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Example 11.3 Let \( Q = {\mathbb{N}}^{2} \) and consider the ideal \( I = \left\langle {{x}^{4},{x}^{2}{y}^{2},{y}^{4}}\right\rangle \) in \( \mathbb{k}\left\lbrack Q\right\rbrack = \mathbb{k}\left\lbrack {x, y}\right\rbrack \) . The following sequence is a minimal irreducible resolution: | \[ 0 \rightarrow \mathbb{k}\left\lbrack {x, y}\right\rbrack /I \rightarrow \mathbb{k}\left\lbrack {x, y}\right\rbrack /\left\langle {{x}^{4},{y}^{2}}\right\rangle \oplus \mathbb{k}\left\lbrack {x, y}\right\rbrack /\left\langle {{x}^{2},{y}^{4}}\right\rangle \rightarrow \mathbb{k}\left\lbrack {x, y}\right\rbrack /\left\... | Yes |
Corollary 11.5 Every monomial ideal \( I \subseteq \mathbb{k}\left\lbrack Q\right\rbrack \) has a unique irredundant expression \( I = {W}_{1} \cap \cdots \cap {W}_{r} \) as an intersection of irreducible ideals \( {W}_{j} \) . | Proof. If \( {\bar{W}}^{ \bullet } \) is a minimal irreducible resolution of \( \mathbb{k}\left\lbrack Q\right\rbrack /I \), then choose \( r = {\mu }^{0} \) and \( {W}_{j} = {W}^{0j} \) . The kernel of the composite homomorphism \( \mathbb{k}\left\lbrack Q\right\rbrack \rightarrow \mathbb{k}\left\lbrack Q\right\rbrack... | Yes |
Proposition 11.9 An indecomposable injective \( \mathbb{k}\{ \mathbf{a} + F - Q\} \) is isomorphic to \( \mathbb{k}\{ \mathbf{b} + F - Q\} \) as a \( \mathbb{k}\left\lbrack Q\right\rbrack \) -module if and only if \( \mathbf{a} + \mathbb{Z}F = \mathbf{b} + \mathbb{Z}F \) . | Proof. The two modules are isomorphic if and only if \( \mathbf{a} \in \mathbf{b} + F - Q \) and \( \mathbf{b} \in \mathbf{a} + F - Q \) . This condition is equivalent to \( \mathbf{a} - \mathbf{b} \in F - Q \) and \( \mathbf{b} - \mathbf{a} \in F - Q \), which is the same as \( \mathbf{a} - \mathbf{b} \in \left( {F - ... | Yes |
Lemma 11.12 The inclusion \( \mathbb{k}\{ F\} \subset \mathbb{k}\{ F - Q\} \) is an essential extension. | Proof. Each element \( \mathbf{u} \in F - Q \) can be expressed as \( \mathbf{u} = \mathbf{f} - \mathbf{a} \) for some \( \mathbf{a} \in Q \) and \( \mathbf{f} \in F \) . The equation \( \mathbf{a} + \mathbf{u} = \mathbf{f} \in F \) translates into \( {\mathbf{t}}^{\mathbf{a}}{\mathbf{t}}^{\mathbf{u}} = \) \( {\mathbf{... | Yes |
Theorem 11.13 A monomial ideal \( W \) is irreducible if and only if the \( Q \) - graded part of some indecomposable injective module \( E \) satisfies \( {E}_{Q} = \bar{W} \) . | Proof. First we prove the \ | No |
Lemma 11.16 \( \underline{\operatorname{Hom}}\left( {M,{N}^{ \vee }}\right) = {\left( M \otimes N\right) }^{ \vee } \) . | Proof. The result is a consequence of the adjointness between Hom and \( \otimes \) that holds for arbitrary \( {\mathbb{Z}}^{d} \) -graded \( \mathbb{k} \) -algebras \( R \) and \( R \) -modules \( M, N \) :\n\n\[ \n{\underline{\operatorname{Hom}}}_{\mathbb{k}}\left( {M{ \otimes }_{R}N,\mathbb{k}}\right) = {\underline... | Yes |
Proposition 11.17 The \( \mathbb{k} \) -vector space \( \underline{\operatorname{Hom}}{\left( \mathbb{k}\{ \mathbf{a} + F - Q\} ,\mathbb{k}\{ \mathbf{b} + G - Q\} \right) }_{\mathbf{0}} \) is either zero or 1-dimensional. The following conditions are equivalent.\n\n1. \( \operatorname{Hom}{\left( \mathbb{k}\{ \mathbf{a... | Proof. Lemma 11.16 implies the first equality below. The second uses the same lemma with the roles of \( M \) and \( N \) switched:\n\n\[ \underline{\operatorname{Hom}}\left( {\mathbb{k}\{ \mathbf{a} + F - Q\} ,\mathbb{k}\{ \mathbf{b} + G - Q\} }\right) \]\n\n\[ = {\left( \mathbb{k}\{ \mathbf{a} + F - Q\} \otimes \math... | Yes |
Theorem 11.19 Monomial matrices represent maps of injective modules: Two monomial matrices represent the same map of injectives (with fixed direct sum decompositions) if and only if (i) their scalar entries are equal, (ii) the corresponding faces \( {F}^{r} \) are equal, where \( r = p, q \), and (iii) the correspondin... | Proof. Proposition 11.17 immediately implies the first sentence. The second sentence is the content of Proposition 11.9. | No |
