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Proposition 2.3.5. Let \( I \) be a nonzero ideal of \( S \) and \( \mathcal{G} = \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \) a system of generators of \( I \) . Consider the \( S \) -module epimorphism \( \epsilon : {S}^{s} \rightarrow \) \( \left( {\operatorname{in}\left( {g}_{1}\right) ,\ldots ,\operatorname{in}\le... | Proof. (a) Set \( {u}_{i} = \operatorname{in}\left( {g}_{i}\right) \) and \( \deg {e}_{i} = \deg {u}_{i} = {\mathbf{a}}_{i} \) for \( i = 1,\ldots, s \) . Then \( \epsilon \) is a \( {\mathbb{Z}}^{n} \) -graded \( S \) -module homomorphism, and hence \( \operatorname{Ker}\left( \epsilon \right) \) is generated by \( {\... | Yes |
Example 2.3.6. We continue Example 2.1.6. Let \( S = K\left\lbrack {{x}_{1},{x}_{2},\ldots ,{x}_{7}}\right\rbrack \) and \( { < }_{\text{lex }} \) the lexicographic order on \( S \) induced by \( {x}_{1} > {x}_{2} > \cdots > {x}_{7} \) . Let \( f = \) \( {x}_{1}{x}_{4} - {x}_{2}{x}_{3} \) and \( g = {x}_{4}{x}_{7} - {x... | Now, as a remainder of \( S\left( {f, g}\right) = {x}_{7}f - {x}_{1}g = \) \( {x}_{1}{x}_{5}{x}_{6} - {x}_{2}{x}_{3}{x}_{7} \) with respect to \( f \) and \( g \), we choose \( S\left( {f, g}\right) \) itself. Let \( h = \) \( {x}_{1}{x}_{5}{x}_{6} - {x}_{2}{x}_{3}{x}_{7} \) with \( {\operatorname{in}}_{{ < }_{\text{le... | Yes |
Proposition 2.3.7. Let \( I \subset S \) be an ideal and \( < \) a monomial order on \( S \) .\n\n(a) If \( I \) is graded, then the reduced Gröbner basis of \( I \) with respect to \( < \) consists of homogeneous polynomials.\n\n(b) If \( I \) is a binomial ideal, then the reduced Gröbner basis of \( I \) consists of ... | Proof. (a) If \( f \) and \( g \) are homogeneous polynomials, then the \( S \) -polynomial \( S\left( {f, g}\right) \) is again homogeneous. In the division algorithm, if \( {g}_{1},\ldots ,{g}_{s} \) and \( f \) are homogeneous polynomials, then a remainder \( {f}^{\prime } \) of \( f \) with respect to \( {g}_{1},\l... | Yes |
Lemma 3.1.1. Given a monomial order \( < \) and a finite number of pairs of monomials \( \left( {{u}_{1},{v}_{1}}\right) ,\ldots ,\left( {{u}_{m},{v}_{m}}\right) \) such that \( {u}_{i} > {v}_{i} \) for all \( i \) . Then there exists a weight \( \mathbf{w} \) such that \( {\deg }_{\mathbf{w}}{u}_{i} > {\deg }_{\mathbf... | Proof. Let \( {u}_{i} = {\mathbf{x}}^{{\mathbf{a}}_{i}} \) and \( {v}_{i} = {\mathbf{x}}^{{\mathbf{b}}_{i}} \) for \( i = 1,\ldots, m \) . We are looking for an integral vector \( \mathbf{w} \in {\mathbb{N}}^{n} \) such that \( \left\langle {{\mathbf{a}}_{i} - {\mathbf{b}}_{i},\mathbf{w}}\right\rangle > 0 \) for all \(... | Yes |
Theorem 3.1.2. Given an ideal \( I \subset S \) and a monomial order \( < \), there exists a weight \( \mathbf{w} \) such that \[ {\operatorname{in}}_{ < }\left( I\right) = {\operatorname{in}}_{\mathbf{w}}\left( I\right) \] | Proof. Let \( {g}_{1},\ldots ,{g}_{m} \) be a Gröbner basis of \( I \) . We consider all pairs \( \left( {{\operatorname{in}}_{ < }\left( {g}_{i}\right), u}\right) \) where \( u \in \operatorname{supp}\left( {g}_{i}\right) \) and \( u \neq {\operatorname{in}}_{ < }\left( {g}_{i}\right) \) . There are finitely many such... | Yes |
Lemma 3.2.1. Let \( f \in S\left\lbrack t\right\rbrack \) be homogeneous. Then \( f \in {I}^{h} \) if and only if \( f = {t}^{m}{g}^{h} \) for some \( g \in I \) and some \( m \in {\mathbb{Z}}_{ + } \) . | Proof. The \ | No |
Proposition 3.2.2. Let \( I \subset S \) be an ideal and \( \mathcal{G} = \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \) a Gröbner basis of \( I \) with respect to a monomial order \( < \) which is graded with respect to \( \mathbf{w} \). Then \( {\mathcal{G}}^{h} = \left\{ {{g}_{1}^{h},\ldots ,{g}_{s}^{h}}\right\} \) is... | Proof. Since \( {I}^{h} \) is homogeneous it suffices to show that for any homogeneous element \( f \in {I}^{h} \) one has \( {\operatorname{in}}_{{ < }^{\prime }}\left( f\right) \in \left( {{\operatorname{in}}_{{ < }^{\prime }}\left( {g}_{1}^{h}\right) ,\ldots ,{\operatorname{in}}_{{ < }^{\prime }}\left( {g}_{s}^{h}\r... | Yes |
Proposition 3.2.4. \( S\left\lbrack t\right\rbrack /{I}^{h} \) is a free \( K\left\lbrack t\right\rbrack \) -module. | Proof. Let \( < \) be monomial order which is graded with respect to \( \mathbf{w} \) . According to Proposition 3.2.2, \( \left\{ {{g}_{1}^{h},\ldots ,{g}_{s}^{h}}\right\} \) is Gröbner basis of \( {I}^{h} \) provided \( \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \) is Gröbner basis of \( I \) . Moreover, one has \( {\... | Yes |
For all \( a \in K \), the element \( t - a \) is a nonzero divisor of \( S\left\lbrack t\right\rbrack /{I}^{h} \) . | Proof. We know from Proposition 3.2.4 that \( S\left\lbrack t\right\rbrack /{I}^{h} \) is a free \( K\left\lbrack t\right\rbrack \) -module. Let \( {\left( {e}_{j}\right) }_{j \in J} \) be a \( K\left\lbrack t\right\rbrack \) -basis, and suppose \( \left( {t - a}\right) f = 0 \) for some \( f \in S\left\lbrack t\right\... | Yes |
Corollary 3.2.6. Let \( I \subset S \) be an ideal, and let \( \mathbf{w} \) be a weight. Then there exists a one parameter flat family of \( K \) -algebras whose special fibre is isomorphic to \( S/{\operatorname{in}}_{\mathbf{w}}\left( I\right) \) and whose general fibres are all isomorphic to \( S/I \) . | Proof. The one parameter flat family is defined by the graded flat \( K \) -algebra homomorphism \( K\left\lbrack t\right\rbrack \rightarrow S\left\lbrack t\right\rbrack /{I}^{h} \) . It is clear that the substitution \( t \mapsto 0 \) maps \( {I}^{h} \) to in \( \mathbf{w}\left( I\right) \) . Thus the special fibre of... | Yes |
Example 3.3.2. Let \( I = \left( {{x}_{1}{x}_{2} - {x}_{3}^{2}, - {x}_{1}{x}_{3} + {x}_{2}^{2},{x}_{1}^{2} - {x}_{2}{x}_{3}}\right) \subset S = K\left\lbrack {{x}_{1},{x}_{2},{x}_{3}}\right\rbrack \) . Then with respect to the lexicographic order we obtain the Gröbner basis | \[ \left\{ {-{x}_{2}^{3} + {x}_{3}^{3},{x}_{1}{x}_{2} - {x}_{3}^{2}, - {x}_{1}{x}_{3} + {x}_{2}^{2},{x}_{1}^{2} - {x}_{2}{x}_{3}}\right\} . \] | Yes |
Theorem 3.3.4. Let \( I \subset S \) be a graded ideal and \( < \) a monomial order on \( S \) . Then\n\n(a) \( \dim S/I = \dim S/{\operatorname{in}}_{ < }\left( I\right) \) ; | Proof. (a) follows from the fact that the residue classes of the monomials which do not belong to \( {\operatorname{in}}_{ < }\left( I\right) \) form a \( K \) -basis of \( S/I \) . Indeed, this implies that \( S/I \) and \( S/{\operatorname{in}}_{ < }\left( I\right) \) have the same Hilbert function and hence the same... | Yes |
Corollary 3.3.5. \( S/I \) is Cohen-Macaulay (resp. Gorenstein) if \( S/{\mathrm{{in}}}_{ < }\left( I\right) \) has the corresponding property. | Proof. Since, by definition, a finitely generated graded \( S \) -module \( M \) is Cohen-Macaulay if and only if \( \dim M = \operatorname{depth}M \), Theorem 3.3.4 implies the statement about Cohen-Macaulayness.\n\nConcerning the Gorenstein property we use the fact (see A.6.6) that for a graded ideal \( J \subset S \... | Yes |
