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Theorem 6.24 Fix a monomial ideal \( I \) and a generic deformation \( \epsilon \) . Define \( {\Delta }_{I}^{\epsilon } \) by relabeling each face \( \sigma \) in the Scarf complex \( {\Delta }_{{I}_{\epsilon }} \) by \( {m}_{\sigma } \) instead of \( \operatorname{lcm}\left( {{m}_{i}{\mathbf{x}}^{{\epsilon }_{i}} \mi... | Proof. Given any vector \( \mathbf{b} \in {\mathbb{N}}^{n} \), the (unlabeled) simplicial subcomplex \( {\left( {\Delta }_{I}^{\epsilon }\right) }_{ \preccurlyeq \mathbf{b}} \) can also be expressed as \( {\left( {\Delta }_{{I}_{\epsilon }}\right) }_{ \preccurlyeq {\mathbf{b}}^{\prime }} \) for the least common multipl... | Yes |
Theorem 6.26 Fix an ideal I generated by monomials dividing \( {\mathbf{x}}^{\mathbf{u}} \), and set \( {I}^{ * } = I + {\mathfrak{m}}^{\mathbf{u} + \mathbf{1}} \). The following are equivalent.\n\n(a) I is generic.\n\n(b) \( {\mathcal{F}}_{{\Delta }_{{I}^{ * }}} \) is a minimal free resolution of \( S/{I}^{ * } \).\n\... | Proof. The scheme of the proof is\n\n\[ \left( b\right) \Rightarrow \left( c\right) \Rightarrow \left( d\right) \Rightarrow \left( e\right) \Rightarrow \left( b\right) \text{ and }\left( c\right) \Rightarrow \left( f\right) \Rightarrow \left( a\right) \Rightarrow \left( g\right) \Rightarrow \left( h\right) \Rightarrow ... | Yes |
Lemma 6.27 If \( \mathbf{b} \in {\mathbb{N}}^{n} \) and \( {\beta }_{i,\mathbf{b}}\left( {S/{I}^{ * }}\right) \neq 0 \) for some \( i \), then there is an irreducible component \( {\mathfrak{m}}^{\mathbf{c}} \) of \( S/{I}^{ * } \) such that \( \mathbf{b} \preccurlyeq \mathbf{c} \) . | Proof. If \( {\beta }_{i,\mathbf{b}}\left( {S/{I}^{ * }}\right) \) is nonzero, then the upper Koszul simplicial complex \( {K}^{\mathbf{b}}\left( I\right) \) is not the whole simplex \( {2}^{\left\lbrack n\right\rbrack } \), so \( {\mathbf{x}}^{\mathbf{b} - \mathbf{1}} \) lies outside of \( {I}^{ * } \) . Since \( S/{I... | Yes |
Theorem 6.29 The number \( {\beta }_{i}\left( I\right) \) of minimal \( {i}^{\text{th }} \) syzygies of any monomial ideal \( I \) with \( r \) generators in \( n \) variables is bounded above by the number \( {C}_{i, n, r} \) of \( i \) -dimensional faces of the cyclic n-polytope with \( r \) vertices. If \( i = n - 1... | Proof. The number of \( i \) -faces of the hull complex hull \( \left( I\right) \) equals \( {\beta }_{i}\left( I\right) \) . Consider the polytope \( {\widetilde{\mathcal{Q}}}_{t} = \operatorname{conv}\left\{ {{t}^{\mathbf{a}} \mid {\mathbf{x}}^{\mathbf{a}} \in \min \left( I\right) }\right\} \) that appears as a Minko... | Yes |
Example 6.30 \( \left( {n = 4, r = {12}}\right) \) For the generic monomial ideal\n\n\[ I = \left\langle {{a}^{9},{b}^{9},{c}^{9},{d}^{9},{a}^{6}{b}^{7}{c}^{4}d,{a}^{2}{b}^{3}{c}^{8}{d}^{5},{a}^{5}{b}^{8}{c}^{3}{d}^{2},}\right.\n\n\[ a{b}^{4}{c}^{7}{d}^{6},{a}^{8}{b}^{5}{c}^{2}{d}^{3},{a}^{4}b{c}^{6}{d}^{7},{a}^{7}{b}^... | every pairwise first syzygy is minimal. The minimal free resolution of \( I \) is\n\n\[ 0 \leftarrow I \leftarrow {S}^{12} \leftarrow {S}^{66} \leftarrow {S}^{108} \leftarrow {S}^{53} \leftarrow 0. \] | Yes |
Corollary 6.31 The number of irreducible components of an ideal generated by \( r \) monomials in \( n \) variables is at most \( {C}_{n - 1, n, r + n} - 1 \) . | Proof. We assume that \( I \) is generic, as the number of irreducible components can only rise under generic deformation (the reader is asked to prove this in Exercise 6.9). Now apply Corollary 6.20: The artinian ideal \( {I}^{ * } \) has at most \( n + r \) generators, and its Scarf complex \( {\Delta }_{{I}^{ * }} \... | No |
Lemma 6.37 A monomial ideal \( I = \left\langle {{\mathbf{x}}^{{\mathbf{b}}_{1}},\ldots ,{\mathbf{x}}^{{\mathbf{b}}_{r}}}\right\rangle \) is cogeneric if and only if its Alexander dual \( {I}^{\left\lbrack \mathbf{a}\right\rbrack } \) for any (hence every) vector \( \mathbf{a} \succcurlyeq \mathop{\bigvee }\limits_{j}{... | Proof. If \( {I}_{i} = {\mathfrak{m}}^{\mathbf{a} \smallsetminus {\mathbf{b}}_{i}},{I}_{j} = {\mathfrak{m}}^{\mathbf{a} \smallsetminus {\mathbf{b}}_{j}} \), and \( {I}_{\ell } = {\mathfrak{m}}^{\mathbf{a} \smallsetminus {\mathbf{b}}_{\ell }} \), then \( {I}_{\ell } \subseteq {I}_{i} + {I}_{j} \) if and only if \( {\mat... | Yes |
Example 6.38 The permutohedron ideal \( I \) in Section 4.3.3 is cogeneric. It is the Alexander dual, with respect to \( \mathbf{a} = \left( {n + 1,\ldots, n + 1}\right) \), of the tree ideal \( {I}^{ \star } \) in Section 4.3.4. Hence the permutohedron ideal \( I \) is the intersection of the irreducible ideals \( \le... | Since the tree ideal \( {I}^{ \star } \) is generic, by Example 6.6, its minimal free resolution is the Scarf complex \( {\Delta }_{{I}^{ \star }} \) . By Theorem 6.13, the Scarf complex \( {\Delta }_{{I}^{ \star }} \) coincides with the hull complex hull \( \left( {I}^{ \star }\right) \) . Applying Alexander duality t... | No |
Theorem 6.40 For a cogeneric monomial ideal \( I \), the algebraic coScarf complex \( {\Delta }^{I,\mathbf{a}} \) is a minimal cellular free resolution of \( I \) . | Proof. Apply Theorem 5.37 to Theorem 6.13. | No |
Corollary 6.41 The interior faces of the Scarf complex \( {\Delta }_{{I}^{ * }} \) minimally resolve \( S/I \) . This resolution coincides with the coScarf resolution \( {\Delta }^{I,\mathbf{a}} \) . | Proof. The identification between \( {\Delta }_{{I}^{ * }} \), labeled as described earlier, and the cohull complex cohull \( \left( {I}^{\left\lbrack \mathbf{a}\right\rbrack }\right) \) is seen by tracing through the constructions of Section 5.4. Then apply Theorem 6.40. | No |
Suppose we are given the task of computing the minimal generators and the free resolution of the trivariate monomial ideal\n\n\[ I = \left\langle {x,{y}^{2},{z}^{3}}\right\rangle \cap \left\langle {{x}^{2},{y}^{3}, z}\right\rangle \cap \left\langle {{x}^{3}, y,{z}^{2}}\right\rangle . \] | Then what we do is to draw the Scarf complex \( {\Delta }_{{I}^{ * }} \) for \( {I}^{ * } = {I}^{\left\lbrack \mathbf{a}\right\rbrack } + \) \( \left\langle {{x}^{{a}_{1} + 1},{y}^{{a}_{2} + 1},{z}^{{a}_{3} + 1}}\right\rangle \) . This has been done in Section 3.2, with \( \mathbf{a} = \left( {3,3,3}\right) \) . Now re... | Yes |
Corollary 6.43 The number of minimal generators of an intersection of \( r \) irreducible monomial idels in \( n \) variables is at most \( {C}_{n - 1, n, r + n} - 1 \) . | For example, if we intersect 9 irreducible monomial ideals \( \left\langle {{a}^{i},{b}^{j},{c}^{k},{d}^{l}}\right\rangle \) in \( \mathbb{k}\left\lbrack {a, b, c, d}\right\rbrack \), then the number of minimal generators is at most 53 . That the bound is tight is seen by taking the Alexander dual of Example 6.32. | No |
Theorem 7.3 The semigroup ring \( \mathbb{k}\left\lbrack Q\right\rbrack \) is isomorphic to the quotient \( S/{I}_{L} \) . | Proof. Let \( {\mathbf{t}}^{{\mathbf{a}}_{1}},\ldots ,{\mathbf{t}}^{{\mathbf{a}}_{n}} \) denote the generators of the semigroup ring \( \mathbb{k}\left\lbrack Q\right\rbrack \) corresponding to the given generators of the semigroup \( Q \) . Then \( \mathbb{k}\left\lbrack Q\right\rbrack \) is the free \( \mathbb{k} \) ... | Yes |
