}}\left( \begin{matrix} m\left( u\ri... | Yes |
What is \( {\left| {A}_{n, d}\right| }^{\gamma } \) We associate each monomial \( u \in {A}_{n, d} \) with \( {u}^{\sigma } \in {B}_{n + d - 1, d} \) . The map \( u \rightarrow {u}^{\sigma } \) gives a bijection between \( {A}_{n, d} \) and \( {B}_{n + d - 1, d} \) . | The map \( u \rightarrow {u}^{\sigma } \) gives a bijection between \( {A}_{n, d} \) and \( {B}_{n + d - 1, d} \) . Its inverse is the map which associate each squarefree monomial \( v = {x}_{{i}_{1}}\cdots {x}_{{i}_{d}} \) of \( {B}_{n + d - 1, d} \), where \( 1 \leq \) \( {i}_{1} < \cdots < {i}_{d} \leq n + d - 1 \),... | Yes |
Lemma 11.2.6. If \( I \subset S \) is a strongly stable ideal, then \( {\beta }_{{ii} + j}\left( I\right) = {\beta }_{{ii} + j}\left( {I}^{\sigma }\right) \) for all \( i \) and \( j \) . | Proof. The desired formula follows from (11.1) together with Corollary 7.2.3 and Corollary 7.4.2 | No |
Lemma 11.2.8. Let \( J \subset K\left\lbrack {{x}_{1},\ldots ,{x}_{m}}\right\rbrack \) be a graded ideal, where \( m \leq n \) . Then \( {\operatorname{gin}}_{{ < }_{\text{rev }}}\left( J\right) S = {\operatorname{gin}}_{{ < }_{\text{rev }}}\left( {JS}\right) . | Proof. We may assume \( m < n \) . Let \( I = {JS} \) . There exists a nonempty Zariski open set \( U \subset \mathrm{{GL}}\left( {n;K}\right) \) such that \( {\operatorname{in}}_{{ < }_{\mathrm{{rev}}}}\left( {\alpha \left( I\right) }\right) = \operatorname{gin}{ < }_{\mathrm{{rev}}}\left( I\right) \) for all \( \alph... | Yes |
Proposition 11.2.9. Let \( I \) be a strongly stable monomial ideal. Then one has \( {\operatorname{gin}}_{{ < }_{\mathrm{{rev}}}}\left( {I}^{\sigma }\right) = I \) . In particular, the squarefree operator establishes a bijection between the strongly stable ideals and the squarefree strongly stable ideals. | Proof. Let \( J = {\operatorname{gin}}_{{ < }_{\text{rev }}}\left( {I}^{\sigma }\right) \) . Then \( J \) is strongly stable and by Theorem 11.2.7 one has \( {J}^{\sigma } = {I}^{\sigma } \) . Therefore \( G\left( {J}^{\sigma }\right) = G\left( {I}^{\sigma }\right) \) . By Lemma 11.2.5 it follows that \( G\left( J\righ... | Yes |
Theorem 11.3.1. Let \( \\Delta \) be a simplicial complex on the vertex set \( \\left\\lbrack n\\right\\rbrack \) . Then\n\n\[ \n{\\beta }_{ij}\\left( {I}_{\\Delta }\\right) \\leq {\\beta }_{ij}\\left( {I}_{{\\Delta }^{s}}\\right) \\;\\text{ for all }i\\text{ and }j \n\] | Proof. By Corollary 3.3.3 we know that \( {\\beta }_{ij}\\left( I\\right) \\leq {\\beta }_{ij}\\left( {{\\operatorname{gin}}_{{ < }_{\\text{rev }}}\\left( I\\right) }\\right) \) . Hence the theorem follows from Lemma 11.2.6. | No |
Lemma 11.3.2. Let \( {\mathbf{e}}_{G} \in {E}_{d} \) with \( G \in \left( \begin{matrix} \left\lbrack n\right\rbrack \\ d \end{matrix}\right) \) . Then one has \( {\mathbf{e}}_{G} \in {\left( {\operatorname{gin}}_{{ < }_{\text{rev }}}\left( I\right) \right) }_{d} \) if and only if \( \operatorname{rank}\left( {{M}_{G}^... | Proof. One has \( \operatorname{rank}\left( {{M}_{G}^{\prime }\left( {I, d}\right) }\right) < \operatorname{rank}\left( {{M}_{G}\left( {I, d}\right) }\right) \) if and only if the row vector \( \left( {0,\ldots ,0,1}\right) \) with \ | No |
Corollary 11.3.3. The rank of a matrix \( {M}_{G}\left( {I, d}\right), G \in \left( \begin{matrix} \left\lbrack n\right\rbrack \\ d \end{matrix}\right) \), is independent of the choice of \( \varphi \in \mathrm{{GL}}\left( {n;K}\right) \) for which \( {\operatorname{gin}}_{{ < }_{\text{rev }}}\left( I\right) = {\operat... | Proof. Note that \( \operatorname{rank}\left( {{M}_{G}^{\prime }\left( {I, d}\right) }\right) = \operatorname{rank}\left( {{M}_{G}\left( {I, d}\right) }\right) - 1 \) if \( \operatorname{rank}\left( {{M}_{G}^{\prime }\left( {I, d}\right) }\right) < \) \( \operatorname{rank}\left( {{M}_{G}\left( {I, d}\right) }\right) \... | No |
Corollary 11.3.4. Let \( I \subset E \) be a homogeneous ideal and \( \psi \in \mathrm{{GL}}\left( {n;K}\right) \) . Then one has \( \operatorname{rank}\left( {{M}_{G}\left( {I, d}\right) }\right) = \operatorname{rank}\left( {{M}_{G}\left( {\psi \left( I\right), d}\right) }\right) \) for all \( G \in \left( \begin{matr... | Proof. Recall that there is a nonempty Zariski open subset \( U \subset \mathrm{{GL}}\left( {n;K}\right) \) such that \( {\operatorname{gin}}_{{ < }_{\text{rev }}}\left( I\right) = {\operatorname{in}}_{{ < }_{\text{rev }}}\left( {\varphi \left( I\right) }\right) \) for all \( \varphi \in U \) . Similarly, there is a no... | Yes |
Corollary 11.3.5. Let \( i \geq d \) and set \( {F}_{\left( i, d\right) } = \{ i - d + 1, i - d + 2,\ldots, i\} \in \left( \begin{matrix} \left\lbrack n\right\rbrack \\ d \end{matrix}\right) \) . Then given a homogeneous ideal \( I \subset E \) one has \[ {m}_{ \leq i}\left( {{\operatorname{gin}}_{{ < }_{\text{rev }}}\... | Proof. Let \( G \in \left( \begin{matrix} \left\lbrack n\right\rbrack \\ d \end{matrix}\right) \) . Then \( m\left( {\mathbf{e}}_{G}\right) \leq i \) if and only if \( {\mathbf{e}}_{{F}_{\left( i, d\right) }}{ \leq }_{\text{rev }}{\mathbf{e}}_{G} \) . On the other hand, Corollary 11.3.3 says that \( \operatorname{rank}... | Yes |