Of course it may happen, and in indeed in most cases it does, that \( S/I \) is Cohen-Macaulay but \( S/{\operatorname{in}}_{ < }\left( I\right) \) is not. For example, consider the ideal \( I = \left( {{x}_{1}^{2} - {x}_{2}{x}_{3},{x}_{1}{x}_{2} - {x}_{3}^{2},{x}_{2}^{2} - {x}_{1}{x}_{3}}\right) \subset S = K\left\lbr... | and the resolutions are\n\n\[ 0 \rightarrow S{\left( -3\right) }^{2} \rightarrow S{\left( -2\right) }^{3} \rightarrow I \rightarrow 0, \]\n\nand\n\n\[ 0 \rightarrow S\left( {-4}\right) \rightarrow S{\left( -3\right) }^{3} \oplus S\left( {-4}\right) \rightarrow S{\left( -2\right) }^{3} \oplus S\left( {-3}\right) \righta... | Yes |
Proposition 3.3.7. Let \( I \subset S \) be a graded ideal and suppose that \( {\operatorname{in}}_{\mathbf{w}}\left( I\right) \) is a prime (resp. a radical) ideal. Then \( I \) is a prime (resp. a radical) ideal. In particular, if \( {\operatorname{in}}_{ < }\left( I\right) \) is a squarefree monomial ideal, then \( ... | Proof. Let \( {I}^{h} \in S\left\lbrack t\right\rbrack \) be the homogenization of \( I \) with respect to the weight \( \mathbf{w} \) . Then \( {I}^{h} \) is a graded ideal in \( S\left\lbrack t\right\rbrack \), if we set \( \deg {x}_{i} = {w}_{i} \) and \( \deg t = \) 1. We claim that \( {I}^{h} \) is a prime ideal (... | Yes |
Lemma 4.1.1. Let \( {U}_{1},\ldots ,{U}_{r} \subset {K}^{m} \) be nonempty Zariski open sets. Then \( {U}_{1} \cap \ldots \cap {U}_{r} \neq \varnothing \) . | Proof. It is enough to show that \( U \cap {U}^{\prime } \neq \varnothing \), if \( U \) and \( {U}^{\prime } \) are nonempty Zariski open sets of \( {K}^{m} \) . Let \( A = {K}^{m} \smallsetminus U \) and \( {A}^{\prime } = {K}^{m} \smallsetminus {U}^{\prime } \), and assume that \( A \) is the common set of zeroes of... | Yes |
Lemma 4.1.4. Let \( {w}_{1},\ldots ,{w}_{t} \) be monomials in \( {S}_{t} \) with \( {w}_{1} > {w}_{2} > \cdots > {w}_{t} \) . The following conditions are equivalent:\n\n(a) the monomials \( {w}_{1},\ldots ,{w}_{t} \) form a \( K \) -basis of \( {\operatorname{in}}_{ < }\left( V\right) \) ;\n\n(b) if \( {w}_{i} = {\op... | Proof. (a) \( \Rightarrow \) (b): Let \( 0 \neq f \in V \) ; then \( {\operatorname{in}}_{ < }\left( f\right) = {w}_{i} \) for some \( i \) . Hence there exists \( a \in K \) such that \( f - a{g}_{i} = 0 \) or \( {\operatorname{in}}_{ < }\left( {f - a{g}_{i}}\right) < {w}_{i} \) . Arguing by induction on \( i \) we se... | Yes |
Lemma 4.1.5. Let \( {w}_{1} \land \cdots \land {w}_{t} \) be the largest standard exterior monomial of \( \mathop{\bigwedge }\limits^{t}{S}_{d} \) with the property that there exists \( \alpha \in {\mathrm{{GL}}}_{n}\left( K\right) \) with\n\n\[{\operatorname{in}}_{ < }\left( {\alpha \left( {f}_{1}\right) \land \cdots ... | Proof. By its definition, \( U \neq \varnothing \) . Let \( p\left( \alpha \right) \) be the coefficient of \( {w}_{1} \land \cdots \land {w}_{t} \) in the presentation of \( \alpha \left( {f}_{1}\right) \land \cdots \land \alpha \left( {f}_{t}\right) \) as a linear combination of standard exterior monomials. Then \( \... | Yes |
Let \( S = K\left\lbrack {{x}_{1},{x}_{2}}\right\rbrack \), and \( < \) the lexicographic monomial order on \( S \). Then the standard exterior monomials in \( \mathop{\bigwedge }\limits^{2}{S}_{2} \) are:\n\n\[ \n{x}_{1}^{2} \land {x}_{1}{x}_{2} > {x}_{1}^{2} \land {x}_{2}^{2} > {x}_{1}{x}_{2} \land {x}_{2}^{2}.\n\] | Let \( {f}_{1} = {x}_{1}^{2},{f}_{2} = {x}_{2}^{2} \) and \( \alpha \in {\mathrm{{GL}}}_{2}\left( K\right) \). Then \( \alpha \left( {f}_{1}\right) = {\alpha }_{11}^{2}{x}_{1}^{2} + 2{\alpha }_{11}{\alpha }_{21}{x}_{1}{x}_{2} + {\alpha }_{21}^{2}{x}_{2}^{2} \) and \( \alpha \left( {f}_{2}\right) = {\alpha }_{12}{x}_{1}... | Yes |
Proposition 4.1.7. Let \( t = {\dim }_{K}{I}_{d} \) and let \( {w}_{1} \land {w}_{2} \land \cdots \land {w}_{t} \) be the standard exterior monomial generating \( \mathop{\bigwedge }\limits^{t}{\operatorname{gin}}_{ < }{\left( I\right) }_{d} \) . Then\n\n\[ \n{w}_{1} \land {w}_{2} \land \cdots \land {w}_{t} = \max \lef... | How can \( {\operatorname{gin}}_{ < }\left( I\right) \) be computed? The nonempty Zariski open set \( U \) with \( {\operatorname{gin}}_{ < }\left( I\right) = {\operatorname{in}}_{ < }\left( {\alpha I}\right) \) for all \( \alpha \in U \) is a dense subset of \( {K}^{m} \) with respect to the Zariski topology. (Here \(... | No |
Lemma 4.2.3. Let \( I \) be a monomial ideal. Suppose of all \( u \in G\left( I\right) \), and for all integers \( 1 \leq i < j \leq n \) such that \( {x}_{j} \) divides \( u \) one has \( {x}_{i}\left( {u/{x}_{j}}\right) \in I \) . Then \( I \) is strongly stable. | Proof. Let \( v \in I \) be a monomial and \( 1 \leq i < j \leq n \) integers such that \( {x}_{j} \) divides \( v \) . There exists \( u \in G\left( I\right) \) and a monomial \( w \) such that \( v = {uw} \) . If \( {x}_{j} \mid u \) , then \( {x}_{i}\left( {u/{x}_{j}}\right) \in I \) by assumption, and so \( {x}_{i}... | Yes |
Proposition 4.2.4. (a) Let \( I \subset S \) be a graded ideal. Then \( I \) is a monomial ideal, if \( I \) is Borel-fixed. | Proof. (a) We show that if \( f \in I \) is a nonzero homogeneous polynomial, and \( u \in \operatorname{supp}\left( f\right) \), then there exists a homogeneous polynomial \( g \in I \) with \( \operatorname{supp}\left( g\right) = \operatorname{supp}\left( f\right) \smallsetminus \{ u\} . \n\nSuppose \( f = {a}_{u}u +... | Yes |
Lemma 4.2.5. (a) Let \( u, v \in \operatorname{Mon}\left( {S}_{d}\right) \) with \( u{ > }_{\text{Borel }}v \). Then\n\n(i) there exist monomials \( {w}_{1},\ldots ,{w}_{r} \in \operatorname{Mon}\left( {S}_{d}\right) \) with \( {w}_{1} = u \) and \( {w}_{r} = v \), and such that for each \( k \) there exist integers \(... | Proof. (a)(i) Let \( u = {x}_{{i}_{1}}{x}_{{i}_{2}}\cdots {x}_{{i}_{d}} \) with \( {i}_{1} \leq {i}_{2} \leq \cdots \leq {i}_{d} \) and \( v = {x}_{{j}_{1}}{x}_{{j}_{2}}\cdots {x}_{{j}_{d}} \) with \( {j}_{1} \leq {j}_{2} \leq \cdots \leq {j}_{d} \). Since \( u{ > }_{\text{Borel }}v \), there exists an integer \( k \) ... | Yes |
Proposition 4.2.6. Let \( I \subset S \) be a graded ideal and \( < \) a monomial order on S. Then the following holds:\n\n(a) \( {\operatorname{gin}}_{ < }\left( I\right) \) is strongly stable, if \( \operatorname{char}K = 0 \) . | Proof. (a) By Theorem 4.2.1 the ideal \( {\sin }_{ < }\left( I\right) \) is Borel-fixed. Thus the statement follows from Proposition 4.2.4. | No |