Theorem 7.4 The following are equivalent.\n\n1. The lattice ideal \( {I}_{L} \) is prime.\n\n2. The semigroup ring \( \mathbb{k}\left\lbrack Q\right\rbrack \) is an integral domain (has no zerodivisors).\n\n3. The group generated by \( Q \) inside of \( A \) is free abelian.\n\n4. The semigroup \( Q \) is an affine sem... | Proof. Replacing \( A \) with the subgroup generated by \( Q \) if necessary, we may as well assume that \( Q \) generates \( A \) . The third and fourth conditions are equivalent because every free abelian group is isomorphic to \( {\mathbb{Z}}^{d} \) for some \( d \) . The equivalence of the first two conditions come... | Yes |
Proposition 7.5 The Krull dimension of \( \mathbb{k}\left\lbrack Q\right\rbrack \) equals \( n - \operatorname{rank}\left( L\right) \) . | Proof. As the statement does not involve \( A \), we again replace \( A \) with its subgroup generated by \( Q \) . The inclusion \( \mathbb{k}\left\lbrack Q\right\rbrack \subseteq \mathbb{k}\left\lbrack A\right\rbrack \) is the localization map inverting the elements \( {\mathbf{t}}^{{\mathbf{a}}_{1}},\ldots ,{\mathbf... | Yes |
Lemma 7.6 The lattice ideal \( {I}_{L} \) is computed from \( {I}_{\mathbf{L}} \) by taking the saturation with respect to the product of all the variables:\n\n\[ \n{I}_{L} = \left( {{I}_{\mathbf{L}} : {\left\langle {x}_{1}\cdots {x}_{n}\right\rangle }^{\infty }}\right) \n\]\n\nwhich by definition is the ideal \( \left... | Proof. Clearly \( {I}_{\mathbf{L}} \) is contained in \( {I}_{L} \) . On the other hand, consider any generator \( {\mathbf{x}}^{\mathbf{u}} - {\mathbf{x}}^{\mathbf{v}} \) of \( {I}_{L} \) . We can write \( \mathbf{u} - \mathbf{v} \) as a \( \mathbb{Z} \) -linear combination of the columns of \( \mathbf{L} \) ; hence \... | Yes |
Lemma 7.10 A subset \( F \) of \( Q \) is a face if and only if \( {P}_{F} \) is a prime ideal. | Proof. The subspace \( {P}_{F} \) is an ideal if and only if the implication \ | No |
Lemma 7.12 The map \( F \mapsto {\mathbb{R}}_{ \geq 0}F \) is a bijection from the set of faces of the semigroup \( Q \) to the set of faces of the cone \( {\mathbb{R}}_{ \geq 0}Q \) . In particular, the semigroup \( Q \) is pointed if and only if the associated cone \( {\mathbb{R}}_{ \geq 0}Q \) is pointed. | Proof. Let \( F \) be a subset of \( Q \) and consider the following linear system of equations and inequalities in an indeterminate vector \( \mathbf{w} \in {\mathbb{R}}^{d} \) :
\[
\mathbf{w} \cdot \mathbf{a} = 0\text{ for }\mathbf{a} \in F\;\text{ and }\;\mathbf{w} \cdot \mathbf{b} > 0\text{ for }\mathbf{b} \in Q \... | Yes |
Every monomial ideal \( I \) in any affine semigroup ring \( \mathbb{k}\left\lbrack Q\right\rbrack \) is an intersection of monomial ideals \( {I}_{F} \), at most one for each face \( F \), with \( {I}_{F} \) primary to \( {P}_{F} \) . | We will prove this in Corollary 11.5, which rests mainly on Proposition 8.11, where we indicate how to derive a more general statement from [Eis95, Exercise 3.5]. | No |
Every principal ideal in the ring \( \mathbb{k}\left\lbrack Q\right\rbrack \) from the previous example is pure of codimension 1. | In the semigroup ring\n\n\[\n\mathbb{k}\left\lbrack {Q}^{\prime }\right\rbrack = \mathbb{k}\left\lbrack {{s}^{4},{s}^{3}t, s{t}^{3},{t}^{4}}\right\rbrack = \mathbb{k}\left\lbrack {a, b, c, d}\right\rbrack /\left\langle {{bc} - {ad},{c}^{3} - b{d}^{2}, a{c}^{2} - {b}^{2}d,{b}^{3} - {a}^{2}c}\right\rangle ,\n\]\n\nthe pr... | Yes |
Proposition 7.15 Any pointed affine semigroup \( Q \) has a unique finite minimal generating set \( {\mathcal{H}}_{Q} \) . | Proof. Every pointed semigroup \( Q \) can be regarded as a partially ordered set (poset) via \( \mathbf{a} \preccurlyeq \mathbf{b} \) if \( \mathbf{b} - \mathbf{a} \in Q \) . Moreover, since \( \{ \mathbf{0}\} \) is a face of \( Q \), we can fix a vector \( \mathbf{w} \in {\mathbb{Z}}^{d} \) such that \( \mathbf{w} \c... | Yes |
Theorem 7.16 (Gordan’s Lemma) If \( C \) is a rational cone in \( {\mathbb{R}}^{d} \), then \( C \cap A \) is an affine semigroup for any subgroup \( A \) of \( {\mathbb{Z}}^{d} \) . | Proof. Since the intersection of \( C \) with the real subspace spanned by \( A \) is again a rational cone (with respect to the lattice \( A \) ), we may as well assume that \( A = {\mathbb{Z}}^{d} \) . What we are claiming is that \( C \cap {\mathbb{Z}}^{d} \) is finitely generated over \( \mathbb{N} \) . Since \( C ... | Yes |
Let \( C \) be a rational pointed cone in \( {\mathbb{R}}^{2} \). The Hilbert basis \( {\mathcal{H}}_{C} \) is constructed geometrically as follows. Let \( {\mathcal{P}}_{C} \) denote the unbounded polygon in \( {\mathbb{R}}^{2} \) obtained by taking the convex hull of all nonzero integer points in \( C \). The polygon... | \[ \det \left( {{\mathbf{a}}_{i},{\mathbf{a}}_{i + 1}}\right) = 1\;\text{ for }i = 1,\ldots, n - 1 \] because the triangle with vertices \( \left\{ {\mathbf{0},{\mathbf{a}}_{i},{\mathbf{a}}_{i + 1}}\right\} \) has no other lattice points in it (Exercise 7.11). It follows that there exists \( {\lambda }_{i} \in \mathbb{... | No |
Proposition 7.20 Assume \( C \subset {\mathbb{R}}^{d} \) is a pointed cone, and let \( {\nu }_{1},\ldots ,{\nu }_{m} \) be the primitive integer inner normals to the facets of \( C \) . Define the map \( \nu : {\mathbb{R}}^{d} \rightarrow {\mathbb{R}}^{m} \) sending \( \mathbf{a} \in {\mathbb{R}}^{d} \) to \( \left( {{... | Proof. The map \( \nu \) is injective precisely because \( C \) is pointed: the intersection of the kernels of \( {\nu }_{1},\ldots ,{\nu }_{m} \) is by definition the lineality space of \( C \) , which is zero for pointed cones. Moreover, a point \( \mathbf{a} \in {\mathbb{R}}^{d} \) lies in \( C \) if and only if all... | Yes |
Theorem 7.21 A vector \( \mathbf{a} \in {\mathbb{Z}}^{d} \) lies in the Hilbert basis \( {\mathcal{H}}_{C} \) if and only if the binomial \( {\mathbf{x}}^{\nu \cdot \mathbf{a}} - {\mathbf{y}}^{\nu \cdot \mathbf{a}} \) appears among the minimal generators of \( {I}_{\Lambda } \) . | Proof. We will equivalently prove that \( \mathbf{u} \in {\mathcal{H}}_{\nu \left( C\right) } \) if and only if \( {\mathbf{x}}^{\mathbf{u}} - {\mathbf{y}}^{\mathbf{u}} \) appears among the minimal generators of \( {I}_{\Lambda } \) . Consider a nonzero vector \( \mathbf{u} \) in \( {\mathbb{N}}^{m} \cap V \) . If \( \... | Yes |
Let us find all nonnegative integer solutions to the equation\n\n\[ 2{u}_{1} + 7{u}_{2} = 3{u}_{3} + 5{u}_{4} \] | The lattice of all integer solutions to this equation has the basis\n\n\[ \left( {-1,0,1, - 1}\right) ,\left( {-1,1,0,1}\right) ,\left( {2,1,2,1}\right) \text{.} \]\n\nUsing this basis we express the corresponding Lawrence ideal \( {I}_{\Lambda } \) as follows:\n\n\[ \left( {\left\langle {{x}_{1}{x}_{4}{y}_{3} - {x}_{3... | Yes |