Lemma 11.3.6. (a) If \( t \neq 0 \), then there is \( {\lambda }_{ij}^{t} \in \mathrm{{GL}}\left( {n;K}\right) \) with \( {I}_{ij}\left( t\right) = \) \( {\lambda }_{ij}^{t}\left( I\right) \) . In particular, the subspace \( {I}_{ij}\left( t\right) \) is an ideal of \( E \) . | Proof. (a) Let \( {\lambda }_{ij}^{t} \in \mathrm{{GL}}\left( {n;K}\right) \) defined by\n\n\[ \n{\lambda }_{ij}^{t}\left( {e}_{k}\right) = \left\{ \begin{array}{ll} {e}_{k}, & \text{ if }k \neq j \\ {e}_{i} + t{e}_{j}, & \text{ if }k = j \end{array}\right. \n\]\n\nWe claim \( {I}_{ij}\left( t\right) = {\lambda }_{ij}^... | Yes |
Corollary 11.3.7. With the same notation as in Corollary 11.3.5 one has \[ \operatorname{rank}\left( {{M}_{{F}_{\left( i, d\right) }}\left( {{J}_{{\operatorname{Shift}}_{ij}\left( \Delta \right) }, d}\right) }\right) \leq \operatorname{rank}\left( {{M}_{{F}_{\left( i, d\right) }}\left( {{J}_{\Delta }, d}\right) }\right... | Proof. Let \( r\left( t\right) \) be the rank of the matrix \( {M}_{{F}_{\left( i, d\right) }}\left( {{\left( {J}_{\Delta }\right) }_{ij}\left( t\right), d}\right) \) . By Corollary 11.3.4 we have \( r\left( t\right) = \operatorname{rank}\left( {{M}_{{F}_{\left( i, d\right) }}\left( {{J}_{\Delta }, d}\right) }\right) \... | Yes |
Corollary 11.3.8. Let \( \Delta \) be a simplicial complex on \( \left\lbrack n\right\rbrack \) . Then for all \( i \) and \( d \) one has\n\n\[ \n{m}_{ \leq i}\left( {{J}_{{\Delta }^{e}}, d}\right) \geq {m}_{ \leq i}\left( {{J}_{{\Delta }^{c}}, d}\right) .\n\] | Proof. Corollary 11.3.5 together with Corollary 11.3.7 guarantees that\n\n\[ \n{m}_{ \leq i}\left( {\operatorname{gin}\left( {J}_{\Delta }\right), d}\right) \geq {m}_{ \leq i}\left( {\operatorname{gin}\left( {J}_{{\text{Shift }}_{ij}\left( \Delta \right) }\right), d}\right) .\n\]\n\n(11.5)\n\nHence \( {m}_{ \leq i}\lef... | Yes |
Proposition 11.3.9. Let \( \Delta \) and \( {\Delta }^{\prime } \) be shifted simplicial complexes on \( \left\lbrack n\right\rbrack \) with \( f\left( \Delta \right) = f\left( {\Delta }^{\prime }\right) \) and suppose that\n\n\[ \n{m}_{ \leq i}\left( {{J}_{\Delta }, j}\right) \geq {m}_{ \leq i}\left( {{J}_{{\Delta }^{... | Proof. Since \( f\left( \Delta \right) = f\left( {\Delta }^{\prime }\right) \), one has \( {m}_{ \leq n}\left( {{I}_{\Delta }, j}\right) = {m}_{ \leq n}\left( {{I}_{{\Delta }^{\prime }}, j}\right) \) for all \( j \) , see Subsection 6.2. Proposition 7.4.3 then yields the inequalities \( {\beta }_{{ii} + j}\left( {I}_{\... | Yes |
Theorem 11.3.10. Let \( \Delta \) be a simplicial complex, \( {\Delta }^{e} \) the exterior algebraic shifted complex of \( \Delta \) and \( {\Delta }^{c} \) a combinatorial shifted complex of \( \Delta \) . Then \[ {\beta }_{{ii} + j}\left( {I}_{{\Delta }^{e}}\right) \leq {\beta }_{{ii} + j}\left( {I}_{{\Delta }^{c}}\... | Proof. Corollary 11.3.8 guarantees \( {m}_{ \leq i}\left( {{J}_{{\Delta }^{c}}, j}\right) \leq {m}_{ \leq i}\left( {{J}_{{\Delta }^{e}}, j}\right) \) for all \( i \) and \( j \) . Thus by virtue of Proposition 11.3.9 the required inequalities \( {\beta }_{{ii} + j}\left( {I}_{{\Delta }^{e}}\right) \leq \) \( {\beta }_{... | Yes |
Lemma 11.3.11. For all \( k,{\dim }_{K}{\widetilde{H}}_{k}\left( {\Delta ;K}\right) \leq {\dim }_{K}{\widetilde{H}}_{k}\left( {\Gamma ;K}\right) \) . | Proof. By considering an extension field of \( K \) if necessary, we may assume that \( K \) is infinite. Let \( {\Delta }^{e} \) denote the exterior algebraic shifted complex of \( \Delta \) . By Proposition 11.4.7 we have \( {\widetilde{H}}_{k}\left( {\Delta ;K}\right) \cong {\widetilde{H}}_{k}\left( {{\Delta }^{e};K... | Yes |
Lemma 11.3.12. Suppose that \( j = i + 1 \) . Then one has\n\n\[ \operatorname{Ker}\left( {\partial }_{1, k}^{\prime }\right) \subset \operatorname{Ker}\left( {\partial }_{1, k}\right) \]\n\nfor all \( k \) . | Proof. Let \( \left\lbrack a\right\rbrack \in \operatorname{Ker}\left( {\partial }_{1, k}^{\prime }\right) \), where \( a \in {\widetilde{\mathcal{C}}}_{k}\left( {\Gamma }_{W}\right) \) . Since \( \left( {\left\lbrack a\right\rbrack ,\left\lbrack a\right\rbrack }\right) \in {\widetilde{H}}_{k}\left( {{\Gamma }_{1};K}\r... | Yes |
Lemma 11.3.13. Fix \( 1 \leq p < q \leq n \) . Let \( \Delta \) be a simplicial complex on \( \left\lbrack n\right\rbrack \) and \( \Gamma = {\operatorname{Shift}}_{pq}\left( \Delta \right) \) . Then\n\n\[ \n{\beta }_{{ii} + j}\left( {I}_{\Delta }\right) \leq {\beta }_{{ii} + j}\left( {I}_{\Gamma }\right) \n\] \n\nfor ... | Proof. Let \( \pi \) be a permutation on \( \left\lbrack n\right\rbrack \) with \( \pi \left( p\right) < \pi \left( q\right) \) . Then \( \pi \) naturally induce the automorphism of \( S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) by setting \( {x}_{i} \mapsto {x}_{\pi \left( i\right) } \) . Write \( \pi ... | Yes |