Proposition 4.2.6 is false in positive characteristic. For example, assume char \( K = p > 0 \), and consider the ideal \( I = \left( {{x}_{1}^{p},{x}_{2}^{p}}\right) \subset K\left\lbrack {{x}_{1},{x}_{2}}\right\rbrack \) . Let \( \alpha \in {\mathrm{{GL}}}_{n}\left( K\right) \) be any element, say, \( \alpha \left( {... | Then\n\n\[\n\alpha \left( {x}_{i}^{p}\right) = \alpha {\left( {x}_{i}\right) }^{p} = {\left( {a}_{i1}{x}_{1} + {a}_{i2}{x}_{2}\right) }^{p} = {a}_{i1}^{p}{x}_{1}^{p} + {a}_{i2}^{p}{x}_{2}^{p}.\n\]\n\nSince the matrix with entries \( {a}_{ij}^{p} \) is also nonsingular, we see that \( {\alpha I} = I \) . Hence \( I \) i... | Yes |
Theorem 4.2.10 (Bayer-Stillman). Borel-fixed ideals are of Borel type. | Proof. We know from Proposition 4.2.4(a) that \( I \) is a monomial ideal. We will show that \( I \) satisfies condition (c) of Proposition 4.2.9. Let \( u \in I \) with \( a = {\nu }_{i}\left( u\right) \), and let \( 1 \leq j < i \) . We want to find an integer \( t \) such that \( {x}_{j}^{t}\left( {u/{x}_{i}^{a}}\ri... | Yes |
Lemma 4.3.1. Let \( M \) be a finitely generated graded \( S \) -module. Then the set \[ U = \left\{ {y \in {S}_{1} : y\text{ is almost regular on }M}\right\} \] is a nonempty Zariski open subset of \( {S}_{1} \) . | Proof. Let \( N \) be the graded submodule of \( M \) consisting of those elements of \( M \) which are annihilated by some power of \( \mathfrak{m} = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) . (Note that \( N \) is just the 0th local cohomology module of \( M \) . But we will not use this fact.) Obviously, \( N = \m... | Yes |
Proposition 4.3.3. Let \( I \subset S \) be a monomial ideal of Borel type. Then \( {x}_{n},{x}_{n - 1},\ldots ,{x}_{1} \) is an almost regular sequence on \( S/I \) . | Proof. By using an induction argument it suffices to show that \( {x}_{n} \) is almost regular. But this is obvious since by Proposition 4.2.9(d) the element \( {x}_{n} \) does not belong to any associated prime ideal of \( S/I \) which is different from \( \mathfrak{m} = \) \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \... | Yes |
Proposition 4.3.5. Let \( M \) be a finitely generated graded \( S \) -module, and \( \mathbf{y} = {y}_{1},\ldots ,{y}_{r} \) a sequence of elements in \( {S}_{1} \) . The following conditions are equivalent:\n\n(a) \( \mathbf{y} \) is an almost regular sequence on \( M \) .\n\n(b) \( {H}_{j}\left( {{y}_{1},\ldots ,{y}... | Proof. (a) \( \Rightarrow \) (b): We prove the assertion by induction on \( i \) . We have \( {H}_{j}\left( {{y}_{1};M}\right) = 0 \) for \( j > 1 \) and \( {H}_{1}\left( {{y}_{1};M}\right) \cong 0{ : }_{M}{y}_{1} \) . This module is of finite length by assumption.\n\nNow let \( i > 1 \) . Then there is the long exact ... | Yes |
Lemma 4.3.8. Let \( U \subset \mathrm{{GL}}\left( {n;K}\right) \) be a Zariski open subset, and \( \sigma \in \mathrm{{GL}}\left( {n;K}\right) \) . We set \( {U}^{-1} = \left\{ {{\varphi }^{-1} : \varphi \in U}\right\} \) and \( {U\sigma } = \{ {\varphi \sigma } : \varphi \in U\} \) . Then \( {U}^{-1} \) and \( {U\sigm... | Proof. Let \( \xi = \left( {x}_{ij}\right) \) be an \( n \times n \) matrix of indeterminates. We write \( U = {\mathrm{{GL}}}_{n}\left( K\right) \smallsetminus A \) where \( A \) is the common set of zeroes of the polynomials \( {f}_{1}\left( \xi \right) ,\ldots ,{f}_{m}\left( \xi \right) \) in the variables \( {x}_{i... | Yes |
Proposition 4.3.11. Let \( I \subset S \) be a graded ideal. Then\n\n\[{\alpha }_{ij}\left( {S/I}\right) = {\alpha }_{ij}\left( {S/{\operatorname{gin}}_{{ < }_{\text{rev }}}\left( I\right) }\right) . | Proof. Let \( i < n \) . According to the definition of the generic annihilator numbers we have \( {\alpha }_{ij}\left( {S/I}\right) = {\dim }_{K}{A}_{i}\left( {{x}_{n},{x}_{n - 1},\ldots ,{x}_{1};S/{\operatorname{gin}}_{{ < }_{\text{rev }}}\left( I\right) }\right) \), and\n\n\[{\alpha }_{ij}\left( {S/{\operatorname{gi... | Yes |
Proposition 4.3.12. Let \( M \) be a graded \( S \) -module and let \( \mathbf{y} = {y}_{1},\ldots ,{y}_{n} \) be a \( K \) -basis of \( {S}_{1} \) which is almost regular on \( M \) . Then \[ {\beta }_{i, i + j}\left( M\right) \leq \mathop{\sum }\limits_{{k = 0}}^{{n - i}}\left( \begin{matrix} n - k - 1 \\ i - 1 \end{... | Proof. After a suitable change of coordinates we may assume that \( {x}_{1},{x}_{2},\ldots ,{x}_{n} \) is almost regular on \( M \) . To simplify notation we set \( {H}_{i}{\left( r\right) }_{j} = {H}_{i}{\left( {x}_{1},\ldots ,{x}_{r};M\right) }_{j} \) and \( {h}_{ij}\left( r\right) = \) \( {\dim }_{K}{H}_{i}{\left( r... | Yes |
Corollary 5.1.4. (a) The functor \( M \mapsto {M}^{ \vee } \) is contravariant and exact. In particular, \( E \) is an injective object in \( \mathcal{G} \) . | Proof. All statements follow from Theorem 5.1.3 and the fact that the functor \( M \mapsto {M}^{ * } \) obviously has all the desired properties. | No |
Proposition 5.1.5. Let \( \Delta \) be a simplicial complex on the vertex set \( \left\lbrack n\right\rbrack \) . Then one has\n\n(a) \( 0 : {}_{E}{J}_{\Delta } = {J}_{{\Delta }^{ \vee }} \) ;\n\n(b) \( K\{ \Delta {\} }^{ \vee } = {J}_{{\Delta }^{ \vee }} \) and \( {\left( {J}_{\Delta }\right) }^{ \vee } = K\left\{ {\D... | Proof. (a) Since \( {J}_{\Delta } \) is a monomial ideal, it follows that \( 0 : {}_{E}{J}_{\Delta } \) is again a monomial ideal. Then by using (5.2) we see that \( {\mathbf{e}}_{F} \in 0 : {}_{E}{J}_{\Delta } \) if and only if \( F \cap G \neq \varnothing \) for all \( G \notin \Delta \) . This is the case if and onl... | Yes |
Proposition 5.1.6. Let \( M \) be a graded E-module. Then \[ {H}^{i}\left( {{M}^{ \vee }, v}\right) \cong {H}_{n - i}\left( {M, v}\right) \;\text{ for all }\;i. \] | Proof. Consider the following diagram \[ {\left( {M}^{ \vee }\right) }_{i - 1}\xrightarrow[]{{\alpha }_{i - 1}}{\operatorname{Hom}}_{K}\left( {{M}_{n - i + 1}, K}\right) \] \[ {\left( {M}^{ \vee }\right) }_{i}\xrightarrow[]{{\alpha }_{i}}{\operatorname{Hom}}_{K}\left( {{M}_{n - i}, K}\right) \] where the horizontal map... | Yes |
Proposition 5.1.8. Let \( {\Delta }_{1} \) and \( {\Delta }_{2} \) be two simplicial complexes on \( \left\lbrack n\right\rbrack \), and let \( \Delta = {\Delta }_{1} \cup {\Delta }_{2} \) and \( \Gamma = {\Delta }_{1} \cap {\Delta }_{2} \). Then\n\n(a) \( {J}_{\Delta } = {J}_{{\Delta }_{1}} \cap {J}_{{\Delta }_{2}} \)... | Proof. (a) One has \( {e}_{F} \in {J}_{\Delta } \) if and only if \( F \notin \Delta \), and this is the case if and only if \( F \notin {\Delta }_{1} \) and \( F \notin {\Delta }_{2} \). The last condition is equivalent to saying that \( {e}_{F} \in {J}_{{\Delta }_{1}} \) and \( {e}_{F} \in {J}_{{\Delta }_{2}} \), whi... | Yes |