The following sequence of maps is cellular, supported on a line segment. The vector labels are all zero. The vertices have face labels \( X \) and \( Y \), the interior has face label \( {\mathbb{N}}^{2} \), and the empty set has face label \( \mathcal{O} \). | This sequence of maps is actually a complex, and it would be exact except that the kernel of the first map \( \mathbb{k}\left\{ {\mathbb{Z}}^{2}\right\} \rightarrow \mathbb{k}\left\{ {X - {\mathbb{N}}^{2}}\right\} \oplus \mathbb{k}\left\{ {Y - {\mathbb{N}}^{2}}\right\} \) is isomorphic to \( \mathbb{k}\left\{ {\left( {... | No |
Lemma 11.23 \( N \) is flat if and only if \( {N}^{ \vee } \) is homologically injective. | Proof. \( M \mapsto M \otimes N \) is exact if and only if \( M \mapsto {\left( M \otimes N\right) }^{ \vee } \) is. Now use the equality \( {\left( M \otimes N\right) }^{ \vee } = \underline{\operatorname{Hom}}\left( {M,{N}^{ \vee }}\right) \) of Lemma 11.16.\n\nThus \ | No |
Proposition 11.24 Indecomposable injectives are homologically injective. | Proof. Since \( \mathbb{k}{\left\lbrack Q - F\right\rbrack }^{ \vee } = \mathbb{k}\{ F - Q\} \), this follows from Lemma 11.23. | No |
Lemma 11.26 Arbitrary \( {\mathbb{Z}}^{d} \) -graded products of homologically injective modules are homologically injective. | Proof. The natural map \( \operatorname{Hom}\left( {N,{}^{ * }\mathop{\prod }\limits_{{p \in P}}{M}^{p}}\right) \rightarrow {}^{ * }\mathop{\prod }\limits_{{p \in P}}\operatorname{Hom}\left( {N,{M}^{p}}\right) \) is an isomorphism (write out carefully what it means to be a homogeneous element of degree a on each side).... | No |
Proposition 11.27 Every module \( M \) is isomorphic to a submodule of a homologically injective module. If \( M \) is finitely generated, then \( M \) is isomorphic to a submodule of a finite direct sum of indecomposable injectives. | Proof. Homogeneous elements \( y \in M \) generate finitely generated submodules. Using Proposition 8.11 and Lemma 7.10, pick a face \( F \) such that \( {P}_{F} \) is associated to \( M \), so \( \left\langle {{\mathbf{t}}^{\mathbf{a}}y}\right\rangle \cong \mathbb{k}\left\{ {{\mathbf{u}}_{y} + {F}^{y}}\right\} \) for ... | Yes |
Lemma 11.28 Let \( J \) be homologically injective and \( E \) any module.\n\n1. If \( E \) is a direct summand of \( J \), then \( E \) is homologically injective.\n\n2. If \( J \subseteq E \), then \( J \) is a direct summand of \( E \) . | Proof. To prove the first part, let \( J = {J}^{\prime } \oplus {J}^{\prime \prime } \) and apply \( \underline{\operatorname{Hom}}\left( {\_, J}\right) = \) \( \operatorname{Hom}\left( {\_ ,{J}^{\prime }}\right) \oplus \operatorname{Hom}\left( {\_ ,{J}^{\prime \prime }}\right) \) to any exact sequence. For the second ... | Yes |
Proposition 11.29 A module \( J \) is homologically injective if and only if \( J \) has no proper essential extensions. | Proof. First assume \( J \) is homologically injective. If \( J \subseteq M \) is an essential extension, then writing \( M = J \oplus N \) for some \( N \) by the second part of Lemma 11.28, it must be that \( N = 0 \), so \( J = M \). Now assume \( J \) has no proper essential extension. Use Proposition 11.27 to find... | Yes |
Theorem 11.30 A module is homologically injective if and only if it is injective in the combinatorial sense of Definition 11.10. | Proof. Finite direct sums of indecomposable injectives are homologically injective by Proposition 11.24 and Lemma 11.26. Now let \( J \) be an arbitrary direct sum of indecomposable injectives, and suppose that \( J \subseteq E \) is an essential extension. If \( x \in E \), then \( \langle x\rangle \cap J \) is isomor... | Yes |
Theorem 11.32 Injective hulls exist and are unique up to isomorphism. | Proof. Existence: Choose an injection \( M \hookrightarrow J \) with \( J \) injective using Proposition 11.27, and let \( E \subseteq J \) be maximal among essential extensions of \( M \) contained in \( J \) ; these exist by Zorn’s Lemma. Suppose \( E \subseteq {E}^{\prime } \) is an essential extension. Lemma 11.22 ... | Yes |
Every module has an injective resolution. Minimal injective resolutions are unique up to isomorphism; in fact, if \( {J}^{ \bullet } \) and \( {E}^{ \bullet } \) are injective resolutions of \( M \) with \( {J}^{ \bullet } \) minimal, \( {E}^{ \bullet } \) contains \( {J}^{ \bullet } \) as a subcomplex. | Proof. Use Theorem 11.32 and Lemma 11.28 to show by induction on cohomological degree \( j \) that \( {E}^{j} \cong {J}^{j} \oplus {\widetilde{E}}^{j} \) for an injective resolution \( {\widetilde{E}}^{ \bullet } \) of 0 . \( ▱ \) | No |