Corollary 7.23 Every d-dimensional pointed affine semigroup can be embedded inside \( {\mathbb{N}}^{d} \) . | Proof. Given a pointed cone \( C \), define \( V \subseteq {\mathbb{R}}^{m} \) as in Proposition 7.20. Choose \( m - d \) standard basis vectors \( {\mathbf{e}}_{{i}_{1}},\ldots ,{\mathbf{e}}_{{i}_{m - d}} \) so that their images modulo \( V \) form a basis for \( {\mathbb{R}}^{m}/V \) . Then the coordinate subspace \(... | Yes |
The semigroup ring \( \mathbb{k}\left\lbrack {Q}_{\text{sat }}\right\rbrack \) of the saturation \( {Q}_{\text{sat }} \) is the normalization of the affine semigroup ring \( \mathbb{k}\left\lbrack Q\right\rbrack \) . | Proof. As earlier, we may as well forget the original \( {\mathbb{Z}}^{d} \) and instead refer to the subgroup \( A \) generated by \( Q \) as \( {\mathbb{Z}}^{d} \), after choosing a basis for it. Let \( {H}^{1},\ldots ,{H}^{r} \) be hyperplanes whose associated closed half-spaces \( {H}_{ > 0}^{i} \) intersect precis... | Yes |
Example 7.26 (Computing the saturation of an affine semigroup) For the semigroup generated by the underlined vectors in Example 7.22,\n\n\[ \nQ = \mathbb{N} \cdot \{ \left( {0,3,7,0}\right) ,\left( {0,5,0,7}\right) ,\left( {3,0,2,0}\right) ,\left( {5,0,0,2}\right) \} \subset {\mathbb{N}}^{4}, \n\]\n\nthe semigroup ring... | Our four vectors generate the rank 3 lattice defined by (7.7), and the cone \( {\mathbb{R}}_{ \geq 0}Q \) is the cone of nonnegative real solutions to (7.7). Therefore the saturation \( {Q}_{\text{sat }} \) is precisely the semigroup that was computed in Example 7.22. Its Hilbert basis consists of the 18 listed vectors... | No |
Theorem 7.27 (Bruns and Gubeladze, 1999) There exists a pointed saturated affine semigroup in \( {\mathbb{Z}}^{6} \) lacking the Carathéodory property. | Proof. Let \( C \) be the semigroup generated by the columns of the matrix\n\n\[ \mathbf{A} = \left\lbrack \begin{matrix} 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 1 & 2 \\ 0 & 1 & 0 & 0 & 0 & 2 & 0 & 2 & 1 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1 & 2 & 0 & 2 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 2 & 0 ... | Yes |
Proposition 7.28 The Krull dimension of \( S/I \) equals \( \dim \left( {{\Delta }_{w}\left( I\right) }\right) + 1 \) . | Proof. The three algebras \( S/I, S/{\operatorname{in}}_{w}\left( I\right) \), and \( S/\operatorname{rad}\left( {{\operatorname{in}}_{w}\left( I\right) }\right) \) have the same Krull dimension the first two because their Hilbert series are equal and the latter two because their zero sets are equal. The result follows... | Yes |
Example 7.31 Let \( {I}_{L} \) be the lattice ideal of a two-dimensional Hilbert basis as in Example 7.19. Choose \( w \in {\mathbb{R}}^{n} \) so that \( {w}_{i - 1} + {w}_{i + 1} > {\lambda }_{i} \cdot {w}_{i} \) for \( i = \) \( 2,\ldots, n - 1 \), where \( {\lambda }_{i} \) is the positive integer defined by (7.5). ... | For all indices \( i \) and \( j \) with \( j - i \geq 2 \) there exists a unique relation \[ {\mathbf{a}}_{i} + {\mathbf{a}}_{j} = \mu {\mathbf{a}}_{k} + \nu {\mathbf{a}}_{k + 1}\;\text{with}\;\mu ,\nu \in \mathbb{N}\text{and}i < k < k + 1 \leq j\text{.} \] The convexity in our construction implies that \( {x}_{i}{x}_... | Yes |
Lemma 7.32 \( {\mathcal{P}}_{w} \) is a simple polyhedron if \( {\operatorname{in}}_{w}\left( {I}_{L}\right) \) is a monomial ideal. | Proof. Let \( m = \operatorname{rank}\left( L\right) \), so that \( {\mathcal{P}}_{w} \) has dimension \( n - m \) . We are claiming that every vertex of \( {\mathcal{P}}_{w} \) misses exactly \( m \) facets. Suppose this is not the case. Then there is a vertex \( \mathbf{u} \in {\mathcal{P}}_{w} \) such that \( \left|... | Yes |
Theorem 7.33 The initial complex \( {\Delta }_{w}\left( {I}_{L}\right) \) of the lattice ideal \( {I}_{L} \) equals the simplicial complex polar to the boundary of the simple polyhedron \( {\mathcal{P}}_{w} \) . | Proof. Our assertion is equivalent to the following statement: a subset \( F \subseteq \{ 1,\ldots, n\} \) lies in \( {\Delta }_{w}\left( {I}_{L}\right) \) if and only if there exists \( \mathbf{u} \in {\mathcal{P}}_{w} \) such that \( \operatorname{supp}\left( \mathbf{u}\right) = \{ 1,\ldots, n\} \smallsetminus F \) .... | Yes |
Corollary 7.35 Let \( \mathbf{L} \) be a nonnegative integer \( m \times n \) matrix with no zero column, and let \( \mathbf{b} \) be a generic point in the cone spanned by the columns of \( \mathbf{L} \) . Then \( \mathcal{P} \) is a simple polytope. Its boundary complex is polar to the initial complex, with respect t... | Proof. The assumption that \( \mathbf{L} \) is nonnegative and has no zero column implies that \( {I}_{\mathbf{L}} = {I}_{L} \) . The corollary now follows from Theorem 7.33. | No |
We compute the polytope \( \mathcal{P} \) in (7.10) for\n\n\[ \mathbf{L} = \left\lbrack \begin{array}{llllll} 4 & 3 & 0 & 0 & 0 & 1 \\ 0 & 1 & 4 & 3 & 0 & 0 \\ 0 & 0 & 0 & 1 & 4 & 3 \end{array}\right\rbrack \;\text{ and }\;\mathbf{b} = \left\lbrack \begin{array}{l} 7 \\ 8 \\ 5 \end{array}\right\rbrack .\n\] | For \( w = \left\lbrack {1,1,1,1,1,0}\right\rbrack \in \mathcal{P} \), the ideal \( {I}_{\mathbf{L}} = {I}_{L} \) has the Gröbner basis\n\n\[ \left\{ {\underline{{x}_{1}^{4}{x}_{2}^{3}} - {x}_{6}^{2}{x}_{4}{x}_{5}^{4},\underline{{x}_{1}^{4}{x}_{2}^{2}{x}_{5}^{4}{x}_{6}^{4}} - {x}_{3}^{4}{x}_{4}^{2},\underline{{x}_{2}{x... | Yes |
The Hilbert basis for \( L \cap {\mathbb{N}}^{3} \) in Example 8.2 consists of the vectors \( \left( {4,2,0}\right) ,\left( {1,1,1}\right) \), and \( \left( {0,2,4}\right) \) . Hence the degree \( \mathbf{0} \) component is\n\n\[ \n{S}_{\left( 0,0\right) } = \mathbb{k}\left\lbrack {{x}^{4}{y}^{2},{xyz},{y}^{2}{z}^{4}}\... | This normal affine semigroup ring is isomorphic to \( \mathbb{k}\left\lbrack {u, v, w}\right\rbrack /\left\langle {{uw} - {v}^{4}}\right\rangle \) . | Yes |
Proposition 8.4 Each graded component \( {S}_{\mathbf{a}} \) of a multigraded polynomial ring is a finitely generated module over the normal semigroup ring \( {S}_{\mathbf{0}} \) . | Proof. We may assume that \( \mathbf{a} \in Q \) and that we are given one monomial \( {\mathbf{x}}^{\mathbf{u}} \) lying in \( {S}_{\mathbf{a}} \) . Here is an algorithm that computes a finite generating set for \( {S}_{\mathbf{a}} \) as a module over \( {S}_{\mathbf{0}} \) . Form the sublattice \( {L}_{\mathbf{a}} \)... | Yes |
Let us illustrate for Example 8.2 the procedure in the proof of Proposition 8.4 in degree \( \mathbf{a} = \left( {4,1}\right) \) . A typical monomial of that degree is \( {\mathbf{x}}^{\mathbf{u}} \) with \( \mathbf{u} = \left( {5,2,3}\right) \), so \( {L}_{\mathbf{a}} \) is the rank 3 sublattice of \( {\mathbb{Z}}^{4}... | The Hilbert basis of \( {L}_{\mathbf{a}} \cap {\mathbb{N}}^{3 + 1} \) consists of the nine vectors\n\n\[ \left( {0,2,4,0}\right) ,\left( {4,2,0,0}\right) ,\left( {1,1,1,0}\right) ,\left( {0,1,6,1}\right) ,\left( {1,0,3,1}\right) \text{,} \]\n\n\[ \left( {3,0,1,1}\right) ,\left( {6,1,0,1}\right) ,\left( {0,0,8,2}\right)... | Yes |