It is known from algebraic topology that \( {\widetilde{H}}^{i}\left( {\Delta ;K}\right) = 0 \) for all \( i \), if the geometric realization of \( \Delta \) is a contractible topological space. In particular, if \( \Delta \) is a simplex, say, \( \mathcal{F}\left( \Delta \right) = \{ \left\lbrack n\right\rbrack \} \),... | We can see this directly. Indeed, \( {\widetilde{H}}^{i}\left( {\Delta ;K}\right) = {H}^{i}\left( {E, e}\right) \). After applying a linear automorphism, we may assume that \( e = {e}_{1} \). Obviously the complex \( \left( {E,{e}_{1}}\right) \) is exact. Hence the conclusion. | Yes |
Proposition 5.1.10 (Alexander duality). Let \( \Delta \) be a simplicial complex on \( \left\lbrack n\right\rbrack \) . Then for each \( i \) one has a functorial isomorphism\n\n\[ \n{\widetilde{H}}^{i - 2}\left( {{\Delta }^{ \vee };K}\right) \cong {\widetilde{H}}_{n - i - 1}\left( {\Delta ;K}\right) \n\] | Proof. By using Proposition 5.1.5 and Proposition 5.1.6 we see that\n\n\[ \n{\widetilde{H}}^{i - 2}\left( {{\Delta }^{ \vee };K}\right) = {H}^{i - 1}\left( {K\left\{ {\Delta }^{ \vee }\right\}, e}\right) \cong {H}^{i - 1}\left( {{\left( {J}_{\Delta }\right) }^{ \vee }, e}\right) \cong {H}_{n - i + 1}\left( {{J}_{\Delta... | Yes |
Lemma 5.1.12. Let \( \Delta \) be a simplicial complex. Then \[ {\widetilde{H}}^{i - 1}\left( {\Delta ;K}\right) \cong {H}^{i}\left( {K\{ \Delta \} }\right) \;\text{ for all }\;i. \] | Proof. The assertion follows once we can show that \( e \) is generic on \( K\{ \Delta \} \) . The subset \( U \subset V \) of elements \( v = \mathop{\sum }\limits_{i}^{n}{a}_{i}{e}_{i} \in {E}_{1} \) with \( \mathop{\prod }\limits_{i}^{n}{a}_{i} \neq 0 \) is a nonempty Zariski open subset of \( V \) . We note that th... | Yes |
Let \( E \) be the exterior algebra of the \( K \) -vector space \( V \) with basis \( {e}_{1},{e}_{2},{e}_{3},{e}_{4} \), and consider the ideal \( J \subset E \) generated by the element \( {e}_{1} \land {e}_{2} + {e}_{3} \land {e}_{4} \) . One has \( {J}_{i} = {E}_{i} \) for \( i = 3,4 \) . Therefore if \( < \) deno... | Thus, in contrast to the polynomial case, the initial ideal of a principal ideal in the exterior algebra need not be principal. | Yes |
Proposition 5.2.10. The generic initial of a graded ideal \( J \subset E \) is strongly stable. | Proof. Suppose \( {\operatorname{gin}}_{ < }\left( J\right) \) is not strongly stable. Then there exists a monomial \( {\mathbf{e}}_{F} \in {\operatorname{gin}}_{ < }\left( J\right) \) and numbers \( i < j \) with \( j \in F \) such that \( {e}_{i} \land {\mathbf{e}}_{F\smallsetminus \{ j\} } \notin {\operatorname{gin}... | Yes |
Let \( S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) be the polynomial ring in \( n \) variables. The monomials of degree \( i \) in \( S \) form a \( K \) -basis of \( {S}_{i} \). | It follows that\n\n\[ H\left( {S, i}\right) = \left( \begin{matrix} n + i - 1 \\ i \end{matrix}\right) = \left( \begin{matrix} n + i - 1 \\ n - 1 \end{matrix}\right) \;\text{ and }\;{H}_{S}\left( t\right) = \frac{1}{{\left( 1 - t\right) }^{n}}. \] | No |
Theorem 6.1.3 (Hilbert). Let \( K \) be a field, \( R \) a standard graded \( K \) -algebra and \( M \) a nonzero finitely generated graded \( R \) -module of dimension \( d \) . Then\n\n(a) there exists a Laurent-polynomial \( {Q}_{M}\left( t\right) \in \mathbb{Z}\left\lbrack {t,{t}^{-1}}\right\rbrack \) with \( {Q}_{... | Proof. (a) After a base field extension we may assume that \( K \) is infinite. We proceed by induction on \( \dim M \) . If \( \dim M = 0 \), then \( {M}_{i} = 0 \) for \( i \gg 0 \), and the assertion is trivial.\n\nSuppose now that \( d = \dim M > 0 \) . We choose \( y \in {R}_{1} \) such that \( y \in \) \( \mathfr... | Yes |
Theorem 6.1.4 (Macaulay). The set of monomials \( \operatorname{Mon}\left( S\right) \smallsetminus \operatorname{Mon}\left( {{\operatorname{in}}_{ < }\left( I\right) }\right) \) form a \( K \) -basis of \( S/I \) . | Proof. Let \( \mathcal{G} = \left\{ {{g}_{1},\ldots ,{g}_{r}}\right\} \) be a Gröbner basis of \( I \), and let \( f \in S \) . Then by Lemma 2.2.3, \( f \) has a unique remainder \( {f}^{\prime } \) with respect to \( \mathcal{G} \) . The residue class of \( f \) modulo \( I \) is the same as that of \( {f}^{\prime } ... | Yes |
Corollary 6.1.6. Let \( \mathcal{G} = \left\{ {{g}_{1},\ldots ,{g}_{r}}\right\} \) be a homogeneous system of generators of \( I \), and let \( J = \left( {{\operatorname{in}}_{ < }\left( {g}_{1}\right) ,\ldots ,{\operatorname{in}}_{ < }\left( {g}_{r}\right) }\right) \) . Then \( \mathcal{G} \) is a Gröbner basis of \(... | Proof. We have \( J \subset {\operatorname{in}}_{ < }\left( I\right) \), so that \( H\left( {S/J, i}\right) \geq H\left( {S/{\operatorname{in}}_{ < }\left( I\right), i}\right) = H\left( {S/I, i}\right) \n\nfor all \( i \) . Equality holds if and only if \( J = {\operatorname{in}}_{ < }\left( I\right) \) . | Yes |
Proposition 6.2.1. Let \( \Delta \) be a simplicial complex of dimension \( d - 1 \) with \( f \) -vector \( \left( {{f}_{0},{f}_{1},\ldots ,{f}_{d - 1}}\right) \) . Then\n\n\[ \n{H}_{K\left\lbrack \Delta \right\rbrack }\left( t\right) = \frac{\mathop{\sum }\limits_{{i = 0}}^{d}{f}_{i - 1}{t}^{i}{\left( 1 - t\right) }^... | Proof. Write \( K\left\lbrack \Delta \right\rbrack = S/{I}_{\Delta } \) where \( S = k\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) . By Corollary 1.1.4 the monomials not belonging to \( {I}_{\Delta } \) form a \( K \) -basis of \( K\left\lbrack \Delta \right\rbrack \) . For a monomial \( u = \) \( {\mathbf{x}... | Yes |
Theorem 6.3.1. Let \( I \subset S \) be a graded ideal. Then there exists a unique lexsegment ideal, denoted \( {I}^{\text{lex }} \), such that \( S/I \) and \( S/{I}^{\text{lex }} \) have the same Hilbert function. | The idea of the proof is simple: say, \( I \subset S \) is a graded ideal. For each graded component \( {I}_{j} \) of \( I \), and let \( {I}_{j}^{\text{lex }} \) be the \( K \) -vector space spanned by the (unique) lexsegment \( {\mathcal{L}}_{j} \) with \( \left| {\mathcal{L}}_{j}\right| = {\dim }_{K}{I}_{j} \) . The... | No |
Lemma 6.3.2. Let \( \mathcal{N} \subset {\operatorname{Mon}}_{d}\left( S\right) \) be a stable set of monomials. Then \( \operatorname{Shad}\left( \mathcal{N}\right) \) is again a stable set and\n\n(a) \( {m}_{i}\left( {\operatorname{Shad}\left( \mathcal{N}\right) }\right) = {m}_{ \leq i}\left( \mathcal{N}\right) \) ;\... | Proof. (b) is of course a consequence of (a). For the proof of (a) we note that the map\n\n\[ \phi : \{ u \in \mathcal{N} : m\left( u\right) \leq i\} \rightarrow \{ u \in \operatorname{Shad}\left( \mathcal{N}\right) : m\left( u\right) = i\}, u \mapsto u{x}_{i} \]\n\nis a bijection. In fact, \( \phi \) is clearly inject... | Yes |
Lemma 6.3.4. Let \( j \) be a positive integer. Then each positive integer a has a unique expansion\n\n\[ a = \left( \begin{matrix} {a}_{j} \\ j \end{matrix}\right) + \left( \begin{matrix} {a}_{j - 1} \\ j - 1 \end{matrix}\right) + \cdots + \left( \begin{matrix} {a}_{k} \\ k \end{matrix}\right) \]\n\nwith \( {a}_{j} > ... | Proof. We choose \( {a}_{j} \) maximal such that \( a \geq \left( \begin{matrix} {a}_{j} \\ j \end{matrix}\right) \) . If equality holds, then this is the desired expansion. Otherwise let \( {a}^{\prime } = a - \left( \begin{matrix} {a}_{j} \\ j \end{matrix}\right) \) . Then \( {a}^{\prime } > 0 \) and by using inducti... | Yes |