Theorem 11.3.14. Let the base field be arbitrary. Let \( \Delta \) be a simplicial complex and \( {\Delta }^{c} \) a combinatorial shifted complex of \( \Delta \) . Then\n\n\[ \n{\beta }_{{ii} + j}\left( {I}_{\Delta }\right) \leq {\beta }_{{ii} + j}\left( {I}_{{\Delta }^{c}}\right) \n\]\n\nfor all \( i \) and \( j \) . | Let \( {\Delta }^{\prime } \) be a shifted simplicial complex with the same \( f \) -vector as \( \Delta \) and \( {\Delta }^{\text{lex }} \) the unique lexsegment simplicial complex with the same \( f \) -vector as \( \Delta \) . By Theorem 7.4.3 we have that \( {\beta }_{{ii} + j}\left( {I}_{{\Delta }^{\prime }}\righ... | Yes |
Theorem 11.4.1. Let \( \Delta \) be a simplicial complex and \( {I}_{\Delta } \subset K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) its Stanley-Reisner ideal, where \( K \) is an infinite field which we assume to be of characteristic 0 in the statements concerning \( {\Delta }^{s} \). (a) For all \( i \) and ... | Proof. In the case of symmetric algebraic shifting the statements in (a) and (b) are direct consequences of Theorem 4.3.17 and Lemma 11.2.6. For exterior algebraic shifting they follow from the subsequent considerations. | No |
Corollary 11.4.6. The Betti number \( {\beta }_{{ii} + j}\left( {S/I}\right) \) is extremal if and only if \( \left( {n + 1 - i - j, j}\right) \) is a distinguished pair for \( E/J \) . Moreover, if the equivalent conditions of Lemma 11.4.5 hold, then \( {\beta }_{{ii} + j}\left( {S/I}\right) = {\dim }_{K}{H}_{0}{\left... | Proof. We know from Corollary 11.4.2 that \( {\beta }_{{ii} + j}\left( {S/I}\right) \) is an extremal Betti number if and only if \( {d}_{{j}^{\prime }}\left( {E/J}\right) - {d}_{j}\left( {E/J}\right) < {j}^{\prime } - j \) for all \( {j}^{\prime } > j \) . By Proposition 11.4.4 this condition is equivalent to\n\n\[ \n... | Yes |
Corollary 11.4.7. Let \( \Delta \) be a simplicial complex and let \( K \) be a field. Then\n\n\[ \n{\widetilde{H}}_{i}\left( {\Delta ;K}\right) \cong {\widetilde{H}}_{i}\left( {{\Delta }^{e};K}\right) \;\text{ for all }\;i.\n\]\n\nMoreover if \( \operatorname{char}K = 0 \), then we also have \( {\widetilde{H}}_{i}\lef... | Proof. Hochster's formula (Theorem 8.1.1) implies that\n\n\[ \n{\beta }_{n - i - 1, n}\left( {S/{I}_{\Gamma }}\right) = {\dim }_{K}{\widetilde{H}}_{i}\left( {\Gamma ;K}\right)\n\]\n\n(11.24)\n\nfor all \( i \), for a simplicial complex \( \Gamma \) on the vertex set \( \left\lbrack n\right\rbrack \) . Thus the assertio... | No |
Proposition 11.4.8. Let \( \Delta \) be a simplicial complex on the vertex set \( \left\lbrack n\right\rbrack \) such that \( {I}_{\Delta } \) is squarefree strongly stable. Then\n\n\[ \n{\dim }_{K}{\widetilde{H}}_{i}\left( {\Delta ;K}\right) = \left| \left\{ {u \in G{\left( {I}_{\Delta }\right) }_{i + 2} : m\left( u\r... | Proof. The first equation follows from (11.24) and Corollary 7.4.2, while the second equation follows trivially from the definitions. | No |
Corollary 11.4.9. Let \( \Delta \) be a simplicial complex and let \( K \) be a field as in Theorem 11.4.1. Then the following conditions are equivalent:\n\n(a) \( \Delta \) is Cohen-Macaulay over \( K \) ;\n\n(b) \( {\Delta }^{e} \) (resp. \( {\Delta }^{s} \) ) is Cohen-Macaulay;\n\n(c) \( {\Delta }^{e} \) (resp. \( {... | Proof. (a) \( \Leftrightarrow \) (b): Since shifting operators preserve \( f \) -vectors, it follows that \( \dim K\left\lbrack \Delta \right\rbrack = \dim K\left\lbrack {\Delta }^{e}\right\rbrack = \dim K\left\lbrack {\Delta }^{s}\right\rbrack \) . Now Theorem 11.4.1 implies that \( \operatorname{projdim}K\left\lbrack... | Yes |
Theorem 11.5.2. Let \( K \) be a field of characteristic 0, and let \( f = \left( {{f}_{-1},{f}_{0}}\right. \) , \( \left. {\ldots ,{f}_{m - 1}}\right) \) and \( b = \left( {{b}_{-1},{b}_{0},\ldots ,{b}_{m - 1}}\right) \) be sequences of non-negative integers. The following conditions are equivalent:\n\n(a) there exist... | Proof. (a) \( \Rightarrow \) (b) Let \( < \) be the reverse lexicographic order. Since by Theorem 4.3.17 the extremal Betti numbers are preserved when we pass from \( I \) to \( {\operatorname{gin}}_{ < }\left( I\right) \), it follows that \( I \) and \( {\operatorname{gin}}_{ < }\left( I\right) \) have the same highes... | Yes |
Lemma 12.1.2. Let \( \mathbf{x},\mathbf{y} \in {\mathbb{R}}_{ + }^{n} \) . Then\n\n\[ \xi \left( \mathbf{x}\right) + \xi \left( \mathbf{y}\right) \geq \xi \left( {\mathbf{x} \vee \mathbf{y}}\right) + \xi \left( {\mathbf{x} \land \mathbf{y}}\right) \] | Proof. Let \( \mathbf{a} \in \mathcal{P} \) be a maximal independent subvector of \( \mathbf{x} \land \mathbf{y} \) . Since \( \mathbf{a} \leq \) \( \mathbf{x} \vee \mathbf{y} \), there exists a maximal independent subvector \( \mathbf{b} \in \mathcal{P} \) of \( \mathbf{x} \vee \mathbf{y} \) with \( \mathbf{a} \leq \m... | Yes |