Theorem 11.36 If \( J \) is injective, then the localization \( \underline{\operatorname{Hom}}\left( {\mathbb{k}\left\lbrack F\right\rbrack, J}\right) \left\lbrack {\mathbb{Z}F}\right\rbrack \) is a free module over \( \mathbb{k}\left\lbrack {\mathbb{Z}F}\right\rbrack \) . Its \( {\mathbb{Z}}^{d} \) -graded piece in de... | Proof. The submodule \( N = \underline{\operatorname{Hom}}\left( {\mathbb{k}\left\lbrack F\right\rbrack ,\mathbb{k}\{ \mathbf{a} + G - Q\} }\right) \) of elements inside \( \mathbb{k}\{ \mathbf{a} + G - Q\} \) annihilated by \( \mathbb{k}\left\lbrack F\right\rbrack \) is zero unless \( F \subseteq G \) . Subsequently l... | Yes |
Lemma 11.38 Any irreducible resolution \( {\bar{W}}^{ \bullet } \) of a Q-graded module \( M \) can be expressed as the \( Q \) -graded part \( {J}_{Q}^{ \bullet } \) of an injective resolution \( {J}^{ \bullet } \) of \( M \) . | Proof. Since \( M \) is \( Q \) -graded, \( M \hookrightarrow {W}^{0} \hookrightarrow E\left( {W}^{0}\right) \) and \( E{\left( {W}^{0}\right) }_{Q} = {W}^{0} \) . Having chosen \( {\bar{W}}^{i} \hookrightarrow {J}^{i} \) such that \( {\bar{W}}^{i} = {J}_{Q}^{i} \), let \( N = {J}^{i}/{J}^{i - 1} \) and \( {J}^{i + 1} ... | Yes |
Proposition 11.39 Let \( M \) be a finitely generated Q-graded module. The Q-graded part of a minimal injective resolution of \( M \) is a finite minimal irreducible resolution of \( M \) . | Proof. Let \( {J}^{ \bullet } \) be a minimal injective resolution of \( M \) . That \( {\bar{W}}^{ \bullet } = {\left( {J}^{ \bullet }\right) }_{Q} \) is an irreducible resolution follows from Theorem 11.13, so it remains to demonstrate minimality. For each \( j \), the number of indecomposable summands in \( {\bar{W}... | No |
Look back at the illustration for \( I = \left\langle {{x}^{4},{x}^{2}{y}^{2},{y}^{4}}\right\rangle \subset \) \( \mathbb{k}\left\lbrack {x, y}\right\rbrack \) in Example 11.3. The injective hull of \( \mathbb{k}\left\lbrack {x, y}\right\rbrack /I \) is the direct sum \( \mathbb{k}{\left\lbrack x, y\right\rbrack }^{ \v... | Proof of Theorem 11.4. Proposition 11.39 says that minimal irreducible resolutions exist as \( Q \) -graded parts of minimal injective resolutions. By Lemma 11.38 and Corollary 11.35, every minimal irreducible resolution can be expressed this way. | No |
The Ehrhart polynomial of the unit 3-cube \( \operatorname{conv}\left( {\{ 0,1{\} }^{3}}\right) \) is the cube of the Ehrhart polynomial of the unit segment: | \[ {E}_{\text{cube }}\left( m\right) = {m}^{3} + 3{m}^{2} + {3m} + 1 = {\left( m + 1\right) }^{3}. \] | Yes |
Lemma 12.4 \( {E}_{\mathcal{P}} \) is the \( \mathbb{N} \) -graded Hilbert function of \( \mathbb{k}\left\lbrack {Q}_{\text{sat }}\right\rbrack \) :\n\n\[ \n{E}_{\mathcal{P}}\left( m\right) = {\dim }_{\mathbb{k}}\left( {\mathbb{k}{\left\lbrack {Q}_{\text{sat }}\right\rbrack }_{m}}\right) \n\] | Proof. The intersection of the cone \( C \) with the hyperplane at height \( m \) is a copy of \( m \cdot \mathcal{P} \) by construction. The lattice points in this copy of \( m \cdot \mathcal{P} \) correspond to the monomials of degree \( m \) in \( \mathbb{k}\left\lbrack {Q}_{\text{sat }}\right\rbrack \) by Definitio... | Yes |
Lemma 12.5 If the polytope \( \mathcal{P} \) is a lattice simplex, then the Hilbert function of the \( \mathbb{N} \) -graded module \( \mathbb{k}\left\lbrack {Q}_{\text{sat }}\right\rbrack \) equals its Hilbert polynomial; that is, \( {E}_{\mathcal{P}}\left( m\right) \) is a polynomial for all nonnegative integers \( m... | Proof. Let \( \mathcal{P} \) be the simplex with vertices \( {\mathbf{a}}_{1},\ldots ,{\mathbf{a}}_{d + 1} \) in \( {\mathbb{Z}}^{d} \), and define \( L \) as the sublattice of \( {\mathbb{Z}}^{d + 1} \) spanned by \( \left( {1,{\mathbf{a}}_{1}}\right) ,\ldots ,\left( {1,{\mathbf{a}}_{d + 1}}\right) \) . This lattice \... | Yes |