Lemma 6.3.5. Let \( a \geq b \) and \( j \) be positive integers. Then \( {a}^{\langle j\rangle } \geq {b}^{\langle j\rangle } \) . | Proof. We may assume that \( a > b \) . By the construction of the binomial expansions it follows that there exists an integer \( l \) such that\n\n\[ \n{a}_{j} = {b}_{j},\;{a}_{j - 1} = {b}_{j - 1},\ldots ,{a}_{l - 1}{b}_{l - 1},{a}_{l} > {b}_{l}.\n\]\n\nSince\n\n\[ \n\left( \begin{matrix} {a}_{l} \\ l \end{matrix}\ri... | No |
Lemma 6.3.6. Let \( u \in {\operatorname{Mon}}_{j}\left( S\right), u = {x}_{{k}_{1}}{x}_{{k}_{2}}\cdots {x}_{{k}_{j}} \) with \( {k}_{1} \leq {k}_{2} \leq \cdots \leq {k}_{j} \). Then\n\n\[ \n{\operatorname{Mon}}_{j}\left( S\right) \smallsetminus {\mathcal{L}}_{u} = \mathop{\bigcup }\limits_{{i = 1}}^{j}{\left\{ {x}_{{... | Proof. We notice that\n\n\[ \n{\operatorname{Mon}}_{j}\left( S\right) \smallsetminus {\mathcal{L}}_{u} = {\left\{ {x}_{{k}_{1} + 1},\ldots ,{x}_{n}\right\} }^{j} \cup \left( {{\operatorname{Mon}}_{j - 1}\left( S\right) \smallsetminus {\mathcal{L}}_{u{x}_{{k}_{1}}^{-1}}}\right) {x}_{{k}_{1}}.\n\]\n\nBy using induction o... | No |
Proposition 6.3.7. Let \( \mathcal{L} \subset {\operatorname{Mon}}_{j}\left( S\right) \) be a lexsegment with \( a = \mid {\operatorname{Mon}}_{j}\left( S\right) \smallsetminus \) \( \mathcal{L} \mid \) . Then \[ \left| {{\operatorname{Mon}}_{j + 1}\left( S\right) \smallsetminus \operatorname{Shad}\left( \mathcal{L}\ri... | Proof. Let \( u \in {\operatorname{Mon}}_{j}\left( S\right) \) be such that \( \mathcal{L} = {\mathcal{L}}_{u} \) . Then \( \operatorname{Shad}\left( \mathcal{L}\right) = {\mathcal{L}}_{u{x}_{n}} \), and the desired equation follows immediately from Lemma 6.3.6. | Yes |
Theorem 7.2.1. Let \( \mathbf{y} \) be a generic sequence on \( M \) . Then the following conditions are equivalent:\n\n(a) \( M \) has maximal Betti numbers.\n\n(b) For all \( j > 0 \) and all \( i \) the multiplication maps\n\n\[ {y}_{i} : {H}_{j}\left( {{y}_{1},\ldots ,{y}_{i - 1};M}\right) \left( {-1}\right) \right... | Proof. (a) \( \Leftrightarrow \) (b): To simplify notation we set \( {H}_{j}{\left( i\right) }_{k} = {H}_{j}{\left( {y}_{1},\ldots ,{y}_{i};M\right) }_{k} \) and \( A{\left( i\right) }_{k} = {A}_{i}{\left( \mathbf{y};M\right) }_{k} \) for all \( i, j \) and \( k \) . Inspecting the proof of Proposition 4.3.12 we see th... | Yes |
Theorem 7.2.2. Let \( I \subset S \) be a monomial ideal, and let \( \mathfrak{m} = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be the graded maximal ideal of \( S \) . Then the following conditions are equivalent:\n\n(a) \( I \) is a stable monomial ideal.\n\n(b) \( \mathfrak{m}{H}_{j}\left( {{x}_{n},{x}_{n - 1},\ldots... | Proof. The implication (b) \( \Rightarrow \) (c) is trivial.\n\n(c) \( \Rightarrow \) (a): Since for each \( i \), the annihilator module\n\n\[ \n\left( {\left( {I,{x}_{i + 1},\ldots ,{x}_{n}}\right) { : }_{S}{x}_{i}}\right) /\left( {I,{x}_{i + 1},\ldots ,{x}_{n}}\right)\n\]\n\nis a factor module of \( {H}_{1}\left( {{... | Yes |
Corollary 7.2.3 (Eliahou-Kervaire). Let \( I \subset S \) be a stable ideal. Then\n\n(a) \( {\beta }_{i, i + j}\left( I\right) = \mathop{\sum }\limits_{{u \in G{\left( I\right) }_{j}}}\left( \begin{matrix} m\left( u\right) - 1 \\ i \end{matrix}\right) \) ;\n\n(b) \( \operatorname{projdim}S/I = \max \{ m\left( u\right) ... | For a stable monomial ideal \( I \subset S \), let \( {m}_{kj} \) be the number of monomials in \( u \in G{\left( I\right) }_{j} \) with \( m\left( u\right) = k \) . Then for \( i > 0 \) the Eliahou-Kervaire formula for the Betti numbers implies\n\n\[ \n{\beta }_{i, i + j}\left( {S/I}\right) = {\beta }_{i - 1, i - 1 + ... | No |
Theorem 7.5.1. Let \( \Delta \) be a simplicial complex on \( \left\lbrack n\right\rbrack, a \in {\mathbb{N}}^{n} \) and \( W = \) \( \operatorname{supp}\left( \mathbf{a}\right) \) . Then, for all \( i \geq 0 \), we have\n\n\[ \n{\operatorname{Tor}}_{i}^{E}{\left( K,{J}_{\Delta }\right) }_{\mathbf{a}} \cong {\widetilde... | Proof. By Theorem A.8.2, \( {\operatorname{Tor}}_{i + 1}^{E}{\left( K, K\{ \Delta )\right) }_{\mathbf{a}} \) may be identified with the component of multidegree \( \mathbf{a} \) of the Cartan homology \( {H}_{i + 1}\left( {{e}_{1},\ldots ,{e}_{n};K\{ \Delta \} }\right) \) . A basis of \( {C}_{i + 1}{\left( {e}_{1},\ldo... | Yes |
Lemma 8.1.3. Let \( \Delta \) be a simplicial complex on \( \left\lbrack n\right\rbrack \) and \( W \subset \left\lbrack n\right\rbrack \) with \( W \notin \) \( \Delta \) . Let \( F = \left\lbrack n\right\rbrack \smallsetminus W \in {\Delta }^{ \vee } \) . Then\n\n\[{\operatorname{link}}_{{\Delta }^{ \vee }}F = {\left... | Proof. Each of \( \mathop{\operatorname{link}}\limits_{{\Delta }^{ \vee }}F \) and \( {\left( {\Delta }_{W}\right) }^{ \vee } \) is a simplicial complex on \( W \) . Let \( G \subset W \) . Then \( G \in {\left( {\Delta }_{W}\right) }^{ \vee } \) if and only if \( W \smallsetminus G \notin \Delta \) . On the other hand... | Yes |
Corollary 8.1.4. Let \( \Delta \) be a simplicial complex, \( \mathbf{a} \in {\mathbb{Z}}^{n} \) be squarefree and \( F = \left\lbrack n\right\rbrack \smallsetminus \operatorname{supp}\left( \mathbf{a}\right) \) . Then\n\n\[ \n{\operatorname{Tor}}_{i}^{S}{\left( K,{I}_{\Delta }\right) }_{\mathbf{a}} \cong {\widetilde{H... | Proof. Lemma 8.1.3 and (8.3) yield the desired isomorphism. The formula for \( {\beta }_{ij}\left( {I}_{\Delta }\right) \) follows from the first part since \( \mathop{\operatorname{lik}}\limits_{{\Delta }^{ \vee }}F = \varnothing \), if \( F \notin {\Delta }^{ \vee } \) . | Yes |
Lemma 8.1.5. Every Cohen-Macaulay simplicial complex is pure. | Proof. Let \( \Delta \) be Cohen-Macaulay over \( K \) . According to Lemma 1.5.4 the minimal prime ideals of \( {I}_{\Delta } \) correspond to the facets of \( \Delta \) . Hence \( \Delta \) is pure if and only if all minimal prime ideals of \( {I}_{\Delta } \) have the same height. However this is guaranteed by the a... | Yes |
Corollary 8.1.8. Let \( \Delta \) be a Cohen-Macaulay complex and \( F \) is a face of \( \Delta \) . Then \( {\operatorname{link}}_{\Delta }F \) is Cohen-Macaulay. | Proof. Let \( G \) be a face of \( {\operatorname{link}}_{\Delta }F \) . Then\n\n\[ \n{\operatorname{link}}_{{\operatorname{link}}_{\Delta }F}G = {\operatorname{link}}_{\Delta }\left( {F \cup G}\right) .\n\]\n\nHence Reisner’s criterion says that \( {\operatorname{link}}_{\Delta }F \) is Cohen-Macaulay. | Yes |