Theorem 8.6 The following conditions are equivalent for a polynomial ring \( S \) multigraded by \( A \), with image semigroup \( Q = \deg \left( {\mathbb{N}}^{n}\right) \). 1. There exists \( \mathbf{a} \in Q \) such that the vector space \( {S}_{\mathbf{a}} \) is finite-dimensional. 2. The only polynomials of degree ... | Proof. \( \left( 1\right) \Rightarrow \left( 2\right) \) : For any \( \mathbf{a} \in Q \) there exists some nonconstant monomial \( {\mathbf{x}}^{\mathbf{u}} \) of degree \( \mathbf{a} \). Multiplication by \( {\mathbf{x}}^{\mathbf{u}} \) defines a monomorphism of vector spaces over \( \mathbb{k} \) from \( {S}_{\mathb... | Yes |
Corollary 8.8 If \( S \) is positively graded by \( A \), then the semigroup \( Q = \) \( \deg \left( {\mathbb{N}}^{n}\right) \) is pointed, and hence can be embedded in \( {\mathbb{N}}^{d} \) for \( d = \operatorname{rank}\left( A\right) \) . | Proof. This statement follows from Corollary 7.23 and Theorem 8.6.6. | No |
Example 8.9 Consider the subsemigroup \( Q \) generated by the two elements \( \mathbf{a} = \left( {1,1}\right) \) and \( \mathbf{b} = \left( {1,0}\right) \) in the group \( A = \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \), so that \( \mathbf{a} - \mathbf{b} \) has order 2 in \( A \) . The degree map \( {\mathbb{N}}^{2}... | But there can be no embedding of \( Q \) into \( {\mathbb{N}}^{d} \) for any \( d \), because although no element of \( Q \) itself has finite order, the group \( A \) contains the torsion element \( \mathbf{a} - \mathbf{b} \) . | Yes |
In the situation of Example 8.2, the polynomial \( {y}^{2}{z}^{4} - 1 \) is homogeneous for the given grading by \( \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \). However, the ideal it generates equals the intersection | \[ \left\langle {{y}^{2}{z}^{4} - 1}\right\rangle = \left\langle {y{z}^{2} - 1}\right\rangle \cap \left\langle {y{z}^{2} + 1}\right\rangle \] of two principal prime ideals neither of whose generators are homogeneous. Indeed, \( \deg \left( {y{z}^{2}}\right) = \left( {0,1}\right) \), whereas \( \deg \left( 1\right) = \l... | Yes |
Proposition 8.11 Let \( S \) be multigraded by a torsion-free abelian group \( A \) . All associated primes of multigraded S-modules are multigraded. | Proof. This is [Eis95, Exercise 3.5]. The proof, based on that of the corresponding \( \mathbb{Z} \) -graded statement in [Eis95, Section 3.5], is essentially presented in the aforementioned exercise from [Eis95]. It works because torsion-free grading groups \( A \cong {\mathbb{Z}}^{d} \) can be totally ordered, for in... | No |
Example 8.13 Let \( S = \mathbb{k}\left\lbrack {a, b, c, d}\right\rbrack \) be multigraded by \( {\mathbb{Z}}^{2} \), with \( \deg \left( a\right) = \) \( \left( {4,0}\right) ,\deg \left( b\right) = \left( {3,1}\right) ,\deg \left( c\right) = \left( {1,3}\right) \), and \( \deg \left( d\right) = \left( {0,4}\right) \) ... | Although the semigroup \( {Q}^{\prime } \) does not generate \( {\mathbb{Z}}^{2} \) as a group, the \( {\mathbb{Z}}^{2} \) -graded translate \( M = \mathbb{k}\left\lbrack {Q}^{\prime }\right\rbrack \left( {-\left( {1,1}\right) }\right) \) of the semigroup ring \( \mathbb{k}\left\lbrack {Q}^{\prime }\right\rbrack \) is ... | No |
Lemma 8.15 Elements in the completion \( \mathbb{Z}\left\lbrack \left\lbrack Q\right\rbrack \right\rbrack \) for a pointed semigroup \( Q \) can be expressed uniquely as formal series \( \mathop{\sum }\limits_{{\mathbf{a} \in Q}}{c}_{\mathbf{a}}{\mathbf{t}}^{\mathbf{a}} \) with \( {c}_{\mathbf{a}} \in \mathbb{Z} \) . | Proof. The lemma is standard when \( Q = {\mathbb{N}}^{n} \), as \( \mathbb{Z}\left\lbrack \left\lbrack {\mathbb{N}}^{n}\right\rbrack \right\rbrack = \mathbb{Z}\left\lbrack \left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \right\rbrack \) is an honest power series ring. For a general pointed semigroup \( Q \), write... | Yes |
Lemma 8.16 The Hilbert series for the multigraded polynomial ring \( S \) is\n\n\[ H\left( {S;\mathbf{t}}\right) = \frac{1}{\left( {1 - {\mathbf{t}}^{{\mathbf{a}}_{1}}}\right) \cdots \left( {1 - {\mathbf{t}}^{{\mathbf{a}}_{n}}}\right) } \]\n\nas an element in \( \mathbb{Z}\left\lbrack \left\lbrack Q\right\rbrack \right... | Proof. Let \( H\left( {S;\mathbf{x}}\right) \) be the Hilbert series of \( S \) in the fine multigrading by \( {\mathbb{N}}^{n} \) . Viewed as a power series, the image of \( H\left( {S;\mathbf{x}}\right) \) under the surjection \( \mathbb{Z}\left\lbrack \left\lbrack {\mathbb{N}}^{n}\right\rbrack \right\rbrack \rightar... | Yes |
Let \( A = {\mathbb{Z}}^{2} \), and consider the subsemigroup \( Q \subset A \) generated by the four vectors \( \left( {4,0}\right) ,\left( {3,1}\right) ,\left( {1,3}\right) \), and \( \left( {0,4}\right) \) . [This is the semigroup from Examples 7.14 and 8.13, where it was called \( {Q}^{\prime } \) .] In this case, | \[ \mathbb{Z}\left\lbrack \left\lbrack Q\right\rbrack \right\rbrack \cong \mathbb{Z}\left\lbrack \left\lbrack {a, b, c, d}\right\rbrack \right\rbrack /\left\langle {{bc} - {ad},{c}^{3} - b{d}^{2}, a{c}^{2} - {b}^{2}d,{b}^{3} - {a}^{2}c}\right\rangle \] is the ring of power series supported on \( Q \) . The subgroup \( ... | Yes |
Proposition 8.18 Every finitely generated multigraded module \( M \) over \( S \) has a finite multigraded resolution by multigraded free modules of the form \( S\left( {-{\mathbf{b}}_{1}}\right) \oplus \cdots \oplus S\left( {-{\mathbf{b}}_{r}}\right) \), even if the grading is not positive. | Proof. Calculate a finite free resolution of \( M \) using Gröbner bases. The reduced Gröbner basis for a graded submodule of a multigraded free module is homogeneous for the given multigrading. Therefore each free module in the resolution has the desired form automatically. | Yes |
Theorem 8.20 The Hilbert series of a finitely generated graded module \( M \) over a polynomial ring positively multigraded by \( A \) is a Laurent series supported on finitely many translates of \( Q = \deg \left( {\mathbb{N}}^{n}\right) \) . More precisely, there is a unique Laurent polynomial \( \mathcal{K}\left( {M... | Proof. By the obvious equality \( H\left( {M\left( {-\mathbf{a}}\right) ;\mathbf{t}}\right) = {\mathbf{t}}^{\mathbf{a}}H\left( {M;\mathbf{t}}\right) \) for graded translates of arbitrary finitely generated modules \( M \) and the fact that Hilbert series are additive on direct sums, we deduce from Lemma 8.16 that\n\n\[... | Yes |
Proposition 8.23 If \( M \) is a finitely generated positively multigraded module, then the \( K \) -polynomial records the alternating sum of its Betti numbers:\n\n\[ \mathcal{K}\left( {M;\mathbf{t}}\right) = \mathop{\sum }\limits_{\substack{{\mathbf{a} \in A} \\ {i \geq 0} }}{\left( -1\right) }^{i}{\beta }_{i,\mathbf... | Proof. Use the proof of Theorem 8.20 on a minimal free resolution of \( M \) : if the \( {i}^{\text{th }} \) homological degree of this resolution is \( S\left( {-{\mathbf{b}}_{1}}\right) \oplus \cdots \oplus S\left( {-{\mathbf{b}}_{r}}\right) \) , then (8.2) contributes \( {\left( -1\right) }^{i}\mathop{\sum }\limits_... | Yes |