Example 12.6 The lattice tetrahedron \( \mathcal{P} \) with vertex set\n\n\[ A = \{ \left( {0,0,0}\right) ,\left( {1,0,0}\right) ,\left( {0,1,0}\right) ,\left( {1,1,2}\right) \} \]\n\nis not normal, since \( \left( {1,1,1}\right) \) lies in \( 2\mathcal{P} \) but not in \( 2\left( {\mathcal{P} \cap {\mathbb{Z}}^{d}}\ri... | The semigroup ring \( \mathbb{k}\left\lbrack {Q}_{\text{sat }}\right\rbrack \) is minimally generated by the five monomials \( {t}_{0},{t}_{0}{t}_{1},{t}_{0}{t}_{2} \) , \( {t}_{0}{t}_{1}{t}_{2}{t}_{3}^{2} \), and \( {t}_{0}^{2}{t}_{1}{t}_{2}{t}_{3} \) . Over the polynomial ring \( \mathbb{k}\left\lbrack {{t}_{0}{\math... | Yes |
Lemma 12.10 \( {\mathcal{P}}_{\mathbf{a}} \) is a contractible union of facets whenever \( \mathbf{a} \notin \operatorname{relint}\left( C\right) \) . | Proof. Since \( {\mathcal{P}}_{\mathbf{a}} \) is the void complex (no faces at all, not even the empty face \( \varnothing \) ) when \( \mathbf{a} \in - C \), we assume \( \mathbf{a} \notin - C \) . Note that \( {\mathcal{P}}_{\mathbf{a}} = {\mathcal{P}}_{\mathbf{a} + \mathbf{b}} \) whenever b is in the lineality space... | No |
Theorem 12.11 If \( Q \) is a saturated affine semigroup, then the dualizing complex \( {\Omega }_{Q}^{ \bullet } \) is a minimal injective resolution of the canonical module \( {\omega }_{Q} \) . | Proof. Write \( {\Omega }^{ \bullet } = {\Omega }_{Q}^{ \bullet } \) for the dualizing complex of \( \mathbb{k}\left\lbrack Q\right\rbrack \) . In any \( {\mathbb{Z}}^{d} \) -graded degree \( \mathbf{a} \in \operatorname{relint}\left( C\right) \), where \( C = {\mathbb{R}}_{ \geq 0}Q \), the degree \( \mathbf{a} \) pie... | Yes |
Lemma 12.12 If \( Q \subseteq {\mathbb{Z}}^{d} \) is a saturated semigroup and \( \mathbf{a} \in {\mathbb{Z}}^{d} \), then \( \bar{F} \in {\mathcal{P}}_{\mathbf{a}} \) if and only if \( \mathbf{a} \notin F - Q \) . | Proof. Note the contrast with (12.2), which has the condition \( \mathbf{a} \notin F - C \) rather than \( \mathbf{a} \notin F - Q \) . Replacing \( {\mathbb{Z}}^{d} \) by the subgroup that \( Q \) generates, we may assume that \( Q \) generates \( {\mathbb{Z}}^{d} \) . For facets \( G \) of \( Q \), we then have \( G ... | Yes |
Lemma 12.15 Let \( F \) be a nonempty face of \( \mathcal{P} \) . If \( {T}_{F} \) denotes the injective hull of the face of \( Q \) corresponding to \( F \), then the Hilbert series of \( \mathbb{k}\left\{ {T}_{F}\right\} \) is summable. More precisely, the Hilbert series \( H\left( {\mathbb{k}\left\{ {T}_{F}\right\} ... | Proof. Translation by \( m \cdot \left( {1,\mathbf{w}}\right) \) for any vector \( \mathbf{w} \in F \) gives a bijection \( {T}_{F} \cap \left( {0 \times {\mathbb{Z}}^{d}}\right) \rightarrow {T}_{F} \cap \left( {m \times {\mathbb{Z}}^{d}}\right) \) between the parts of \( {T}_{F} \) at levels 0 and \( m \) . Thus \( H\... | Yes |
For the one-dimensional polytope \( \mathcal{P} = \left\lbrack {2,3}\right\rbrack \) in Example 12.14, substituting \( t = 1 \) into expression (12.6) yields the Ehrhart polynomial \( {E}_{\mathcal{P}}\left( m\right) = m + 1 \) . Hence | \[ {\left( -1\right) }^{\dim \left( \mathcal{P}\right) } \cdot {E}_{\mathcal{P}}\left( {-m}\right) = - \left( {\left( {-m}\right) + 1}\right) = m - 1 \] is the number of interior lattice points in \( m \cdot \mathcal{P} = \left\lbrack {{2m},{3m}}\right\rbrack \) . | Yes |
Theorem 12.18 (Barvinok's Theorem) Suppose that the dimension \( d \) is fixed and \( \mathcal{P} \) is a lattice polytope in \( {\mathbb{R}}^{d} \) . Then the lattice point enumerator\n\n\[ \n{\Phi }_{\mathcal{P}}\left( \mathbf{t}\right) = \mathop{\sum }\limits_{{\mathbf{v} \in \mathcal{P}}}{\mathbf{t}}^{\mathbf{v}} \... | From the rational function on the right-hand side of Theorem 12.18, one can read off the number of lattice points in \( \mathcal{P} \) . This amounts to substituting \( \mathbf{t} = \left( {1,\ldots ,1}\right) \) while being careful to avoid the poles. The basic idea is to substitute a numerical vector close to \( \lef... | Yes |