Theorem 8.1.9 (Eagon-Reiner). Let \( \Delta \) be a simplicial complex on \( \left\lbrack n\right\rbrack \) and let \( K \) be a field. Then the Stanley-Reisner ideal \( {I}_{\Delta } \subset K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) has a linear resolution if and only if \( K\left\lbrack {\Delta }^{ \vee... | Proof. Let \( K\left\lbrack {\Delta }^{ \vee }\right\rbrack \) be Cohen-Macaulay with \( \dim {\Delta }^{ \vee } = d - 1 \) . Let \( F \) be a face of \( {\Delta }^{ \vee } \) with \( \left| F\right| = n - j \) . Reisner’s theorem says that \( {\widetilde{H}}_{i - 1}\left( {{\operatorname{link}}_{{\Delta }^{ \vee }}F;K... | Yes |
Proposition 8.1.10. Let \( \Delta \) be a simplicial complex. Then\n\n\[ \n\text{ projdim }{I}_{\Delta } = \operatorname{reg}K\left\lbrack {\Delta }^{ \vee }\right\rbrack \text{. }\n\] | Proof. The regularity of a finitely generated graded \( S \) -module in terms of local cohomology is given by\n\n\[ \n\operatorname{reg}\left( M\right) = \max \left\{ {j : {H}_{\mathfrak{m}}^{i}{\left( M\right) }_{j - i} \neq 0\text{ for some }i}\right\}\n\]\n\nsee Appendix A.7. Then (8.4) and (8.5) applied to \( {\Del... | Yes |
Proposition 8.2.1. Suppose \( I \subset S \) is a graded ideal generated in degree \( d \) and that \( I \) has linear quotients. Then \( I \) has a d-linear resolution. | Proof. Let \( {f}_{1},\ldots ,{f}_{m} \) be a system of generators of \( I \) where each \( {f}_{j} \) is of degree \( d \), and assume that for all \( k,{L}_{k} = \left( {{f}_{1},\ldots ,{f}_{k - 1}}\right) : {f}_{k} \) is generated by linear forms. We show by induction on \( k \) that \( {I}_{k} = \left( {{f}_{1},\ld... | Yes |
Corollary 8.2.2. Let \( I \subset S \) be a graded ideal with linear quotients generated in one degree. Then with the notation introduced one has\n\n\[ \n{\beta }_{i}\left( I\right) = \mathop{\sum }\limits_{{k = 1}}^{n}\left( \begin{matrix} {r}_{k} \\ i \end{matrix}\right)\n\]\n\nIn particular it follows that \( \opera... | Proof. In the long exact sequence (8.7) for \( j = d \)\n\n\[ \n\rightarrow {\operatorname{Tor}}_{i + 1}^{S}{\left( \left( S/{L}_{k}\right) \left( -d\right), K\right) }_{\left( {i + 1}\right) + \left( {d - 1}\right) } \rightarrow {\operatorname{Tor}}_{i}^{S}{\left( {I}_{k - 1}, K\right) }_{i + d} \rightarrow {\operator... | Yes |
Lemma 8.2.3. The monomial ideal \( I \) has linear quotients with respect to the monomial generators \( {u}_{1},{u}_{2},\ldots ,{u}_{m} \) of \( I \) if and only if for all \( j < i \) there exists an integer \( k < i \) and an integer \( \ell \) such that\n\n\[ \frac{{u}_{k}}{\gcd \left( {{u}_{k},{u}_{i}}\right) } = {... | Proof. The assertion follows immediately from the fact that \( \left( {{u}_{1},\ldots ,{u}_{i - 1}}\right) : {u}_{i} \) is generated by the monomials \( {u}_{j}/\gcd \left( {{u}_{j},{u}_{i}}\right), j = 1,\ldots, i - 1 \), see Proposition 1.2.2. | Yes |
Proposition 8.2.5. Let \( \Delta \) be a simplicial complex. The following conditions are equivalent:\n\n(a) \( {I}_{\Delta } \) has linear quotients with respect to a monomial system of generators; (b) the Alexander dual \( {\Delta }^{ \vee } \) of \( \Delta \) is shellable.\n\nMore precisely, if \( G\left( {I}_{\Delt... | Proof. It follows from Lemma 1.5.3 that \( {\bar{F}}_{1},{\bar{F}}_{2},\cdots ,{\bar{F}}_{m} \) are the facets of \( {\Delta }^{ \vee } \) . Since \( {\bar{F}}_{r} \smallsetminus {\bar{F}}_{s} = {F}_{s} \smallsetminus {F}_{r} \) for all \( r \) and \( s \), all assertions follow from Corollary 8.2.4. | Yes |
Theorem 8.2.6. A pure shellable simplicial complex is Cohen-Macaulay over an arbitrary field. | Proof. By Proposition 8.2.5 the simplicial complex \( \Delta \) is shellable if \( {I}_{{\Delta }^{ \vee }} \) is generated in one degree and has linear quotients with respect to a monomial system of generators. This property is independent of the characteristic of the base field. Thus, if \( \Delta \) is shellable, th... | Yes |
Proposition 8.2.7. Let \( {F}_{1},\ldots ,{F}_{m} \) be a shelling of \( \Delta \) . Then\n\n\[ \n\Delta = \mathop{\bigcup }\limits_{{k = 1}}^{m}\left\lbrack {\mathcal{R}\left( {F}_{k}\right) ,{F}_{k}}\right\rbrack \n\] \n\nis a partition of \( \Delta \) . | Proof. Let \( F \in \Delta \), and let \( k \) be the smallest integer such that \( F \subset {F}_{k} \) . We claim that \( \mathcal{R}\left( {F}_{k}\right) \subset F \) . Indeed, let \( i \in \mathcal{R}\left( {F}_{k}\right) \) and suppose that \( i \notin F \) . Since \( {F}_{k} \smallsetminus \{ i\} \in {\Delta }_{k... | Yes |
Proposition 8.2.8. Given an ordering of \( {F}_{1},\ldots ,{F}_{m} \) of the facets of \( \Delta \) and a map \( \mathcal{R} : \left\{ {{F}_{1},\ldots ,{F}_{m}}\right\} \rightarrow \Delta \), the following conditions are equivalent:\n\n(i) \( {F}_{1},\ldots ,{F}_{m} \) is a shelling and \( \mathcal{R} \) is its restric... | Proof. (i) \( \Rightarrow \) (ii) follows from the definition of the restriction map attached to a shelling. In order to prove the implication (ii) \( \Rightarrow \) (i), we show that\n\n\[{\Delta }_{k - 1} \cap \left\langle {F}_{k}\right\rangle = \left\langle {{F}_{k}\smallsetminus \{ i\} : i \in \mathcal{R}\left( {F}... | Yes |
Proposition 8.2.9. Let \( {F}_{1},\ldots ,{F}_{m} \) be a shelling of the \( \left( {d - 1}\right) \) -dimension simplicial complex \( \Delta \) with restriction map \( \mathcal{R} \) . Let \( {F}_{{i}_{1}},{F}_{{i}_{2}},\cdots ,{F}_{{i}_{m}} \) be the rearrangement obtained by taking first all facets of dimension \( d... | Proof. By using Proposition 8.2.8 it suffices to show that\n\n\[ \mathcal{R}\left( {F}_{{i}_{j}}\right) \subset {F}_{{i}_{k}}\;\text{ implies }\;j \leq k. \]\n\nSuppose this condition is not satisfied. Then there exist integers \( r \) and \( s \) such that\n\n\[ r < s,\;\left| {F}_{r}\right| \leq \left| {F}_{s}\right|... | Yes |
Lemma 8.2.10. If \( I \subset S \) is a graded ideal with linear resolution, then \( \mathfrak{m}I \) has again a linear resolution. | Proof. Say that \( I \) is generated in degree \( d \) . Then the least shift in the \( i \) th position of the graded minimal free resolution of \( \mathfrak{m}I \) is at least \( i + d + 1 \) . This implies that \( {\operatorname{Tor}}_{i}^{S}{\left( K, I\right) }_{i + j} = 0 \) for all \( i \geq 0 \) and \( j < d + ... | Yes |
Lemma 8.2.11. Let \( I \subset \) be a componentwise linear ideal. Then \( {I}_{ \leq j} \) is componentwise linear for all \( j \) . | Proof. Let \( k \leq j \), then \( {\left( {I}_{ \leq j}\right) }_{\langle k\rangle } = {I}_{\langle k\rangle } \) . Therefore \( {\left( {I}_{ \leq j}\right) }_{\langle k\rangle } \) has a linear resolution for \( k \leq j \) . Let \( k > j \), then \( {\left( {I}_{ \leq j}\right) }_{\langle k\rangle } = \mathfrak{m}{... | Yes |