For any multigrading of \( S \) by \( A \), the Hilbert series of the residue field \( \mathbb{k} = S/\left\langle {{x}_{1},\ldots ,{x}_{n}}\right\rangle \) is just \( 1 \in \mathbb{Z}\left\lbrack \left\lbrack Q\right\rbrack \right\rbrack \left\lbrack A\right\rbrack \). | This agrees with the calculation of its \( K \) -polynomial from the Koszul complex, which yields\n\n\[ \mathcal{K}\left( {S/\left\langle {{x}_{1},\ldots ,{x}_{n}}\right\rangle ;\mathbf{t}}\right) = \mathop{\sum }\limits_{{\Lambda \subseteq \{ 1,\ldots, n\} }}{\left( -1\right) }^{\left| \Lambda \right| }{\mathbf{t}}^{\... | Yes |
Proposition 8.26 If \( \widetilde{K} \) is the homogenization of a submodule \( K \) of a free module \( \mathcal{F} \) with respect to a weight order \( \left( {w,\varepsilon }\right) \), then \( \widetilde{K}/\left( {y - 1}\right) \widetilde{K} \cong K \) and \( \widetilde{K}/y\widetilde{K} \cong {\operatorname{in}}_... | Proof. The isomorphisms are immediate from Definition 8.25. For freeness, the proof of [Eis95, Theorem 15.17] works here mutatis mutandis. | No |
Lemma 8.27 Let \( R \) be a ring and \( \widetilde{\mathcal{F}} \) . a free resolution of an \( R\left\lbrack y\right\rbrack \) -module \( M \) over \( R\left\lbrack y\right\rbrack \) . If \( y \) is a nonzerodivisor on \( M \), then \( {\widetilde{\mathcal{F}}}_{ \bullet }/y{\widetilde{\mathcal{F}}}_{ \bullet } \) is ... | Proof. The homology of \( {\widetilde{\mathcal{F}}}_{ \bullet }/y{\widetilde{\mathcal{F}}}_{ \bullet } \) is \( {\operatorname{Tor}}_{ \bullet }^{R\left\lbrack y\right\rbrack }\left( {R, M}\right) \), which can also be calculated by tensoring the \( R\left\lbrack y\right\rbrack \) -free resolution\n\n\[ 0 \leftarrow R\... | Yes |
Proposition 8.28 Adopt all of the notation from Definition 8.25, and fix an \( S\left\lbrack y\right\rbrack \) -free resolution \( \widetilde{\mathcal{F}} \) . of \( \mathcal{F}\left\lbrack y\right\rbrack /\widetilde{K} \) that is multigraded by \( A \times \mathbb{Z} \) . Then\n\n1. \( {\widetilde{\mathcal{F}}}_{ \bul... | Proof. Both \( y \) and \( y - 1 \) are nonzerodivisors on \( \mathcal{F}\left\lbrack y\right\rbrack /\widetilde{K} \) by the freeness in Proposition 8.26. Therefore, ignoring the grading for the moment, \( {\widetilde{\mathcal{F}}}_{ \bullet }/y{\widetilde{\mathcal{F}}}_{ \bullet } \) and \( {\widetilde{\mathcal{F}}}_... | Yes |
Theorem 8.29 (Upper-semicontinuity) Fix a graded submodule \( K \) of a graded free module \( \mathcal{F} \) over a positively multigraded polynomial ring. If \( \operatorname{in}\left( K\right) \) is the initial submodule of \( K \) for some term order or weight order, then\n\n\[{\beta }_{i,\mathbf{a}}\left( {\mathcal... | Proof. Assume that the free resolution \( \widetilde{\mathcal{F}} \) . in Proposition 8.28 is minimal, which we can do because the multigrading on \( S\left\lbrack y\right\rbrack \) is positive. Since \( y \) lies in the graded maximal ideal of \( S\left\lbrack y\right\rbrack \), the free resolution \( {\widetilde{\mat... | Yes |
Theorem 8.34 If \( S \) is an arbitrary (perhaps not positively) multigraded polynomial ring and \( M \) is a finitely generated multigraded module, then the \( K \) -polynomials of \( M \) relative to any two finite free resolutions are equal. | Sketch of proof. Define a multigraded free resolution \( {\mathcal{F}}_{ \bullet } \) to dominate another such resolution \( {\mathcal{F}}_{ \bullet }^{\prime } \) if there is a surjection \( {\mathcal{F}}_{ \bullet } \rightarrow {\mathcal{F}}_{ \bullet }^{\prime } \) of resolutions, in the sense that \( {\mathcal{F}}_... | Yes |
Example 8.35 Take \( S = \mathbb{k}\left\lbrack {x, y}\right\rbrack \) and set \( A = \mathbb{Z} \), with \( \deg \left( x\right) = 1 \) and \( \deg \left( y\right) = - 1 \) . If \( I = \langle {xy} - 1\rangle \), which is homogeneous for this multigrading, then \( I \) can also be represented as the ideal \( \left\lan... | Nevertheless, the two free resolutions\n\n\[ 0 \leftarrow S\overset{\left\lbrack xy - 1\right\rbrack }{ \leftarrow }S \leftarrow 0\;\text{ and }\;0 \leftarrow S\overset{\left\lbrack {x}^{2}{y}^{2} - xy\;{x}^{2}{y}^{2} - 1\right\rbrack }{ \leftarrow }{S}^{2}\overset{\left\lbrack \begin{matrix} {xy} + 1 \\ {xy} \end{matr... | Yes |
Theorem 8.36 Fix a term order on a multigraded free module \( \mathcal{F} \) . If \( K \) is a graded submodule of \( \mathcal{F} \) then \( \mathcal{K}\left( {\mathcal{F}/K;\mathbf{t}}\right) = \mathcal{K}\left( {\mathcal{F}/\operatorname{in}\left( K\right) ;\mathbf{t}}\right) \) . | Proof. Immediate from Proposition 8.28 and Theorem 8.34. | No |
Example 8.38 Let \( I = \langle {xy},{yz}\rangle = \langle y\rangle \cap \langle x, z\rangle \), and consider the module \( M = \mathbb{k}\left\lbrack {x, y, z}\right\rbrack /I \), with the multigrading as in Example 8.2. This module is modest, and the Hilbert function \( \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \right... | \[ H\left( {M;s, t}\right) = \frac{{s}^{-2}t}{1 - {s}^{-2}t} + \frac{1}{\left( {1 - {st}}\right) \left( {1 - s}\right) }, \] where the two ratios - which are viewed as lying in the two subgroups above - sum the positive powers of \( y \) and the monomials in \( \mathbb{k}\left\lbrack {x, z}\right\rbrack \) . | Yes |
Theorem 8.41 If \( M \) is finitely generated and modest, then\n\n\[ H\left( {M;\mathbf{t}}\right) \equiv \frac{\mathcal{K}\left( {M;\mathbf{t}}\right) }{\mathop{\prod }\limits_{{i = 1}}^{n}\left( {1 - {\mathbf{t}}^{{\mathbf{a}}_{i}}}\right) } \] | Proof. Express \( M \) as a quotient \( \mathcal{F}/N \) of a multigraded free module. The Hilbert series of the modest quotient \( \mathcal{F}/N \) equals that of \( \mathcal{F}/\operatorname{in}\left( N\right) \) under any term order, because the standard monomials for the initial submodule in \( \left( N\right) \) f... | Yes |
Two open problems concerning Hilbert series and \( K \) -polynomials are\n\n(i) how to represent the kernel of the homomorphism of abelian groups \( \{ \) Hilbert series of modest modules \( \} \rightarrow \{ K \) -polynomials \( \} \) ; and\n\n(ii) how to write down Hilbert series for immodest modules. | The map \{Hilbert series\} \( \rightarrow \{ K \) -polynomials \( \} \) in (i) is never injective when the grading is nonpositive: there can be many modest modules, with very different-looking Hilbert series, that nonetheless have equal \( K \) -polynomials. Such ambiguity does not occur in the positively graded case b... | No |
Theorem 8.44 Exactly one additive degenerative function \( \mathcal{C} \) satisfies\n\n\[ \mathcal{C}\left( {S/\left\langle {{x}_{{i}_{1}},\ldots ,{x}_{{i}_{r}}}\right\rangle ;\mathbf{t}}\right) = \left\langle {{\mathbf{a}}_{{i}_{1}},\mathbf{t}}\right\rangle \cdots \left\langle {{\mathbf{a}}_{{i}_{r}},\mathbf{t}}\right... | Proof. If \( M \) is a finitely generated graded module, then let \( M \cong \mathcal{F}/K \) be a graded presentation, and pick a term order on \( \mathcal{F} \) . Set \( {M}^{\prime } = \mathcal{F}/\operatorname{in}\left( K\right) \) , so \( \mathcal{C}\left( {M;\mathbf{t}}\right) = \mathcal{C}\left( {{M}^{\prime };\... | Yes |