Example 12.20 Let \( \mathcal{P} \) be the triangle in \( {\mathbb{R}}^{2} \) with vertices \( \left( {0,0}\right) ,\left( {a,0}\right) \) , and \( \left( {a,{a}^{2}}\right) \), where \( a \) is a large positive integer. The Ehrhart polynomial is\n\n\[ \n{E}_{\mathcal{P}}\left( m\right) = 1 + \left( {a + \frac{{a}^{2}}... | The last two terms are easily computed in polynomial time:\n\n\[ \n{K}_{\left( a,0\right) } = 1\text{ and }{D}_{\left( a,0\right) } = \left( {1 - {t}_{1}^{-1}}\right) \left( {1 - {t}_{2}}\right)\n\]\n\n\[ \n{K}_{\left( a,{a}^{2}\right) } = 1\;\text{ and }\;{D}_{\left( a,{a}^{2}\right) } = \left( {1 - {t}_{1}^{-1}{t}_{2... | Yes |
Theorem 12.23 For \( d \) fixed, the Hilbert basis \( {\mathcal{H}}_{Q} \) of any saturated affine semigroup \( Q \subset {\mathbb{Z}}^{d} \) can be computed in polynomial time. | The point is that while the size of \( {\mathcal{H}}_{Q} \) can grow exponentially in the bit complexity of the description of \( Q \), we write the Laurent polynomial \( {\mathcal{H}}_{Q} \) as a short rational function requiring only polynomially many bits. A simple example comes from the cone generated by \( \left( ... | Yes |
Example 12.25 Fix \( n = 4 \), set \( d = 2 \), and let \( a \geq 3 \) be a large integer. The input is the matrix \( \mathbf{A} = \left\lbrack \begin{matrix} a & a - 1 & 1 & 0 \\ 0 & 1 & a - 1 & a \end{matrix}\right\rbrack \) . The task is to compute the reduced lexicographic Gröbner basis of \( {I}_{L} \) for the ker... | The output would consist of the rational function \[ G\left( {\mathbf{x},\mathbf{y}}\right) = {x}_{1}{x}_{4}{y}_{2}{y}_{3} + {x}_{2}{x}_{4}^{a - 2}{y}_{3}^{a - 1} + \frac{{x}_{1}{x}_{3}{y}_{2}^{2}\left( {{\left( {x}_{1}{y}_{2}\right) }^{a - 2} - {\left( {x}_{3}{y}_{4}\right) }^{a - 2}}\right) }{{x}_{1}{y}_{2} - {x}_{3}... | Yes |
Lemma 13.3 If \( E = {\bigoplus }_{k \in K}\mathbb{k}\left\{ {{\mathbf{a}}_{k} + {F}^{k} - Q}\right\} \) is an injective module, then \[ {\Gamma }_{{I}_{\Delta }}E = {\bigoplus }_{{F}^{k} \in \Delta }\mathbb{k}\left\{ {{\mathbf{a}}_{k} + {F}^{k} - Q}\right\} \] is obtained by taking only those summands whose support fa... | Proof. First make the (easy) check that \( {\Gamma }_{I} \) commutes with direct sums and \( {\mathbb{Z}}^{d} \) -graded translation. Then use the fact that \( {\Gamma }_{{I}_{\Delta }}\mathbb{k}\{ F - Q\} \) is zero unless \( F \in \Delta \), in which case every element is annihilated by some power of \( {I}_{\Delta }... | No |
Let \( Q \) be the subsemigroup of \( {\mathbb{Z}}^{3} \) from Examples 7.13 and 12.8, generated by \( \left( {1,0,0}\right) ,\left( {1,1,0}\right) ,\left( {1,1,1}\right) \), and \( \left( {1,0,1}\right) \). Its semigroup ring is \( \mathbb{k}\left\lbrack Q\right\rbrack \cong \mathbb{k}\left\lbrack {a, b, c, d}\right\r... | Applying \( {\Gamma }_{\mathfrak{p}} \) to the dualizing complex yields a complex  by Lemma 13.3. Again using the notation of Example 12.8, consider the contributions of the injective hulls \( {T}_{ab},{T}_{a},{T}_{b... | Yes |
Given a module \( M \), its submodule with support on the ideal \( I = \left\langle {{m}_{1},\ldots ,{m}_{r}}\right\rangle \) is the kernel of the homomorphism \( M \rightarrow {\bigoplus }_{i}M \otimes R\left\lbrack {m}_{i}^{-1}\right\rbrack \) (Exercise 13.6). | Since this homomorphism is just the first map in the complex \( M{ \otimes }_{R}{\check{\mathcal{C}}}^{ \bullet }\left( {{m}_{1},\ldots ,{m}_{r}}\right) \), the next result should at least be plausible. | No |
Theorem 13.7 The local cohomology of \( M \) supported on \( I = \left\langle {{m}_{1},\ldots ,{m}_{r}}\right\rangle \) is the cohomology of the Čech complex tensored with \( M \) : | Sketch of proof. One possibility is to use homological algebra as in Exercise 1.12 , where the double complex this time comes from tensoring together an injective resolution \( {E}^{ \bullet } \) of \( M \) and the Čech complex \( {\check{\mathcal{C}}}^{ \bullet } \) . View \( {E}^{ \bullet } \) as going upward and \( ... | No |
Proposition 13.10 The module \( {\underline{\operatorname{Ext}}}_{R}^{i}\left( {N, M}\right) \) is isomorphic to the \( {i}^{\text{th }} \) cohomology of the complex \( {\underline{\operatorname{Hom}}}_{R}\left( {{\mathcal{F}}_{ \bullet }, M}\right) \), for any free resolution \( {\mathcal{F}}_{ \bullet } \) of \( N \)... | The homological algebra used to prove this fact is the same as in Exercise 1.12 , by comparing cohomology to that of the total complex, except that here the tensor product complex is replaced by \( {\underline{\operatorname{Hom}}}_{R}\left( {{\mathcal{F}}_{ \bullet },{E}^{ \bullet }}\right) \), and the directions of on... | No |