Theorem 12.1.3. (a) Let \( \mathcal{P} \subset {\mathbb{R}}_{ + }^{n} \) be a polymatroid on the ground set \( \left\lbrack n\right\rbrack \) and \( \rho \) its ground set rank function. Then \( \rho \) is nondecreasing, i.e. if \( A \subset B \subset \left\lbrack n\right\rbrack \) , then \( \rho \left( A\right) \leq \... | Proof. (a) \( \Rightarrow \) (b): Clearly, \( \rho \) is nondecreasing.\n\nIn general, for \( X \subset \left\lbrack n\right\rbrack \) and for \( \mathbf{y} \in {\mathbb{R}}_{ + } \), we define \( {\mathbf{y}}_{X} \in {\mathbb{R}}_{ + } \) by setting \( {\mathbf{y}}_{X}\left( i\right) = \mathbf{y}\left( i\right), i \in... | Yes |
Theorem 12.1.4. Let \( \mathcal{P} \subset {\mathbb{R}}_{ + }^{n} \) be a polymatroid on the ground set \( \left\lbrack n\right\rbrack \) and \( \rho \) its ground set rank function. Then the vertices of \( \mathcal{P} \) are all points \( \mathbf{v} = \) \( \mathbf{v}\left( {k,\pi }\right) \in {\mathbb{R}}_{ + }^{n} \... | \[ {v}_{{i}_{1}} = \rho \left( {A}_{\pi }^{1}\right) \] \[ {v}_{{i}_{2}} = \rho \left( {A}_{\pi }^{2}\right) - \rho \left( {A}_{\pi }^{1}\right) \] \[ {v}_{{i}_{3}} = \rho \left( {A}_{\pi }^{3}\right) - \rho \left( {A}_{\pi }^{2}\right) \] ... \[ {v}_{{i}_{k}} = \rho \left( {A}_{\pi }^{k}\right) - \rho \left( {A}_{\pi ... | Yes |
Lemma 12.2.3. Let \( P \) be a discrete polymatroid.\n\n(a) Let \( d \leq \operatorname{rank}P \) . Then the set \( {P}^{\prime } = \{ \mathbf{u} \in P : \left| \mathbf{u}\right| \leq d\} \) is a discrete polyma-troid of rank \( d \) with the set of bases \( \{ \mathbf{u} \in P : \left| \mathbf{u}\right| = d\} \) .\n\n... | Proof. (a) Let \( \mathbf{u},\mathbf{v} \in P \) with \( d \geq \left| \mathbf{v}\right| > \left| \mathbf{u}\right| \) . There exists \( \mathbf{w} \in P \) such that \( \mathbf{u} < \mathbf{w} \leq \mathbf{u} \vee \mathbf{v} \) . Since \( \mathbf{w} > \mathbf{u} \), and since \( P \) contains all subvectors of \( \mat... | Yes |
Theorem 12.2.4. Let \( P \) be a nonempty finite set of integer vectors in \( {\mathbb{R}}_{ + }^{n} \) which contains with each \( \mathbf{u} \in P \) all its integral subvectors, and let \( B\left( P\right) \) be the set of vectors \( \mathbf{u} \in P \) with \( \mathbf{u} < \mathbf{v} \) for no \( \mathbf{v} \in P \... | Proof. (a) \( \Rightarrow \) (b): Already shown in the proof of Lemma 12.2.3.\n\n(b) \( \Rightarrow \) (a): Obvious.\n\n(b) \( \Rightarrow \) (c): We have already noted that (c)(i) holds. Thus it remains to prove (c)(ii). Let \( \mathbf{u},\mathbf{v} \in B\left( P\right) \) with \( \mathbf{u}\left( i\right) > \mathbf{v... | Yes |
Lemma 12.2.5. Let \( P \) be a nonempty finite set of integer vectors in \( {\mathbb{R}}_{ + }^{n} \) which contains with each \( \mathbf{u} \in P \) all its integral subvectors. Let \( B\left( P\right) \) denote the set of vectors \( \mathbf{u} \in P \) with \( \mathbf{u} < \mathbf{v} \) for no \( \mathbf{v} \in P \) ... | Proof. By using the inductive argument, we assume that \( d = r - 1 \) . Since all \( \mathbf{u} \in B\left( P\right) \) have the same modulus, it follows that all \( \mathbf{u} \in B\left( {P}^{\prime }\right) \) have the same modulus. Let \( \mathbf{u},\mathbf{v} \in B\left( {P}^{\prime }\right) \) . One has \( {\mat... | Yes |
Proposition 12.2.6. Work with the same situation as in Theorem 12.2.4 and suppose that the condition (c) is satisfied. Then, for \( \mathbf{u},\mathbf{v} \in B\left( P\right) \) with \( \mathbf{u}\left( i\right) < \mathbf{v}\left( i\right) \), there exists \( j \) with \( \mathbf{u}\left( j\right) > \mathbf{v}\left( j\... | Proof. Fix \( i \) with \( \mathbf{u}\left( i\right) < \mathbf{v}\left( j\right) \) . If there is \( {k}_{1} \neq i \) with \( \mathbf{u}\left( {k}_{1}\right) < \mathbf{v}\left( {k}_{1}\right) \), then there is \( {\ell }_{1} \) with \( \mathbf{u}\left( {\ell }_{1}\right) > \mathbf{v}\left( {\ell }_{1}\right) \) such t... | Yes |
Proposition 12.2.7. Let \( P \) be a nonempty finite set of integer vectors in \( {\mathbb{R}}_{ + }^{n} \) which contains with each \( \mathbf{u} \in P \) all its integral subvectors. Then the following conditions are equivalent:\n\n(a) \( P \) is a discrete polymatroid of rank \( d \) on the ground set \( \left\lbrac... | Proof. (a) \( \Rightarrow \) (b): We will show that \( B \) satisfies condition (c) of Theorem 12.2.4. Let \( \mathbf{u},\mathbf{v} \in P, i = d - \left| \mathbf{u}\right|, j = d - \left| \mathbf{v}\right| \) and set \( {\mathbf{u}}^{\prime } = \left( {\mathbf{u}, i}\right) \) and \( {\mathbf{v}}^{\prime } = \left( {\m... | Yes |
Fix positive integers \( {d}_{1},\ldots ,{d}_{n} \) and \( d \) with \( {d}_{1} + \cdots + {d}_{n} \geq d \) . Let \( P \subset {\mathbb{Z}}_{ + }^{n} \) be the set of vectors \( \mathbf{u} \in {\mathbb{Z}}_{ + }^{n} \) with \( \mathbf{u}\left( i\right) \leq {d}_{i} \) for all \( 1 \leq i \leq n \) and with \( \left| \... | To see why \( P \) is a discrete polymatroid, we use Theorem 12.2.4. Let \( B\left( P\right) \) be the set of vectors \( \mathbf{u} \in P \) with \( \mathbf{u} < \mathbf{v} \) for no \( \mathbf{v} \in P \) . Thus \( \mathbf{u} \in P \) belongs to \( B\left( P\right) \) if and only if \( \left| \mathbf{u}\right| = d \) ... | Yes |