Let \( S = \mathbb{k}\left\lbrack {a, b, c, d}\right\rbrack \) be multigraded by \( {\mathbb{Z}}^{2} \), with\n\n\[ \deg \left( a\right) = \left( {2, - 1}\right) ,\deg \left( b\right) = \left( {1,0}\right) ,\deg \left( c\right) = \left( {0,1}\right) ,\text{ and }\deg \left( d\right) = \left( {-1,2}\right) .\n\]\n\nIf \... | because of the Scarf resolution. Gathering \( - {t}_{1}^{2} + {t}_{1}^{2}{t}_{2} = - {t}_{1}^{2}\left( {1 - {t}_{2}}\right) \), we get\n\n\[ \mathcal{K}\left\lbrack \left\lbrack {M;\mathbf{1} - \mathbf{t}}\right\rbrack \right\rbrack = 1 - {\left( 1 - {t}_{1}\right) }^{2}{t}_{2} - {t}_{1}{\left( 1 - {t}_{2}\right) }^{2}... | Yes |
Lemma 8.48 Let \( \mathbf{b} \in {\mathbb{Z}}^{d} \) . If \( K\left( \mathbf{t}\right) = 1 - {\mathbf{t}}^{\mathbf{b}} = 1 - {t}_{1}^{{b}_{1}}\cdots {t}_{d}^{{b}_{d}} \), then substituting \( 1 - {t}_{j} \) for each occurrence of \( {t}_{j} \) yields \( K\left\lbrack \left\lbrack {\mathbf{1} - \mathbf{t}}\right\rbrack ... | Proof. \( K\left\lbrack \left\lbrack {\mathbf{1} - \mathbf{t}}\right\rbrack \right\rbrack = 1 - \mathop{\prod }\limits_{{j = 1}}^{d}{\left( 1 - {t}_{j}\right) }^{{b}_{j}} = 1 - \mathop{\prod }\limits_{{j = 1}}^{d}\left( {1 - {b}_{j}{t}_{j} + O\left( {t}_{j}^{2}\right) }\right) \), and this equals \( 1 - \left( {1 - \ma... | Yes |
Proposition 8.49 \( K \) -polynomials of prime monomial quotients satisfy\n\n\[ \mathcal{K}\left\lbrack \left\lbrack {S/\left\langle {{x}_{{i}_{1}},\ldots ,{x}_{{i}_{r}}}\right\rangle ;\mathbf{1} - \mathbf{t}}\right\rbrack \right\rbrack = \left( {\mathop{\prod }\limits_{{\ell = 1}}^{r}\left\langle {{\mathbf{a}}_{{i}_{\... | Proof. Using the Koszul complex, the \( K \) -polynomial of the quotient module \( M = S/\left\langle {{x}_{{i}_{1}},\ldots ,{x}_{{i}_{r}}}\right\rangle \) is computed to be \( \mathcal{K}\left( {M;\mathbf{t}}\right) = \left( {1 - {\mathbf{t}}^{{\mathbf{a}}_{{i}_{1}}}}\right) \cdots \left( {1 - {\mathbf{t}}^{{\mathbf{a... | Yes |
Example 8.50 Consider the ring \( S = \mathbb{k}\left\lbrack {a, b, c, d}\right\rbrack \) multigraded by \( {\mathbb{Z}}^{2} \) with\n\n\[ \deg \left( a\right) = \left( {3,0}\right) ,\deg \left( b\right) = \left( {2,1}\right) ,\deg \left( c\right) = \left( {1,2}\right) \text{, and}\deg \left( d\right) = \left( {0,3}\ri... | Note that these multidegrees all lie inside the ring\n\n\[ \mathbb{Z}\left\lbrack {3{s}_{1},{s}_{1} + 2{s}_{2},2{s}_{1} + {s}_{2},3{s}_{2}}\right\rbrack = \mathbb{Z}\left\lbrack {{s}_{1} + 2{s}_{2},2{s}_{1} + {s}_{2}}\right\rbrack \]\n\nand not just \( \mathbb{Z}\left\lbrack {{s}_{1},{s}_{2}}\right\rbrack \), since the... | Yes |
Corollary 8.52 If \( M = S/\left\langle {{x}_{{i}_{1}},\ldots ,{x}_{{i}_{r}}}\right\rangle \) then \( \mathcal{C}\left( {M\left( {-\mathbf{b}}\right) ;\mathbf{t}}\right) = \mathcal{C}\left( {M;\mathbf{t}}\right) \) . | Proof. Shifting by \( \mathbf{b} \) multiplies the \( K \) -polynomial by \( {\mathbf{t}}^{\mathbf{b}} \), so \( \mathcal{K}\left( {M\left( {-\mathbf{b}}\right) ;\mathbf{t}}\right) = \) \( {\mathbf{t}}^{\mathbf{b}}\mathcal{K}\left( {M;\mathbf{t}}\right) \) . The degree \( r \) form in \( \mathcal{K}\left\lbrack \left\l... | Yes |
Theorem 9.2 The Betti number \( {\beta }_{j,\mathbf{b}} \) of \( {I}_{L} \) equals the dimension over \( \mathbb{k} \) of the \( {j}^{\text{th }} \) reduced homology group \( {\widetilde{H}}_{j}\left( {{\Delta }_{\mathbf{b}};\mathbb{k}}\right) \) . | Proof. The proof has the same structure as many of the proofs in Part I: find a multigraded complex of free \( S \) -modules with the appropriate homology, and identify the graded pieces of this complex as the desired reduced chain complexes. As in Lemma 1.32, the desired Betti number can be expressed as \( {\beta }_{j... | Yes |
Corollary 9.4 The projective dimension of \( {I}_{L} \) is at most \( {2}^{n - d} - 2 \) . | Proof. Let \( {F}_{1},{F}_{2},\ldots ,{F}_{s} \) denote the distinct facets (maximal faces) of \( {\Delta }_{\mathbf{b}} \) . There exist monomials \( {\mathbf{x}}^{{\mathbf{u}}_{1}},{\mathbf{x}}^{{\mathbf{u}}_{2}},\ldots ,{\mathbf{x}}^{{\mathbf{u}}_{s}} \) of degree \( \mathbf{b} \) such that \( \operatorname{supp}\le... | Yes |
Choose the lattice \( L \) in \( {\mathbb{Z}}^{8} \) with basis given by the rows of\n\n\[ \mathbf{L} = \left\lbrack \begin{array}{rrrrrrrr} 1 & 1 & 1 & 2 & - 2 & - 1 & - 1 & - 1 \\ 1 & 2 & - 2 & - 1 & 1 & 1 & - 1 & - 1 \\ 1 & - 1 & 1 & - 1 & 1 & - 1 & 2 & - 2 \end{array}\right\rbrack \] | This matrix has the properties that all eight sign patterns appear among its columns and its maximal minors are relatively prime. The latter condition ensures that \( Q = {\mathbb{N}}^{8}/L \) is an affine semigroup. The ideal \( {I}_{L} \) has 13 minimal generators, and its minimal free resolution looks like\n\n\[ 0 \... | Yes |
A polyhedral cell complex is locally finite if every face meets finitely many others. In general, hull complexes of Laurent monomial modules need not be locally finite. For example, consider the Laurent monomial module \( M \) over \( \mathbb{k}\left\lbrack {x, y, z}\right\rbrack \) generated by \( y/x \) and \( {\left... | \[ M = \left\langle \frac{y}{x}\right\rangle + \left\langle {\left. {\left( \frac{y}{z}\right) }^{i}\right| \;i \in \mathbb{Z}}\right\rangle . \] The vertex \( y/x \) lies on infinitely many edges of hull \( \left( M\right) \) . Only one of these edges is needed in the minimal free resolution of \( M \) over \( \mathbb... | Yes |
The Laurent monomial module in Example 9.7 is the lattice module \( {M}_{L} \) for the lattice \( L = \ker \left( {1,1}\right) = \left\{ {\left( {u, - u}\right) \in {\mathbb{Z}}^{2} \mid u \in \mathbb{Z}}\right\} \) . More generally, consider the lattice \( L = \ker \left( {1,1,\ldots ,1}\right) \), which consists of a... | Let us write a lattice module \( {M}_{L} \) in terms of generators and relations. There is one generator \( {\mathbf{e}}_{\mathbf{u}} \) for each element \( \mathbf{u} \) in the lattice \( L \), and \( {M}_{L} \) is the free \( S \) -module on the generators \( \left\{ {{\mathbf{e}}_{\mathbf{u}} \mid \mathbf{u} \in L}\... | Yes |
Theorem 9.14 The hull complex of a lattice module is locally finite. | Proof. We claim that the vertex \( \mathbf{0} \in L \) is incident to only finitely many edges of hull \( \left( {M}_{L}\right) \) . This claim implies the theorem because (i) the lattice \( L \) acts transitively on the vertices of hull \( \left( {M}_{L}\right) \), so it suffices to consider the vertex \( \mathbf{0} \... | Yes |
Theorem 9.17 The functor \( \pi : \mathcal{A} \rightarrow \mathcal{B} \) sending \( M \) to \( M/L \) is an equivalence of categories. | Proof. By condition (iii) of [MacL98, Theorem IV.4.1], we must show that\n\n- \( \pi \) is fully faithful, meaning that \( \pi \) induces a natural identification \( {\operatorname{Hom}}_{\mathcal{A}}\left( {M,{M}^{\prime }}\right) = {\operatorname{Hom}}_{\mathcal{B}}\left( {\pi \left( M\right) ,\pi \left( {M}^{\prime ... | No |
Theorem 9.20 The hull resolution of the semigroup ring \( S/{I}_{L} \) is a finite \( {\mathbb{Z}}^{n}/L \) -graded free resolution of length \( \leq n \) . | Proof. The lattice \( L \) acts freely on hull \( \left( {M}_{L}\right) \), which implies that \( {\mathcal{F}}_{\text{hull }\left( {M}_{L}\right) } \) is a free \( S\left\lbrack L\right\rbrack \) -module. Since \( \pi \) (free \( S\left\lbrack L\right\rbrack \) -module) is a free \( S \) -module, the hull resolution o... | Yes |