Lemma 13.11 \( {\underline{\operatorname{Hom}}}_{R}\left( {R/{I}^{t}, M}\right) = \left( {0{ : }_{M}{I}^{t}}\right) = \left\{ {y \in M \mid {I}^{t}y = 0}\right\} \) is the set of elements in \( M \) annihilated by \( {I}^{t} \) . Taking direct limits over \( t \) yields\n\n\[\n\mathop{\lim }\limits_{\overrightarrow{t}}... | Loosely, the union of the homomorphic images of \( R/{I}^{t} \) inside \( M \) for ever-increasing values of \( t \) fills up the part of \( M \) supported on \( I \) . The proof is immediate from the definitions. | No |
Theorem 13.12 Local cohomology with support on I equals the limit\n\n\[ \n{H}_{I}^{i}\left( M\right) \cong \mathop{\lim }\limits_{\underset{t}{ \rightarrow }}{\underline{\operatorname{Ext}}}_{R}^{i}\left( {R/{I}^{t}, M}\right) \n\] | Proof. Apply \( \operatorname{Hom}\left( {R/{I}^{t},\_ }\right) \) to an injective resolution \( {E}^{ \bullet } \) of \( M \) and take the direct limit as \( t \) approaches \( \infty \) . By Lemma 13.11 the limit complex is \( {\Gamma }_{I}{E}^{ \bullet } \) . Since taking cohomology commutes with direct limits [Wei9... | Yes |
Theorem 13.13 The Hilbert series of the \( {i}^{\text{th }} \) maximal-support local cohomology module of a Stanley-Reisner ring satisfies\n\n\[ H\left( {{H}_{\mathfrak{m}}^{i}\left( {S/{I}_{\Delta }}\right) ;\mathbf{x}}\right) = \mathop{\sum }\limits_{{\sigma \in \Delta }}{\dim }_{\mathbb{k}}{\widetilde{H}}^{i - \left... | Proof. Given a vector \( \mathbf{b} \in {\mathbb{Z}}^{n} \), for the duration of this proof we let \( {\mathbf{b}}^{ - } \) and \( {\mathbf{b}}^{ + } \) denote the subsets of \( \{ 1,\ldots, n\} \), where \( \mathbf{b} \) has strictly negative and strictly positive entries, respectively. Having fixed \( \Delta \), defi... | Yes |
Theorem 13.14 Fix a saturated affine semigroup \( Q \) such that \( \dim \left( \mathcal{P}\right) = \) \( r - 1 \) . Let \( \Delta \) be a polyhedral subcomplex of \( {\mathbb{R}}_{ \geq 0}Q \) corresponding to a sub-complex \( \bar{\Delta } \subseteq \mathcal{P} \) . The \( {i}^{\text{th }} \) local cohomology of the... | Proof. By Theorem 12.11, the local cohomology can be calculated using the dualizing complex \( {\Omega }_{Q} \) . The indecomposable injective summand \( \mathbb{k}\{ F - Q\} \) lies inside \( {\Gamma }_{{I}_{\Delta }}{\Omega }_{Q}^{ \bullet } \) if and only if \( F \in \Delta \) . Now use Lemma 12.12 along with the ca... | Yes |
For the polynomial ring \( S = \mathbb{k}\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) and \( \Delta \) a simplicial complex on \( n \) vertices, the \( {\mathbb{Z}}^{n} \) -graded Hilbert series of \( {H}_{{I}_{\Delta }}^{i}\left( S\right) \) is\n\n\[ H\left( {{H}_{{I}_{\Delta }}^{i}\left( S\right) ;\mathbf{x... | Proof. \( {\overline{\mathbb{N}}}_{\mathbf{a}}^{n} \) consists precisely of those faces of \( \bar{\Delta } \) corresponding to the faces \( \sigma \in \Delta \) such that \( i \in \sigma \) whenever \( {a}_{i} \leq 0 \) . These are all of the faces containing the set \( \sigma \left( \mathbf{a}\right) \) of indices \(... | Yes |
Theorem 13.20 There is a sector partition \( \mathcal{S} \vdash {H}_{I}^{i}\left( M\right) \) of the local cohomology of any finitely generated \( {\mathbb{Z}}^{d} \) -graded module over any normal semigroup ring \( \mathbb{k}\left\lbrack Q\right\rbrack \), in which each sector in \( \mathcal{S} \) consists of the latt... | Proof. Treat the cohomological index \( i \) as fixed, and consider the three terms \( {E}^{i - 1} \rightarrow {E}^{i} \rightarrow {E}^{i + 1} \) in a minimal injective resolution of \( M \) . The local cohomology \( {H}_{I}^{i}\left( M\right) \) is the middle cohomology of the complex \( {\Gamma }_{I}{E}^{i - 1} \righ... | Yes |
Lemma 13.22 \( {H}^{0}\left( {M \otimes {\mho }_{Q}^{ \bullet }}\right) = {\Gamma }_{\mathfrak{m}}M \) for all \( \mathbb{k}\left\lbrack Q\right\rbrack \) -modules \( M \) . | Proof. An element in \( M \) is supported at \( \mathfrak{m} \) if and only if its image in every localization \( {M}_{v} \) for vertices \( \bar{v} \) of \( \mathcal{P} \) is zero. | No |