Lemma 12.3.1. If \( \mathcal{P} \subset {\mathbb{R}}_{ + }^{n} \) is an integral polymatroid and if \( \mathbf{u},\mathbf{v} \in \mathcal{P} \cap {\mathbb{Z}}^{n} \) with \( \left| \mathbf{v}\right| > \left| \mathbf{u}\right| \), then there is \( \mathbf{w} \in \mathcal{P} \cap {\mathbb{Z}}^{n} \) such that \( \mathbf{... | Proof. Suppose, on the contrary, that no \( \mathbf{w} \in \mathcal{P} \cap {\mathbb{Z}}^{n} \) satisfies \( \mathbf{u} < \mathbf{w} \leq \mathbf{u} \vee \mathbf{v} \) . Let \( V = \{ i \in \left\lbrack n\right\rbrack : \mathbf{v}\left( i\right) > \mathbf{u}\left( i\right) \} \) . We claim that, for each \( i \in V \),... | Yes |
Lemma 12.3.2. If \( {X}_{1} \subset {X}_{2} \subset \cdots \subset {X}_{s} \subset \left\lbrack n\right\rbrack \) is a sequence of subsets of \( \left\lbrack n\right\rbrack \) , then there is \( \mathbf{u} \in B\left( P\right) \) such that \( \mathbf{u}\left( {X}_{k}\right) = {\rho }_{P}\left( {X}_{k}\right) \) for all... | Proof. We work with induction on \( s \) and suppose that there is \( \mathbf{u} \in B\left( P\right) \) such that \( \mathbf{u}\left( {X}_{k}\right) = {\rho }_{P}\left( {X}_{k}\right) \) for all \( 1 \leq k < s \) . Choose \( \mathbf{v} \in B\left( P\right) \) with \( \mathbf{v}\left( {X}_{s}\right) = \) \( {\rho }_{P... | Yes |
Corollary 12.3.3. The function \( {\rho }_{P} : {2}^{\left\lbrack n\right\rbrack } \rightarrow {\mathbb{R}}_{ + } \) is submodular. | Proof. Let \( A, B \subset \left\lbrack n\right\rbrack \) . By Lemma 12.3.2 there is \( \mathbf{u} \in B\left( P\right) \) such that \( \mathbf{u}(A \cap \) \( B) = {\rho }_{P}\left( {A \cap B}\right) \) and \( \mathbf{u}\left( {A \cup B}\right) = {\rho }_{P}\left( {A \cup B}\right) \) . Hence\n\n\[ \n{\rho }_{P}\left(... | Yes |
Theorem 12.3.4. A nonempty finite set \( P \subset {\mathbb{Z}}_{ + }^{n} \) is a discrete polymatroid if and only if \( \operatorname{Conv}\left( P\right) \subset {\mathbb{R}}_{ + }^{n} \) is an integral polymatroid with \( \operatorname{Conv}\left( P\right) \cap {\mathbb{Z}}^{n} = P \) . | Proof. The \ | No |
Lemma 12.4.2. We work with the same situation as in the proof of Theorem 12.4.1. Suppose that \( \rho \left( {\{ 2,3,\ldots, r\} \cup \{ j\} }\right) = \rho \left( {\{ 2,3,\ldots, r\} }\right) \) for all \( j \in X \) . Then \( \rho \left( {\{ 2,3,\ldots, r\} \cup X}\right) = \rho \left( {\{ 2,3,\ldots, r\} }\right) \)... | Proof. We proceed with induction on \( \left| X\right| \) . Clearly, the assertion is true if \( \left| X\right| = 1 \) . Now, let \( \left| X\right| > 1 \) and fix \( {j}_{0} \in X \) . Let \( Z = \{ 2,3,\ldots, r\} \) . Then, by using the assumption of induction, one has\n\n\[ \rho \left( Z\right) + \rho \left( Z\rig... | Yes |
Theorem 12.5.1. The base ring of a discrete polymatroid is normal. | Proof. Let \( P \) be a discrete polymatroid on \( \left\lbrack n\right\rbrack \) with set of bases \( B = B\left( P\right) \) . Let \( \mathcal{P} = \operatorname{Conv}\left( P\right) \subset {\mathbb{R}}^{n} \) and \( \mathcal{Q} = \operatorname{Conv}\left( B\right) \subset {\mathbb{R}}^{n} \) . Then by using Theorem... | Yes |
Let \( n = 2 \) and let \( {a}_{1},{a}_{2} > 0 \) be integers. Let \( P \subset {\mathbb{Z}}_{ + }^{2} \) denote the discrete polymatroid of rank \( d \) consisting of those \( \mathbf{u} = \left( {{u}_{1},{u}_{2}}\right) \in \) \( {\mathbb{Z}}_{ + }^{2} \) such that \( {u}_{1} \leq {a}_{1},{u}_{2} \leq {a}_{2} \) and ... | If \( P \) is generic, then the bases of \( P \) are \( \left( {{a}_{1}, d - {a}_{1}}\right) ,\left( {{a}_{1} - 1, d - {a}_{1} + 1}\right) ,\ldots ,\left( {d - {a}_{2},{a}_{2}}\right) \) . Thus the base ring \( K\left\lbrack B\right\rbrack \) of \( P \) is Gorenstein if and only if either \( {a}_{1} + {a}_{2} = d + 1 \... | No |
Theorem 12.5.4. (a) Let \( n \geq 3 \) . Let \( P \subset {\mathbb{Z}}_{ + }^{n} \) be a discrete polymatroid of rank \( d \) and suppose that the canonical basis vectors \( {\epsilon }_{1},\ldots ,{\epsilon }_{n} \) of \( {\mathbb{R}}^{n} \) belong to \( P \) . Let \( \rho : {2}^{\left\lbrack n\right\rbrack } \rightar... | Proof. (a) Suppose that a discrete polymatroid \( P \subset {\mathbb{Z}}_{ + }^{n} \) of rank \( d \) is generic and that the base ring \( K\left\lbrack B\right\rbrack \) of \( P \) is Gorenstein. Let \( \mathcal{F} = \operatorname{Conv}\left( B\right) \) . Since \( K\left\lbrack B\right\rbrack \) is Gorenstein, there ... | Yes |
Theorem 12.6.2. A polymatroidal ideal has linear quotients. | Proof. Let \( I \) be a polymatroidal ideal with \( G\left( I\right) = \left\{ {{\mathbf{x}}^{{\mathbf{u}}_{1}},\ldots ,{\mathbf{x}}^{{\mathbf{u}}_{s}}}\right\} \), where \( {\mathbf{x}}^{{\mathbf{u}}_{1}} > \cdots > {\mathbf{x}}^{{\mathbf{u}}_{s}} \) with respect to the reverse lexicographic order. Let \( J = \) \( \l... | Yes |