Example 9.22 Suppose that \( L \) is a unimodular lattice. This means that for all subsets \( \sigma \subseteq \{ 1,\ldots, n\} \), the group \( {\mathbb{Z}}^{n}/\left( {L + \mathop{\sum }\limits_{{i \in \sigma }}\mathbb{Z}{\mathbf{e}}_{i}}\right) \) is torsion-free. This property holds for an affine semigroup \( Q = \... | Consider the Lawrence lifting \( \Lambda \left( L\right) = \left\{ {\left( {\mathbf{u}, - \mathbf{u}}\right) \in {\mathbb{Z}}^{2n} \mid \mathbf{u} \in L}\right\} \), which is also a unimodular lattice, but now in \( {\mathbb{Z}}^{2n} \) . Its corresponding lattice ideal is\n\n\[ \n{I}_{\Lambda \left( L\right) } = \left... | Yes |
Theorem 9.24 For generic Laurent monomial modules \( M \), the following coincide:\n\n1. The Scarf complex of \( M \)\n\n2. The hull resolution of \( M \)\n\n3. The minimal free resolution of \( M \) | Proof. The proof of Theorem 6.13 carries over from monomial ideals to Laurent monomial modules. | No |
Corollary 9.25 The minimal free resolution of a generic lattice ideal \( {I}_{L} \) is its Scarf complex, which is the image under \( \pi \) of the Scarf complex of \( {M}_{L} \) . | The lattice \( L \) in Example 9.21 is generic because all three generators of \[ {I}_{L} = \left\langle {{x}_{1}{x}_{2}^{2} - {x}_{3}^{2},{x}_{1}{x}_{3} - {x}_{2}^{3},{x}_{2}{x}_{3} - {x}_{1}^{2}}\right\rangle \] have full support. The Scarf complex of \( {M}_{L} \) coincides with the hull complex depicted in Fig. 9.1... | Yes |
Example 9.26 Things become much more complicated in four dimensions. The smallest codimension 1 generic lattice module in four variables is determined by the lattice \( L = \ker \left( \left\lbrack \begin{array}{llll} {20} & {24} & {25} & {31} \end{array}\right\rbrack \right) \subset {\mathbb{Z}}^{4} \) . The lattice i... | \[ {M}_{L} = S\left\lbrack L\right\rbrack /\left\langle {{a}^{4} - {bcd}{\mathbf{z}}^{ * },{a}^{3}{c}^{2} - {b}^{2}{d}^{2}{\mathbf{z}}^{ * },{a}^{2}{b}^{3} - {c}^{2}{d}^{2}{\mathbf{z}}^{ * }, a{b}^{2}c - {d}^{3}{\mathbf{z}}^{ * },}\right. \] \[ {b}^{4} - {a}^{2}{cd}{\mathbf{z}}^{ * },{b}^{3}{c}^{2} - {a}^{3}{d}^{2}{\ma... | Yes |
Theorem 9.27 (Barany and Scarf) The closure of \( {\mathcal{T}}_{d, n} \) has measure zero in the closure of \( {\mathcal{S}}_{d, n} \) in \( {\mathbb{R}}^{d \times n} \) . | Proof. Condition (A3) in the article [BaS96] by Barany and Scarf describes an open set of matrices \( \mathbf{L} \) that represent generic lattices. Theorem 1 in [BaS96] shows that the set of all generic lattices with a fixed Scarf complex is an open polyhedral cone. The union of these cones is a dense subset in the cl... | No |
Lemma 10.2 A polynomial \( f \in S \) is a common eigenvector for \( G \) if and only if it is homogeneous under the multigrading by \( A \) . In particular, \( f \in S \) is fixed by \( G \) if and only if it is homogeneous of degree \( \mathbf{0} \), so \( \deg \left( f\right) \in L \cap {\mathbb{N}}^{n} \) . | Proof. If \( \zeta = \left( {{\zeta }_{1},\ldots ,{\zeta }_{n}}\right) \) represents an element in \( G \), then the image of a polynomial \( f\left( {{x}_{1},\ldots ,{x}_{n}}\right) = \sum {c}_{\mathbf{u}}{\mathbf{x}}^{\mathbf{u}} \) under \( \zeta \) can be computed as follows:\n\n\[ f\left( {{\zeta }_{1}{x}_{1},\ldo... | Yes |
Lemma 10.3 An ideal \( I \) inside \( S \) is stable under the action of \( G \) (that is, \( G \cdot I = I \) ) if and only if \( I \) is homogeneous for the multigrading by \( A \) . | Proof. Every homogeneous ideal is generated by homogeneous polynomials, which are simultaneous eigenvectors for all of \( G \) by Lemma 10.2. Therefore such ideals are stable under the action of \( G \) . For the converse, suppose that \( I \) is a \( G \) -stable ideal and \( f \in I \) . It suffices to prove that eve... | Yes |
In Example 10.1, the invariant ring for the action of \( G \cong \) \( {\mathbb{C}}^{ * } \times {\mathbb{Z}}_{2} \) equals | \[ \mathbb{C}{\left\lbrack {x}_{1},{x}_{2},{x}_{3}\right\rbrack }^{G} = \mathbb{C}\left\lbrack {{x}_{1}^{4}{x}_{2}^{2},{x}_{1}{x}_{2}{x}_{3},{x}_{2}^{2}{x}_{3}^{4}}\right\rbrack \cong \mathbb{C}\left\lbrack {u, v, w}\right\rbrack /\left\langle {{uw} - {v}^{4}}\right\rangle . \] The inclusion of this ring into \( \mathb... | Yes |
Example 10.6 (Veronese rings) Fix a positive integer \( p \) and let \( L \) denote the sublattice of \( {\mathbb{Z}}^{n} \) consisting of all vectors whose coordinate sum is divisible by \( p \) . Then \( A = {\mathbb{Z}}^{n}/L \) is isomorphic to the cyclic group \( \mathbb{Z}/p\mathbb{Z} \), and the grading of \( S ... | \[ {\mathcal{H}}_{Q} = \left\{ {\left( {{i}_{1},{i}_{2},\ldots ,{i}_{n}}\right) \in {\mathbb{N}}^{n} \mid {i}_{1} + {i}_{2} + \cdots + {i}_{n} = p}\right\} . \] The ring \( {S}^{G} \) is the \( {p}^{\text{th }} \) Veronese subring of the polynomial ring \( S \) . | Yes |
Proposition 10.9 The map \( P \mapsto P \cap {S}_{0} \) defines a projective morphism from the projective GIT quotient \( {\mathbb{C}}^{n}/{/}_{\mathbf{a}}G \) to the affine GIT quotient \( {\mathbb{C}}^{n}//G \) . \( {\mathbb{C}}^{n}/{/}_{\mathbf{a}}G \) is a projective toric variety if and only if \( S \) is positive... | Proof. The canonical map from the projective spectrum of an \( \mathbb{N} \) -graded ring to the spectrum of its \( \mathbb{N} \) -graded degree zero part is a projective morphism by definition, proving the first statement. For the second, a complex variety is projective over \( \mathbb{C} \) if and only if it admits a... | No |
Proposition 10.10 The \( {S}_{\mathbf{0}} \) -algebra \( {S}_{\left( \mathbf{a}\right) } \) is minimally generated over \( {S}_{\mathbf{0}} \) by the monomials \( {\mathbf{x}}^{\mathbf{u}}{\gamma }^{r} \), where \( \left( {\mathbf{u}, r}\right) \) runs over all vectors in \( {\mathcal{H}}_{ + } \) . | Proof. In the proof of Proposition 8.4, we saw that \( {S}_{\mathbf{0}} \) is minimally generated as a \( \mathbb{C} \) -algebra by the monomials \( {\mathbf{x}}^{\mathbf{u}} \) for \( \mathbf{u} \) in \( {\mathcal{H}}_{0} \) . Likewise, the ring \( {S}_{\left( \mathbf{a}\right) } \) is minimally generated as a \( \mat... | Yes |
Proposition 10.11 The affine toric variety \( \mathcal{U}\left( {{\mathbf{x}}^{\mathbf{u}}{\gamma }^{r}}\right) \) is the spectrum of the semigroup ring over \( \mathbb{C} \) for the semigroup \( \left\{ {\mathbf{w} \in L \mid \left( {\mathbf{w} + \mathbb{N}\mathbf{u}}\right) \cap {\mathbb{N}}^{n} \neq \varnothing }\ri... | Proof. The \( \gamma \) -degree 0 part of the localization \( {S}_{\left( \mathbf{a}\right) }\left\lbrack {{\mathbf{x}}^{-\mathbf{u}}{\gamma }^{-r}}\right\rbrack \) is spanned by all monomials \( {\mathbf{x}}^{\mathbf{v} - s\mathbf{u}} \) for nonnegative integers \( s \) and monomials \( {\mathbf{x}}^{\mathbf{v}} \) of... | Yes |