Proposition 13.23 Let \( F \) be a face of an affine semigroup \( Q \) . The complex \( \mathbb{k}\{ F - Q\} \otimes {\mho }_{Q} \) can only have nonzero cohomology when \( \bar{F} = \varnothing \), in which case \( {H}^{0}\left( {\mathbb{k}\{ - Q\} \otimes {\mho }_{Q}}\right) = \mathbb{k}\{ - Q\} \) and all higher coh... | Proof. When \( \bar{F} = \varnothing \), so that the prime corresponding to \( F \) is \( {P}_{F} = \mathfrak{m} \) , the cohomology is as stated because all localizations of \( \mathbb{k}\{ - Q\} \) at primes corresponding to nonempty faces of \( \mathcal{P} \) are zero. Suppose now that \( \bar{F} \) is nonempty. The... | Yes |
Theorem 13.24 Let \( \mathbb{k}\left\lbrack Q\right\rbrack \) be an affine semigroup ring with multigraded maximal ideal \( \mathfrak{m} \) . The local cohomology of any \( \mathbb{k}\left\lbrack Q\right\rbrack \) -module \( M \) supported at \( \mathfrak{m} \) is the cohomology of the Ishida complex tensored with \( M... | Proof. Apply Fact 13.8, using Lemma 13.22 and Proposition 13.23. | No |
Corollary 13.26 The degree \( \mathbf{b} \) part of the local cohomology of the semigroup ring \( \mathbb{k}\left\lbrack Q\right\rbrack \) supported at \( \mathfrak{m} \) is isomorphic to the cohomology of \( {\nabla }_{Q}\left( \mathbf{b}\right) \) : | \[ {H}_{\mathfrak{m}}^{i}{\left( \mathbb{k}\left\lbrack Q\right\rbrack \right) }_{\mathbf{b}} = {H}^{i}\left( {{\nabla }_{Q}\left( \mathbf{b}\right) ;\mathbb{k}}\right) \] | Yes |
Example 13.29 Start with the triangular, square, and pentagonal minimal cellular resolutions of\n\n\[ S/\langle a, b, c\rangle ,\;S/\langle {ab},{bc},{cd},{ad}\rangle ,\;\text{ and }\;S/\langle {abc},{bcd},{cde},{ade},{abe}\rangle \]\n\nfor appropriate \( S \) in Example 4.12. The associated canonical Čech complexes ha... | \n\nThe empty set is labeled \( 0\cdots 0 \) in all three pictures. The triangle here gives monomial matrices for the usual Čech complex in Example 13.6, whereas the triangle in Example 4.12 is the Koszul complex in ... | Yes |
Proposition 13.30 Suppose that \( {I}_{\Delta } \) is generated by squarefree monomials \( {m}_{1},\ldots ,{m}_{r} \) . If \( \mathcal{F} \) . is the Taylor resolution on these generators, then \( {\check{\mathcal{C}}}_{\mathcal{F}}^{ \bullet } = \) \( {\check{\mathcal{C}}}^{ \bullet }\left( {{m}_{1},\ldots ,{m}_{r}}\r... | This is a key point, and it follows immediately from the definitions. | No |
Theorem 13.31 The local cohomology of \( M \) supported on \( {I}_{\Delta } \) is the cohomology of any generalized Čech complex tensored with \( M \) :\n\n\[ \n{H}_{{I}_{\Delta }}^{i}\left( M\right) = {H}^{i}\left( {M \otimes {\check{\mathcal{C}}}_{\mathcal{F}}^{ \bullet }}\right) \n\] | The proof, at the end of this section, relies on a construction that extends the construction in Definition 13.28 to arbitrary \( {\mathbb{Z}}^{n} \) -graded modules. | No |
Proposition 13.34 If \( \mathcal{F} \) . is a free resolution of \( S/{I}_{\Delta } \) with squarefree row and column labels, then the generalized Čech complex can be expressed as\n\n\[ \n{\check{\mathcal{C}}}_{\mathcal{F}}^{ \bullet } = \left( {\check{\mathcal{C}}{\mathcal{F}}^{ \bullet }}\right) \left( \mathbf{1}\rig... | Proof. Every summand \( S\left( {-\sigma }\right) \) in \( \mathcal{F} \) . becomes a summand \( S\left( {-\bar{\sigma }}\right) \) with generator of degree \( \bar{\sigma } = \mathbf{1} - \sigma \) in \( {\mathcal{F}}^{ \bullet } \) . It is straightforward to check that \( \check{\mathcal{C}}\left( {S\left( {-\bar{\si... | Yes |
Theorem 13.37 Let \( M \) be a finitely generated graded module of dimension \( r \) over a positively \( {\mathbb{Z}}^{d} \) -graded polynomial ring \( S \) with maximal ideal \( \mathfrak{m} \) , and fix a coarsening to a positive \( \mathbb{N} \) -grading. The following are equivalent. | Proof. \( 2 \Leftrightarrow 1 \) by Definition 5.52, which works in any positive multigrading.\n\n\( 3 \Leftrightarrow 2 \) by the Auslander-Buchsbaum formula [BH98, Theorem 1.3.3].\n\n\( 4 \Leftrightarrow 3 \) : For the \( \Leftarrow \) direction, use the fact that every finitely generated module has a homogeneous sys... | Yes |