Lemma 8.2.12. Let \( I \subset S \) be a graded ideal. Then, for all \( k \) and for all \( j \leq k \) , one has\n\n\[{\beta }_{i, i + j}\left( I\right) = {\beta }_{i, i + j}\left( {I}_{ \leq k}\right) .\] | Proof. Let \( {H}_{i}\left( {\mathbf{x};I}\right) \) denote the Koszul homology of \( I \) with respect to the sequence \( \mathbf{x} = {x}_{1},{x}_{2},\ldots ,{x}_{n} \) of the variables. By using an isomorphism of graded \( K \) -vector space \( {\operatorname{Tor}}_{i}^{S}\left( {K, I}\right) \cong {H}_{i}\left( {\m... | Yes |
Theorem 8.2.18 (Björner-Wachs). Let \( \Delta \) be a shellable simplicial complex. Then all skeletons and pure skeletons of \( \Delta \) are shellable. | Proof. Let \( \dim \Delta = d - 1 \), and let \( 0 \leq s \leq d - 1 \) be an integer. We want to show that \( {\Delta }^{\left( s\right) } \) and \( \Delta \left( s\right) \) are shellable. Applying Proposition 8.2.9 the shellability of \( {\Delta }^{\left( s\right) } \) guarantees the shellability of \( \Delta \left(... | Yes |
Theorem 8.2.20. Let \( \Delta \) be a simplicial complex on \( \left\lbrack n\right\rbrack \) . Then \( {I}_{\Delta } \subset S \) is componentwise linear if and only if \( {\Delta }^{ \vee } \) is sequentially Cohen-Macaulay. | Proof. Let \( I = {I}_{\Delta } \) . Then by Proposition 8.2.17 \( I \) is componentwise linear if and only if \( I \) is squarefree componentwise linear. Let \( {\Delta }_{j} \) denote the simplicial complex on \( \left\lbrack n\right\rbrack \) with \( {I}_{{\Delta }_{j}} = {I}_{\left\lbrack j\right\rbrack } \) . Let ... | Yes |
Corollary 8.2.21. Let \( I \) be a squarefree monomial ideal with linear quotients. Then \( I \) is componentwise linear. | Proof. Let \( \Delta \) be the simplicial complex with \( I = {I}_{\Delta } \). Proposition 8.2.5 says \( {\Delta }^{ \vee } \) is nonpure shellable, and hence by Corollary \( {8.2.19}{\Delta }^{ \vee } \) is sequentially Cohen-Macaulay. Thus the assertion follows from Theorem 8.2.20. | Yes |
Theorem 8.2.23. Let \( I \) be a graded ideal generated in degree \( d \) . Then\n\n(a) If \( {\beta }_{i, i + j}\left( {\operatorname{gin}\left( I\right) }\right) \neq 0 \), then \( {\beta }_{{i}^{\prime },{i}^{\prime } + j}\left( {\operatorname{gin}\left( I\right) }\right) \neq 0 \) for all \( {i}^{\prime } < i \) ;\... | Proof. Statement (a) follows from the Eliahou-Kervaire formula in Corollary 4.2.6, since by Proposition 7.2.3 the generic initial ideal is strongly stable. | Yes |
Lemma 8.2.24. Let \( I \) and \( J \) be graded ideals of \( S \) generated in degree \( d \) with the same graded Betti numbers. Then \( {I}_{ \geq d + 1} \) and \( {J}_{ \geq d + 1} \) have the same graded Betti numbers. | Proof. The exact sequence\n\n\[ 0 \rightarrow {I}_{ \geq d + 1} \rightarrow I \rightarrow \cdots \rightarrow K{\left( -d\right) }^{{\beta }_{0, d}} \rightarrow 0 \]\n\ninduces the long exact sequence\n\n\[ \rightarrow {\operatorname{Tor}}_{i + 1}{\left( {I}_{ \geq d + 1}\right) }_{\left( {i + 1}\right) + \left( {j - 1}... | Yes |
Lemma 9.1.1. A finite graph \( G \) is bipartite if and only if every cycle of \( G \) is of even length. In particular every forest is bipartite. | Proof. First, suppose that \( G \) is a bipartite graph with the decomposition \( U \cup V \) of its vertices. Let \( C = \left\{ {\left\{ {{v}_{1},{v}_{2}}\right\} ,\left\{ {{v}_{2},{v}_{3}}\right\} ,\ldots ,\left\{ {{v}_{q - 1},{v}_{q}}\right\} ,\left\{ {{v}_{q},{v}_{1}}\right\} }\right\} \) be a cycle of length \( q... | Yes |
Lemma 9.1.2 (The Marriage Theorem). Let \( G \) be a bipartite graph on the vertex set \( W \cup {W}^{\prime } \) with \( \left| W\right| = \left| {W}^{\prime }\right| \) . For each \( U \subset W \) we write \( N\left( U\right) \) for the set of those \( j \in {W}^{\prime } \) such that \( \{ i, j\} \in E\left( G\righ... | Proof. First, suppose that \( \left| {N\left( U\right) }\right| \geq \left| U\right| + 1 \) for all nonempty proper subsets \( U \subset W \) . Fix an arbitrary edge \( \{ a, b\} \) of \( G \) with \( a \in W \) and \( b \in W \) . Since \( \left| {N\left( U\right) \smallsetminus \{ b\} }\right| \geq \left| U\right| \)... | Yes |
Lemma 9.1.3. A simplicial complex \( \Delta \) is flag if and only if \( \Delta \) is the clique complex of a finite graph. | Proof. Let \( G \) be a finite graph on \( \left\lbrack n\right\rbrack \) and \( \Delta \left( G\right) \) its clique complex. A subset \( F \subset \left\lbrack n\right\rbrack \) is a nonface of \( \Delta \left( G\right) \) if and only if \( F \) is not a clique of \( G \) . Thus if \( F \) is a nonface of \( \Delta \... | Yes |
Lemma 9.1.4. Let \( G \) be a graph on \( \left\lbrack n\right\rbrack \) . A subset \( C = \left\{ {{i}_{1},\ldots ,{i}_{r}}\right\} \subset \left\lbrack n\right\rbrack \) is a vertex cover of \( G \) if and only if the prime ideal \( {P}_{C} = \left( {{x}_{{i}_{1}},\ldots ,{x}_{{i}_{r}}}\right) \) contains \( I\left( ... | Proof. A generator \( {x}_{i}{x}_{j} \) of \( I\left( G\right) \) belongs to \( {P}_{C} \), if and only if \( {x}_{{i}_{k}} \) divides \( {x}_{i}{x}_{j} \) for some \( {i}_{k} \in C \) . This is the case if and only if \( C \cap \{ i, j\} \neq \varnothing \) . Thus \( I\left( G\right) \subset {P}_{C} \) if and only if ... | Yes |
Corollary 9.1.5. The ideal \( {I}_{G} \) is minimally generated by those monomials \( {x}_{C} \) for which \( C \in \mathcal{M}\left( G\right) \) . | Proof. The proof is an immediate consequence of Lemma 9.1.4 together with Corollary 1.5.5. | No |
Theorem 9.1.8. The squarefree monomial ideal \( {H}_{P} \) has linear quotients. Thus in particular \( {H}_{P} \) has a linear resolution. | Proof. Fix a total order \( < \) on \( G\left( {H}_{P}\right) \) with the property that if \( \gamma \subset \alpha \), then \( {u}_{\gamma } < {u}_{\alpha } \) . Let \( \gamma \subset \alpha \) with \( \gamma \neq \alpha \) . Then there is \( p \in \alpha \smallsetminus \gamma \) such that \( \delta = \alpha \smallset... | Yes |
Lemma 9.1.10. Every Cohen-Macaulay graph is unmixed. | Proof. Recall that, for a subset \( C \subset \left\lbrack n\right\rbrack \), the notation \( {P}_{C} \) stands for the monomial prime ideal of \( S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) generated by those variables \( {x}_{i} \) with \( i \in C \) . Let \( {C}_{1},\ldots ,{C}_{s} \) be the minimal ... | Yes |
Lemma 9.1.11. A bipartite graph coming from a poset is Cohen-Macaulay. | Proof. Let \( P = \left\{ {{p}_{1},\ldots ,{p}_{n}}\right\} \) be a finite poset and \( {V}_{n} = \left\{ {{x}_{1},\ldots ,{x}_{n},{y}_{1},\ldots ,{y}_{n}}\right\} \) . The edge ideal \( I\left( {G\left( P\right) }\right) \) is generated by those 2-element subsets \( \left\{ {{x}_{i},{y}_{j}}\right\} \) with \( {p}_{i}... | Yes |
Lemma 9.1.12. Every Cohen-Macaulay complex is connected in codimension one. | Proof. Let \( \Delta \) be a Cohen-Macaulay complex of dimension \( d - 1 \) . If \( d - 1 = 0 \) , the assertions are trivial. Therefore we now assume that \( d - 1 > 0 \) . Let \( F \) and \( G \) be two facets of \( \Delta \) . Since \( \Delta \) is connected (Lemma 8.1.7), there exists a sequence of facets \( F = {... | Yes |