Theorem 12.6.3. Let \( I \) and \( J \) be polymatroidal ideals. Then IJ is again poly-matroidal. | Proof. Let \( P \) and \( Q \) be discrete polymatroids, and let \( B\left( P\right) \) and \( B\left( Q\right) \) be their bases. Theorem 12.3.4 together with Theorem 12.1.5 says that \( \{ \mathbf{u} + \mathbf{v} : \mathbf{u} \in \) \( B\left( P\right) ,\mathbf{v} \in B\left( Q\right) \} \) is the set of a discrete p... | Yes |
Lemma 12.6.6. If \( I \subset S \) is a Cohen-Macaulay polymatroidal ideal, then its radical \( \sqrt{I} \) is squarefree Veronese. | Proof. Let \( I \subset S \) be a Cohen-Macaulay polymatroidal ideal. We may assume that \( \mathop{\bigcup }\limits_{{u \in G\left( I\right) }}\operatorname{supp}\left( u\right) = \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) . Let \( u \in G\left( I\right) \) be a monomial for which \( \left| {\operatorname{supp}\left... | Yes |
Theorem 12.7.2. A weakly polymatroidal ideal I has linear quotients. | Proof. Let \( G\left( I\right) = {u}_{1},\ldots ,{u}_{m} \), where \( {u}_{1} > {u}_{2} > \cdots > {u}_{m} \) in the pure lexicographical order with induced by \( {x}_{1} > {x}_{2} > \cdots > {x}_{n} \) . We show that \( I \) has linear quotients with respect to \( {u}_{1},\ldots ,{u}_{m} \) .\n\nFix a number \( j \) a... | Yes |
Lemma 2.2.1. If \( \left\{ {{a}_{m, n} : \left( {m, n}\right) \in {\left( {\mathbb{Z}}^{ + }\right) }^{2}}\right\} \subseteq \lbrack 0,\infty ) \), then\n\n\[ \mathop{\sum }\limits_{{m = 1}}^{\infty }\mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{m, n} = \mathop{\sum }\limits_{{\left( {m, n}\right) \in {\left( {\mathbb{... | Proof: For each \( M, N \in {\mathbb{Z}}^{ + } \), \n\n\[ \left( {\mathop{\sum }\limits_{{m = 1}}^{\infty }\mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{m, n}}\right) \land \left( {\mathop{\sum }\limits_{{n = 1}}^{\infty }\mathop{\sum }\limits_{{m = 1}}^{\infty }{a}_{m, n}}\right) \geq \mathop{\sum }\limits_{{\{ \left(... | Yes |
Lemma 2.2.16. If \( \Gamma \in {\overline{{\mathcal{B}}_{\mathbb{R}}}}^{{\lambda }_{\mathbb{R}}} \) has positive Lebesgue measure, then the set \( \Gamma - \Gamma \equiv \{ y - x : x, y \in \Gamma \} \) contains an open interval \( \left( {-\delta ,\delta }\right) \) for some \( \delta > 0 \) . | Proof: Without loss of generality, we will assume that \( \Gamma \in {\mathcal{B}}_{\mathbb{R}} \) and that \( {\lambda }_{\mathbb{R}}\left( \Gamma \right) \in \left( {0,\infty }\right) \).\n\nChoose an open set \( G \supseteq \Gamma \) for which \( {\lambda }_{\mathbb{R}}\left( {G \smallsetminus \Gamma }\right) < \fra... | Yes |
Lemma 2.2.24. Suppose that \( \varnothing \neq S \subset {\mathbb{Z}}^{ + } \) . If \( B \in \mathcal{A}\left( S\right) \) and \( B \subseteq H \in \) \( \mathfrak{G}\left( \Omega \right) \), then there is a \( G \in \mathfrak{G}\left( \Omega \right) \cap \mathcal{A}\left( S\right) \) such that \( B \subseteq G \subset... | Proof: Given \( \omega \in B \), note that \( {\Pi }_{S}^{-1}\left( {\{ \omega \upharpoonright S\} }\right) \subset \subset H \) . In particular, there exists an \( n\left( \omega \right) \in {\mathbb{Z}}^{ + } \) for which \( \rho \left( {{\Pi }_{S}^{-1}\left( {\{ \omega \upharpoonright S\} }\right) ,{H}^{\complement ... | Yes |
Theorem 3.2.2 (The Monotone Convergence Theorem). Suppose that \( \left\{ {{f}_{n} : n \geq 1}\right\} \) is a sequence of non-negative, measurable functions on the measure space \( \left( {E,\mathcal{B},\mu }\right) \) and that \( {f}_{n} \nearrow f \) (pointwise) as \( n \rightarrow \infty \) . Then \( \int {fd\mu } ... | Proof: Obviously \( \int {fd\mu } \geq \mathop{\lim }\limits_{{n \rightarrow \infty }}\int {f}_{n}{d\mu } \) . To prove the opposite inequality, for each \( m \geq 1 \) choose a non-decreasing sequence \( \left\{ {{\varphi }_{m, n} : n \geq 1}\right\} \) of nonnegative, measurable, simple functions for which \( {\varph... | Yes |
Theorem 3.2.4 (Lebesgue's Dominated Convergence Theorem). Let \( \left\{ {{f}_{n} : n \geq 1}\right\} \) be a sequence of measurable functions on \( \left( {E,\mathcal{B},\mu }\right) \), and suppose that \( f \) is a measurable function to which \( \left\{ {{f}_{n} : n \geq 1}\right\} \) converges \( \mu \) -almost ev... | Proof: Let \( \widehat{E} \) be the set of \( x \in E \) for which \( f\left( x\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{f}_{n}\left( x\right) \) and \( \mathop{\sup }\limits_{{n \geq 1}}\left| {{f}_{n}\left( x\right) }\right| \leq g\left( x\right) \) . Then \( \widehat{E} \) is measurable and \( \mu \l... | Yes |
Theorem 3.2.5 (Lieb’s Version of Fatou’s Lemma). Let \( \left( {E,\mathcal{B},\mu }\right) \) be a measure space, \( \left\{ {{f}_{n} : n \geq 1}\right\} \cup \{ f\} \subseteq {L}^{1}\left( {\mu ;\mathbb{R}}\right) \), and assume that \( {f}_{n} \rightarrow f \) (a.e., \( \mu \) ). Then\n\n\[ \mathop{\lim }\limits_{{n ... | Proof: Since\n\n\[ \left| {{\begin{Vmatrix}{f}_{n}\end{Vmatrix}}_{{L}^{1}\left( {\mu ;\mathbb{R}}\right) } - \parallel f{\parallel }_{{L}^{1}\left( \mu \right) } - {\begin{Vmatrix}{f}_{n} - f\end{Vmatrix}}_{{L}^{1}\left( {\mu ;\mathbb{R}}\right) }}\right| \leq \int \left| \right| {f}_{n}\left| -\right| f\left| -\right|... | Yes |