Example 10.12 (The two resolutions of the cone over the quadric) Consider the action of \( G = {\mathbb{C}}^{ * } \) on affine 4-space given by \( \left( {{x}_{1},{x}_{2},{x}_{3},{x}_{4}}\right) \mapsto \) \( \left( {z{x}_{1}, z{x}_{2},{z}^{-1}{x}_{3},{z}^{-1}{x}_{4}}\right) \) . This notation should be thought of as i... | Let us compute the map for \( \mathbf{a} = 1 \) in more detail. The ring \( {S}_{\left( \mathbf{a}\right) } \) is\n\n\[ \n{S}_{\left( 1\right) } = {S}_{0}\left\lbrack {\gamma {x}_{1},\gamma {x}_{2}}\right\rbrack = \mathbb{C}\left\lbrack {{x}_{1}{x}_{3},{x}_{1}{x}_{4},{x}_{2}{x}_{3},{x}_{2}{x}_{4},\gamma {x}_{1},\gamma ... | Yes |
Fix a finite directed graph on the vertex set \( V = \{ 1,\ldots, d\} \) . The edge set \( E \) is a subset of \( V \times V \) . Loops and multiple edges are allowed. The torus \( {\left( {\mathbb{C}}^{ * }\right) }^{V} \) with coordinates \( {z}_{i} \) for \( i \in V \) acts on the vector space \( {\mathbb{C}}^{E} \)... | Every directed cycle \( {i}_{1},{i}_{2},\ldots ,{i}_{r},{i}_{1} \) gives a monomial of degree \( \mathbf{0} \) ,\n\n\[ \n{x}_{{i}_{1}{i}_{2}}{x}_{{i}_{2}{i}_{3}}{x}_{{i}_{3}{i}_{4}}\cdots {x}_{{i}_{r}{i}_{1}} \n\]\n\nand these monomials minimally generate the semigroup ring \( {S}_{\mathbf{0}} = K{\left\lbrack {x}_{ij}... | Yes |
Example 10.14 (The 3-dimensional cube) We construct the toric variety \( {X}_{\mathcal{P}} \) associated with the standard 3-dimensional cube \( \mathcal{P} = \operatorname{conv}\{ 0,1{\} }^{3} \) . For the representation (10.5) with \( n = 6 \), we take \( \mathbf{w} = \left( {1,1,1,0,0,0}\right) \) and | \[ L = \mathbb{Z} \cdot \{ \left( {1,0,0, - 1,0,0}\right) ,\left( {0,1,0,0, - 1,0}\right) ,\left( {0,0,1,0,0, - 1}\right) \} . \] This lattice induces the action of \( G = {\left( {\mathbb{C}}^{ * }\right) }^{3} \) on \( {\mathbb{C}}^{6} \) via \[ \left( {{x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6}}\right) \mapsto ... | Yes |
Lemma 10.15 The A-grading is positive if and only if \( C \) equals \( {L}^{ \vee } \otimes \mathbb{R} \) . | Proof. The \( A \) -grading is not positive if and only if there exists a nonzero vector \( \mathbf{u} \) in \( L \cap {\mathbb{N}}^{n} \), by Theorem 8.6. On the other hand, the cone \( C \) fails to equal \( {L}^{ \vee } \otimes \mathbb{R} \) if and only if all functionals \( {\nu }_{i} \) lie on one side of a hyperp... | Yes |
Lemma 10.20 The affine toric variety \( {X}_{\sigma } \) equals the spectrum of the semigroup ring over \( \mathbb{C} \) for the semigroup \( \left\{ {\mathbf{w} \in L \mid \left( {\mathbf{w} + {\mathbb{{Ne}}}_{\bar{\sigma }}}\right) \cap {\mathbb{N}}^{n} \neq \varnothing }\right\} \) of vectors \( \mathbf{w} \) in \( ... | Proof. A Laurent monomial \( {\mathbf{x}}^{\mathbf{w}} \) lies in the localization \( S\left\lbrack {\mathbf{x}}^{-\bar{\sigma }}\right\rbrack \) if and only if \( \mathbf{w} = \mathbf{v} - r{\mathbf{e}}_{\bar{\sigma }} \) for some \( \mathbf{v} \in {\mathbb{N}}^{n} \) and \( r \in \mathbb{N} \) ; in other words, \( \m... | Yes |
Proposition 10.21 The semigroup \( \left\{ {\mathbf{w} \in L \mid \left( {\mathbf{w} + {\mathbb{{Ne}}}_{\sigma }}\right) \cap {\mathbb{N}}^{n} \neq \varnothing }\right\} \) from Lemma 10.20 equals the semigroup \( {\sigma }^{ \vee } \cap L \), where \( {\sigma }^{ \vee } \) is the cone in \( L \otimes \mathbb{R} \) dua... | Proof. Suppose \( \sigma \) is generated as a real cone by \( {\nu }_{{i}_{1}},\ldots ,{\nu }_{{i}_{r}} \) . Let \( {\mathbf{e}}_{i}^{ * } \) be the basis vector of \( {\mathbb{Z}}^{n} \) mapping to \( {\nu }_{i} \) . The subset of \( L \) on which the linear functionals \( {\mathbf{e}}_{{i}_{1}}^{ * },\ldots ,{\mathbf... | Yes |
Corollary 10.23 If \( \tau \) is a face of a cone \( \sigma \in \sum \), then \( {X}_{\tau } \) is an open affine toric subvariety of \( {X}_{\sigma } \) . More precisely, \( S{\left\lbrack {\mathbf{x}}^{-\bar{\tau }}\right\rbrack }_{\mathbf{0}} \) is a localization of \( S{\left\lbrack {\mathbf{x}}^{-\bar{\sigma }}\ri... | Proof. The ring \( S{\left\lbrack {\mathbf{x}}^{-\bar{\tau }}\right\rbrack }_{\mathbf{0}} \) is obtained from \( S{\left\lbrack {\mathbf{x}}^{-\bar{\sigma }}\right\rbrack }_{\mathbf{0}} \) by inverting all monomials \( {\mathbf{x}}^{\mathbf{w}} \) for which the linear functional \( \mathbf{w} \) vanishes on \( \tau \) ... | Yes |
Lemma 10.24 The open subvariety \( {\mathcal{U}}_{\sum } = {\mathbb{C}}^{n} \smallsetminus \mathcal{V}\left( {B}_{\sum }\right) \) of \( {\mathbb{C}}^{n} \) comes endowed with a morphism \( {\mathcal{U}}_{\sum } \rightarrow {X}_{\sum } \) of varieties. | Proof. The gluing used to define \( {\mathcal{U}}_{\sum } \) and \( {X}_{\sum } \) from their open affines \( {\mathcal{U}}_{\sigma } \) and \( {X}_{\sigma } \) commutes with the projections \( {\mathcal{U}}_{\sigma } \rightarrow {X}_{\sigma } \) by Corollary 10.23. | No |
Here is a concrete example demonstrating how the spector of a multigraded ring can depend on the choice of irrelevant ideal. For positive integers \( r \) and \( s \), consider the polytope \( {\mathcal{P}}_{r, s} \) beneath the planes \( z = y \) and \( z = x \), above the \( {xy} \) -plane, and satisfying \( x \leq r... | Given the multigrading, the only extra information we need to define a toric variety is an irrelevant ideal in \( S = \mathbb{k}\left\lbrack {{x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}}\right\rbrack \) . When \( r > s \) , the corresponding polytope \( {\mathcal{P}}_{r > s} \) has toric variety\n\n\[ {X}_{{\mathcal{P}}_{r... | No |
Theorem 10.27 The toric spectrum \( {X}_{\sum } = \operatorname{SpecTor}\left( {S,{B}_{\sum }}\right) \) is the categorical quotient of \( {\mathcal{U}}_{\sum } = {\mathbb{C}}^{n} \smallsetminus \mathcal{V}\left( {B}_{\sum }\right) \) by \( G \) . | Proof. Suppose \( {\mathcal{U}}_{\sum } \rightarrow Y \) is a \( G \) -equivariant morphism. Then any local function on \( Y \) induces a \( G \) -invariant function on an open subset \( U \) of the variety \( {\mathcal{U}}_{\sum } \) . Any \( G \) -invariant function on \( U \) is locally given by elements in a locali... | Yes |
Example 10.32 (Cubes yet again) The three tetrahedra at the end of Example 10.19 arise by embedding the cube into a simplex of dimension 5: each tetrahedron is the intersection of two codimension 1 simplices corresponding to opposite faces of the cube. These nonintersecting pairs of faces correspond to pairs of coordin... | The picture is somewhat simpler one dimension lower down, where the square is expressed as the intersection of a tetrahedron with a 2-plane \( E \) :\n\n\n\nWhen two facets of the tetrahedron intersect \( E \) in opp... | Yes |
Example 10.33 (The five varieties of \( 2 \times 2 \) minors of a \( 2 \times 3 \) matrix) Let \( S \) be the polynomial ring generated by the entries of a \( 2 \times 3 \) matrix \( X = \left( {x}_{ij}\right) \) of variables and consider the ideal of \( 2 \times 2 \) minors | \[ I = \left\langle {{x}_{11}{x}_{22} - {x}_{12}{x}_{21},{x}_{11}{x}_{32} - {x}_{12}{x}_{31},{x}_{21}{x}_{32} - {x}_{22}{x}_{31}}\right\rangle . \] There are five different ways, all very natural, of associating to the prime ideal \( I \) a subvariety \( Y \) of a toric variety \( X \) . In each case, the inclusion of ... | Yes |
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