Theorem 9.1.13. A bipartite graph \( G \) is Cohen-Macaulay if and only if \( G \) comes from a finite poset. | Proof. The \ | No |
Corollary 9.1.14. Let \( G \) be a bipartite graph with vertex partition \( V \cup {V}^{\prime } \). Then the following conditions are equivalent:\n\n(i) \( G \) is a Cohen-Macaulay graph;\n\n(ii) \( \left| V\right| = \left| {V}^{\prime }\right| \) and the vertices \( V = \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) an... | Proof. By Theorem 9.1.13, the graph \( G \) is Cohen-Macaulay if and only if \( G = G\left( P\right) \) for some poset \( P = \left\{ {{p}_{1},\ldots ,{p}_{n}}\right\} \.\n\n(i) \( \Rightarrow \) (ii): We may assume that \( {p}_{i} \leq {p}_{j} \) implies that \( i \leq j \). With this labelling (ii) follows from (i).\... | Yes |
Corollary 9.1.15. Let \( P \) be a finite poset. Then the minimal vertex covers of \( G\left( P\right) \) are precisely the sets \( {\alpha }_{x} \cup {\alpha }_{y} \) with \( \alpha \in \mathcal{J}\left( P\right) \) . | Proof. According to Lemma 9.1.4 the monomial vertex covers of \( G\left( P\right) \) correspond to the generators of \( I{\left( G\left( P\right) \right) }^{ \vee } \) . By Lemma 9.1.9 we have \( {H}_{P}^{ \vee } = I\left( {G\left( P\right) }\right) \) . Therefore, \( I{\left( G\left( P\right) \right) }^{ \vee } = {\le... | Yes |
Corollary 9.1.16. An unmixed bipartite graph \( G \) is Cohen-Macaulay if and only if it is shellable. | Proof. If \( G \) is Cohen-Macaulay, then we may assume by Theorem 9.1.13 that \( G = G\left( P\right) \) for some finite poset \( P \) . Since \( I\left( {G\left( P\right) }\right) = {H}_{P}^{ \vee } \) and since by Theorem 9.1.8, \( {H}_{P} \) has linear quotients, the assertion follows from Proposition 8.2.5. | Yes |
Theorem 9.1.18. Let \( \mathcal{L} \) be a subset of \( {\mathcal{L}}_{n} \) . Then there exists a (unique) unmixed bipartite graph \( G \) on \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \cup \left\{ {{y}_{1},\ldots ,{y}_{n}}\right\} \) such that \( \mathcal{L} = {\mathcal{L}}_{G} \) if and only if \( \varnothing \) a... | Proof. Let \( G \) be an unmixed bipartite graph with \( \mathcal{L} = {\mathcal{L}}_{G} \) . Let \( C = \) \( \left\{ {{x}_{{i}_{1}},\ldots ,{x}_{{i}_{s}},{y}_{{i}_{s + 1}},\ldots ,{y}_{{i}_{n}}}\right\} \) and \( {C}^{\prime } = \left\{ {{x}_{{j}_{1}},\ldots ,{x}_{{j}_{t}},{y}_{{j}_{t + 1}},\ldots ,{y}_{{j}_{n}}}\rig... | Yes |
Theorem 9.1.19. A subset \( \mathcal{L} \) of \( {\mathcal{L}}_{n} \) is a full sublattice of \( {\mathcal{L}}_{n} \) if and only if there exists a Cohen-Macaulay bipartite graph \( G \) on \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \cup \left\{ {{y}_{1},\ldots ,{y}_{n}}\right\} \) with \( \mathcal{L} = {\mathcal{L}}... | Proof. Let \( G \) be a Cohen-Macaulay bipartite graph on the set \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \cup \) \( \left\{ {{y}_{1},\ldots ,{y}_{n}}\right\} \) . Theorem 9.1.13 guarantees the existence of a finite poset \( P \) with \( G = G\left( P\right) \), where \( \left| P\right| = n \) . Corollary 9.1.15 s... | Yes |
Lemma 9.1.21. Let \( G \) be a bipartite graph with bipartition \( \left\{ {{x}_{1},\ldots ,{x}_{m}}\right\} \) and \( \left\{ {{y}_{1},\ldots ,{y}_{n}}\right\} \) . If \( G \) is sequentially Cohen-Macaulay, then there exists a vertex \( v \in V\left( G\right) \) with \( \deg v = 1 \) . | Proof. We may assume that \( m \leq n \), and that \( G \) has no isolated vertices. Let \( {I}_{G} = I{\left( G\right) }^{ \vee } \) be the vertex cover ideal of \( G \), and let \( L = {\left( {I}_{G}\right) }_{\left\lbrack n\right\rbrack } \) be the squarefree part of the \( n \) th component of \( {I}_{G} \) . Sinc... | Yes |
Lemma 9.1.22. Let \( x \) be a vertex of \( G \), and let \( {G}^{\prime } = G \smallsetminus \left( {\{ x\} \cup {N}_{G}\left( x\right) }\right) \) . If \( G \) is sequentially Cohen–Macaulay, then \( {G}^{\prime } \) is sequentially Cohen–Macaulay. | Proof. Let \( \Delta \) be the simplicial complex of independent sets of \( G \) and \( {\Delta }^{\prime } \) the simplicial complex of independent sets of \( {G}^{\prime } \) . We first show that\n\n\[ \n{\Delta }^{\prime } = {\operatorname{link}}_{\Delta }\{ x\} \n\]\n\n(9.2)\n\nLet \( F \in {\operatorname{link}}_{\... | Yes |
Corollary 9.2.2. Let \( G \) be a chordal graph on \( \left\lbrack n\right\rbrack \) and \( \Delta \left( G\right) \) its clique complex. Then \( {\widetilde{H}}_{i}\left( {\Delta \left( G\right) ;K}\right) = 0 \) for all \( i \neq 0 \) . | Proof. We work with induction on the number of vertices on \( G \) . If \( G \) is a complete graph, then \( \Delta \left( G\right) \) is the simplex on \( \left\lbrack n\right\rbrack \) . Thus \( {\widetilde{H}}_{i}\left( {\Delta \left( G\right) ;K}\right) = 0 \) for all \( i \), see Example 5.1.9. Suppose that \( G \... | Yes |
Theorem 9.2.3 (Fröberg). The edge ideal \( I\left( G\right) \) of a finite graph \( G \) has a linear resolution if and only if the complementary graph \( \bar{G} \) of \( G \) is chordal. | Proof. Since \( I\left( G\right) = {I}_{\Delta \left( \bar{G}\right) } \), what we must prove is that the Stanley-Reisner ideal of the clique complex \( \Delta \left( \bar{G}\right) \) of \( \bar{G} \) has a linear resolution if and only if \( \bar{G} \) is chordal.\n\nHochster’s formula (Theorem 8.1.1) says that \( {I... | Yes |
Example 9.2.5. Let \( n = 6 \) and \( I = \left( {{x}_{4}{x}_{5}{x}_{6},{x}_{1}{x}_{5}{x}_{6},{x}_{1}{x}_{2}{x}_{6},{x}_{1}{x}_{2}{x}_{5}}\right) \) . Thus the \( 6 \times 4 \) matrix \( {A}_{I} \) is\n\n\[ \left\lbrack \begin{matrix} {x}_{1} & - {x}_{4} & 0 & 0 \\ {x}_{1}{x}_{2} & 0 & - {x}_{4}{x}_{5} & 0 \\ 0 & {x}_{... | By using Lemma 9.2.4 one has proj \( \dim I = 1 \) . In fact, \( I \) has three Hilbert-Burch matrices\n\n\[ \left\lbrack \begin{matrix} {x}_{1} - {x}_{4} & 0 & 0 & \\ 0 & {x}_{2} & - {x}_{5} & 0 \\ 0 & {x}_{2} & 0 & - {x}_{6} \end{matrix}\right\rbrack ,\left\lbrack \begin{matrix} {x}_{1} - {x}_{4} & 0 & 0 & \\ 0 & {x}... | Yes |
Lemma 9.2.6. A finite graph \( G \) has a perfect elimination ordering if and only if the clique complex \( \Delta \left( G\right) \) of \( G \) is a quasi-forest. | Proof. \ | No |
Lemma 9.2.7. A quasi-forest is a flag complex. | Proof. Let \( \Delta \) be a quasi-forest on \( \left\lbrack n\right\rbrack \) and \( {F}_{1},\ldots ,{F}_{q} \) its leaf order. We work with induction on \( q \) . Let \( q > 1 \) . Since \( {\Delta }^{\prime } = \left\langle {{F}_{1},\ldots ,{F}_{q - 1}}\right\rangle \) is a quasi-forest, it follows that \( {\Delta }... | Yes |
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