Lemma 3.2.13. Let \( \\left\\{ {{f}_{n} : n \\geq 1}\\right\\} \\subseteq {L}^{1}\\left( {\\mu ;\\mathbb{R}}\\right) \) . If\n\n\[ \n\\mathop{\\lim }\\limits_{{m \\rightarrow \\infty }}\\mathop{\\sup }\\limits_{{n \\geq m}}{\\begin{Vmatrix}{f}_{n} - {f}_{m}\\end{Vmatrix}}_{{L}^{1}\\left( {\\mu ;\\mathbb{R}}\\right) } =... | Proof: By (3.1.7) we know that (3.2.11) holds. Hence, we can find a measurable \( f \) for which \( {f}_{n} \\rightarrow f \) in \( \\mu \) -measure; and so, by Fatou’s Lemma,\n\n\[ \n{\\begin{Vmatrix}f - {f}_{m}\\end{Vmatrix}}_{{L}^{1}\\left( {\\mu ;\\mathbb{R}}\\right) } \\leq \\mathop{\\lim }\\limits_{{n \\rightarro... | Yes |
Corollary 3.3.2. Referring to the preceding, for each \( R > 0 \), the set \( \left\{ {x \in \mathbb{R} : {\mathcal{L}}_{ + }F\left( x\right) > R}\right\} \) is open and \[ {\lambda }_{\mathbb{R}}\left( {{\mathcal{L}}_{ + }F > R}\right) \leq \frac{F\left( \infty \right) }{R} \] where \( F\left( \infty \right) = \mathop... | Proof: Set \( {g}_{R}\left( x\right) = F\left( x\right) - {Rx} \) . Then \( {g}_{R} \) is right-continuous and upper semicontinuous, and \( {g}_{R}\left( x\right) \) tends to \( \mp \infty \) as \( x \rightarrow \pm \infty \) . Furthermore, \[ {G}_{R} \equiv \left\{ {x : \exists y > x : {g}_{R}\left( y\right) > {g}_{R}... | Yes |
Lemma 3.3.5. Given \( L \in \lbrack 0,\infty ) \), define \( {F}_{L} : \mathbb{R} \rightarrow \mathbb{R} \) by\n\n\[ \n{F}_{L}\left( x\right) = {\mu }_{F}(\left( {-\infty, x\rbrack \cap \{ \mathcal{L}F \leq L\} }\right) \;\text{ for }x \in \mathbb{R}.\n\]\n\nThen \( {F}_{L} \) is a bounded, non-decreasing function that... | Proof: Clearly, it suffices to prove the final inequality.\n\nFor any \( x < y,{F}_{L}\left( y\right) - {F}_{L}\left( x\right) = {\mu }_{F}\left( {(x, y\rbrack \cap \{ \mathcal{L}F \leq L\} }\right) \) . Thus, if \( (x, y\rbrack \cap \) \( \{ \mathcal{L}F \leq L\} = \varnothing \), then \( {F}_{L}\left( y\right) - {F}_... | Yes |
Corollary 5.2.5. Let \( G \) be an open set in \( {\mathbb{R}}^{N} \) and \( \Phi \in {C}^{2}\left( {G;{\mathbb{R}}^{N}}\right) \) a diffeomorphism, and set\n\n\[ \n{\mu }_{\Phi }\left( \Gamma \right) = {\int }_{\Gamma }{J\Phi }\left( x\right) {dx}\;\text{ for }\;\Gamma \in {\overline{{\mathcal{B}}_{G}}}^{{\lambda }_{{... | As a mnemonic device, it is useful to represent the conclusion of Corollary 5.2.5 as the change of variables statement\n\n\[ \nf\left( y\right) {dy} = f \circ \Phi \left( x\right) {J\Phi }\left( x\right) {dx}\;\text{ when }\;y = \Phi \left( x\right) .\n\] | No |
Corollary 5.3.8. Referring to the preceding, one has\n\n\[ \n{\int }_{G}f{L}_{V}{gd}{\lambda }_{{\mathbb{R}}^{N}} = {\int }_{G}g{L}_{V}^{\top }{fd}{\lambda }_{{\mathbb{R}}^{N}} + {\int }_{\partial G}{fg}{\left( \mathbf{n}, V\right) }_{{\mathbb{R}}^{N}}d{\lambda }_{\partial G} \n\]\n\nfor all \( f, g \in {C}_{\mathrm{b}... | Proof: Simply observe that \( \operatorname{div}\left( {fgV}\right) = g{L}_{V}f - f{L}_{V}^{\top }g \), and apply Theorem 5.3.7 to the vector field \( {fgV} \) . | Yes |
Corollary 7.1.4. If \( S \) is a subset of a real or complex Hilbert space \( H \) , then \( S \) spans a dense subset of \( H \) if and only if \( {S}^{ \bot } = \{ 0\} \) . | Proof: Let \( L \) denote the closure of the subspace of \( H \) spanned by \( S \), and note that \( {S}^{ \bot } = {L}^{ \bot } \) . Hence, without loss in generality, we will assume that \( S = L \) and must show that \( L = H \) if and only if \( {L}^{ \bot } = \{ 0\} \) . But if \( L = H \) and \( x \bot L \), the... | Yes |
Corollary 7.2.3. The family\n\n\[ \n\\{ \\sqrt{2}\\cos \\left( {2\\pi nx}\\right) : n \\in \\mathbb{N}\\} \\cup \\left\\{ {\\sqrt{2}\\sin \\left( {2\\pi nx}\\right) : n \\in {\\mathbb{Z}}^{ + }}\\right\\} \n\]\n\nis an orthonormal basis for \( {L}^{2}\\left( {{\\lambda }_{\\left\\lbrack 0,1\\right\\rbrack };\\mathbb{R}... | Proof: There is no doubt that this family is orthonormal. To prove that it is a basis, suppose that \( f \\in {L}^{2}\\left( {{\\lambda }_{\\mathbb{R}};\\mathbb{R}}\\right) \) is orthogonal to all its members. Then, as an element of \( {L}^{2}\\left( {{\\lambda }_{\\left\\lbrack 0,1\\right\\rbrack };\\mathbb{C}}\\right... | No |
Corollary 7.2.5. If \( \ell \in {\mathbb{Z}}^{ + } \) and \( f \in {C}_{1}^{\ell }\left( {\left\lbrack {0,1}\right\rbrack ;\mathbb{C}}\right) \), then (7.2.4) holds and so the series \( \mathop{\sum }\limits_{{n \in \mathbb{Z}}}{\left( f,{\mathfrak{e}}_{n}\right) }_{{L}^{2}\left( {{\lambda }_{\left\lbrack 0,1\right\rbr... | Proof: Because we already know that \( \mathop{\sum }\limits_{{n \in {\mathbb{Z}}^{ + }}}{\left( f,{\mathfrak{e}}_{n}\right) }_{{L}^{2}\left( {{\lambda }_{\left\lbrack 0,1\right\rbrack };\mathbb{C}}\right) }{\mathfrak{e}}_{n} \) converges to \( f \) in \( {L}^{2}\left( {{\lambda }_{\left\lbrack 0,1\right\rbrack };\math... | Yes |
Lemma 7.3.5. Given \( t \in \left( {0,\infty }\right) \), define \( {g}_{t} : {\mathbb{R}}^{N} \rightarrow \left( {0,\infty }\right) \) so that\n\n\[ \n{g}_{t}\left( x\right) = {t}^{-\frac{N}{2}}\exp \left( {-\frac{\pi {\left| x\right| }^{2}}{t}}\right) ,\;\mathbf{x} \in {\mathbb{R}}^{N}.\n\]\n\nThen, for all \( t > 0,... | Proof: The first part of (7.3.8) is an easy application of (7.3.7), and, given the first part, the second part is another application of (7.3.7). Thus we need only prove (7.3.7). First note that, by Fubini's Theorem, it is enough to handle the case \( N = 1 \) . That is, we have to check that\n\n\( \left( *\right) \)\n... | Yes |
Theorem 1.3. Suppose \( f \) is entire and \( 0 < p < \infty \) . Then\n\n\[{\left| f\left( a\right) \right| }^{p} \leq \frac{1}{2\pi }{\int }_{0}^{2\pi }{\left| f\left( a + r{\mathrm{e}}^{\mathrm{i}\theta }\right) \right| }^{p}\mathrm{\;d}\theta\]\n\nfor all \( a \in \mathbb{C} \) and all \( r \in \lbrack 0,\infty ) \... | Because \( r \) above is arbitrary, we often multiply both sides of (1.1) by some function of \( r \) and then integrate with respect to \( r \) . For example, if we multiply both sides of (1.1) by \( r \) and then integrate from 0 to \( R \), the result is\n\n\[{\left| f\left( a\right) \right| }^{p} \leq \frac{1}{\pi ... | Yes |
Theorem 1.5. Suppose that\n\n(a) \( f \) is analytic on the closed disk \( \\left| z\\right| \\leq r \) ,\n\n(b) \( f \) does not vanish on \( \\left| z\\right| = r \) ,\n\n(c) \( f\\left( 0\\right) = 1 \), and\n\n(d) the zeros of \( f \) in \( \\left| z\\right| < r \) are \( \\left\{ {{z}_{1},\\cdots ,{z}_{N}}\\right\... | \[ \n\\log \\left| {f\\left( 0\\right) }\\right| = - \\mathop{\\sum }\\limits_{{k = 1}}^{N}\\log \\frac{r}{\\left| {z}_{k}\\right| } + \\frac{1}{2\\pi }{\\int }_{0}^{2\\pi }\\log \\left| {f\\left( {r{\\mathrm{e}}^{\\mathrm{i}\\theta }}\\right) }\\right| \\mathrm{d}\\theta \n\]\n\nwhere \( \\left\{ {{z}_{1},\\cdots ,{z}... | Yes |
Lemma 1.12. Let \( \Lambda = \Lambda \left( {\omega ,{\omega }_{1},{\omega }_{2}}\right) \) be any lattice in \( \mathbb{C} \) . For any positive number \( \delta \), there exists a positive constant \( C \) such that\n\n\[ \mathop{\sum }\limits_{{z \in \Lambda }}{\mathrm{e}}^{-\delta {\left| z - w\right| }^{2}} \leq C... | Proof. By translation invariance, it suffices for us to prove the desired inequality for \( w \) in the fundamental region \( {R}_{00} \) of \( \Lambda \) . If \( w \) is in the relatively compact set \( {R}_{00} \), then \( \left| {w/z}\right| < 1/2 \) for all but a finite number of points \( z \in \Lambda \) . For al... | Yes |
Lemma 1.13. With notation from above, we have\n\n\[ \mathbb{C} = \bigcup \left\{ {{R}_{mn} : m \in \mathbb{Z}, n \in \mathbb{Z}}\right\} \]\n\nand\n\n\[ {\int }_{\mathbb{C}}f\left( z\right) \mathrm{d}A\left( z\right) = \mathop{\sum }\limits_{{m, n \in \mathbb{Z}}}{\int }_{{R}_{mn}}f\left( z\right) \mathrm{d}A\left( z\r... | Proof. The decomposition of \( \mathbb{C} \) into the union of congruent parallelograms is obvious. Since any two different \( {R}_{mn} \) only overlap on a set of zero area, the desired integral decomposition follows immediately. | No |
Lemma 1.14. Let \( \Lambda = \Lambda \left( {\omega ,{\omega }_{1},{\omega }_{2}}\right) \) be a lattice in \( \mathbb{C} \) . For any positive number \( R \) , there exists a positive integer \( N \) such that we can decompose \( \Lambda \) into the disjoint union of \( N \) sublattices,\n\n\[ \Lambda = {\Lambda }_{1}... | Proof. Fix a positive integer \( k \) such that \( k\left| {\omega }_{1}\right| > R \) and \( k\left| {\omega }_{2}\right| > R \) . For each \( j = \) \( \left( {{j}_{1},{j}_{2}}\right) \) with \( 0 \leq {j}_{1} \leq k \) and \( 0 \leq {j}_{2} \leq k \), let\n\n\[ {\Lambda }_{j} = \Lambda \left( {\omega + {j}_{1}{\omeg... | Yes |
Lemma 1.15. For any positive \( r \) and \( \sigma \), there exists a positive constant \( C = {C}_{r,\sigma } \) such that\n\n\[ \mathop{\sum }\limits_{{z \in r{\mathbb{Z}}^{2}}}\mathop{\sum }\limits_{{w \in r{\mathbb{Z}}^{2}}}{\mathrm{e}}^{-\sigma {\left| z - w\right| }^{2}}{\chi }_{\gamma \left( {z, w}\right) }\left... | Proof. Without loss of generality, we may assume that \( r = 1 \) . Adjusting the constant \( \sigma \) will then produce the general case.\n\nAlso, it is obvious that\n\n\[ u + \gamma \left( {z, w}\right) = \gamma \left( {u + z, u + w}\right) \] \n\nwhich implies that the sum\n\n\[ S = \mathop{\sum }\limits_{{z \in {\... | Yes |
Proposition 1.17. Each \( \zeta \) is an odd meromorphic function with simple poles at precisely the points of \( \Lambda \) . Furthermore, for \( k = 1,2 \), we have\n\n\[ \zeta \left( {z + {\omega }_{k}}\right) = \zeta \left( z\right) + {\eta }_{k},\;z \in \mathbb{C} - \Lambda ,\]\n\nwhere \( {\eta }_{k} = {2\zeta }\... | Proof. Again we fix any small positive number \( \delta \) and consider the region \( {U}_{\delta } \) defined in the proof of the previous proposition. It is clear that\n\n\[ \frac{1}{z - {\omega }_{mn}} + \frac{1}{{\omega }_{mn}} + \frac{z}{{\omega }_{mn}^{2}} = O\left( \frac{1}{{\left| {\omega }_{mn}\right| }^{3}}\r... | Yes |
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