Theorem 13.42 Let \( \mathbb{k}\left\lbrack Q\right\rbrack \) be an affine semigroup ring, and express it as a quotient \( \mathbb{k}\left\lbrack Q\right\rbrack \cong S/{I}_{L} \), as in Theorem 7.3. Then \( \mathbb{k}\left\lbrack Q\right\rbrack \) is Cohen-Macaulay if and only if its dualizing complex from Definition ... | Proof. Theorem 13.24 plus Criterion 9 of Theorem 13.37 together imply that \( M = \mathbb{k}\left\lbrack Q\right\rbrack \) is Cohen-Macaulay precisely when the Ishida complex \( {\mathcal{O}}_{Q} \) has cohomology only in the latest possible place. This occurs if and only if the Matlis dual of \( {\mathcal{O}}_{Q}^{ \b... | Yes |
Proposition 14.2 The list \( \left( \right. \det \left( {\Theta }_{\sigma }\right) \left| \right| \sigma \subseteq \left\lbrack n\right\rbrack \) and \( \left. {\left| \sigma \right| = d}\right) \) of maximal minors up to scale identifies the row span of \( \Theta \) uniquely. More precisely, a matrix \( {\Theta }^{\pr... | Proof. If the matrices \( {\Theta }^{\prime } \) and \( \Theta \) have the same row span, then \( {\Theta }^{\prime } \) equals \( {\Gamma \Theta } \) for some invertible \( d \times d \) matrix \( \Gamma \) over \( \mathbb{k} \) . It follows that \( \det \left( {\Theta }_{\sigma }^{\prime }\right) = \) \( \det \left( ... | Yes |
We illustrate how the standard p-monomial in (14.6) is reconstructed from the initial \( \mathbf{x} \) -term (14.4) in its expansion under \( {\phi }_{n} \) . Let \( n = 4 \) . The following \( \mathbf{x} \) -monomial satisfies condition (14.5):\n\n\[ m = {x}_{11}^{2}{x}_{12}^{3}{x}_{13}{x}_{14}{x}_{22}{x}_{23}^{3}{x}_... | Indeed, this monomial lies outside of \( {\operatorname{in}}_{ \prec }\left( {I}_{4}\right) \), and \( \operatorname{in}\left( {{\phi }_{4}\left( {\mathbf{p}}^{\mathbf{a}}\right) }\right) = m \) . | Yes |
Lemma 14.13 A monomial \( m \) in \( \mathbb{k}\left\lbrack \mathbf{x}\right\rbrack \) is the initial term of a polynomial in the Plücker algebra \( \operatorname{image}\left( {\phi }_{n}\right) \) if and only if the tableau corresponding to \( m \) is semistandard. | Proof. We have seen in (14.4)-(14.6) that every semistandard tableau is associated to the initial term of \( {\phi }_{n}\left( {\mathbf{p}}^{\mathbf{a}}\right) \) for some monomial \( {\mathbf{p}}^{\mathbf{a}} \) supported on a chain in \( \mathcal{P} \) . On the other hand, Corollary 14.9 implies that every polynomial... | Yes |
Consider the special case of the Grassmannian \( {G}_{2,4} \) . Its homogeneous coordinate ring is generated by the six \( 2 \times 2 \) minors of a \( 2 \times 4 \) matrix of indeterminates, and its presentation ideal is\n\n\[ \ker \left( {\phi }_{4}\right) = \left\langle {{p}_{14}{p}_{23} - {p}_{13}{p}_{24} + {p}_{12... | The presentation ideal of the sagbi degeneration of \( {G}_{2,4} \) is\n\n\[ \ker \left( {\psi }_{4}\right) = \left\langle {{p}_{14}{p}_{23} - {p}_{13}{p}_{24}}\right\rangle \] | No |
Theorem 14.16 The toric ideal \( {J}_{n} = \ker \left( {\psi }_{n}\right) \) equals the initial ideal for the ideal \( {I}_{n} \) of Plücker relations with respect to the partial term order \( \leq \) . The reduced Gröbner basis of \( {J}_{n} \) under the reverse lexicographic term order on \( \mathbb{k}\left\lbrack \m... | Proof. The Gröbner basis constructed in the proof of Theorem 14.6 is minimal (meaning that no element in the Gröbner basis can be omitted) but not reduced. For the following argument we replace it by the reduced Gröbner basis. Consider any quadratic polynomial in the reduced Gröbner basis of \( {I}_{n} \) with respect ... | Yes |
In the above proof, it was essential that we used the reduced Gröbner basis of \( {I}_{n} \) instead of the Gröbner basis of Theorem 14.6. We illustrate the distinction for \( n = 8 \) . For the ideal of the Grassmannian \( {G}_{4,8} \), both Gröbner bases consist of 721 quadrics in 70 unknowns \( {p}_{ijkl} \) . A typ... | \[ \underline{{p}_{1278}{p}_{3456}} + \underline{{p}_{1258}{p}_{3467}} - {p}_{1257}{p}_{3468} - \underline{{p}_{1248}{p}_{3567}} + {p}_{1247}{p}_{3568} \] \[ + \underline{{p}_{1245}{p}_{3678} - {p}_{1238}{p}_{4576}} - {p}_{1237}{p}_{4568} - {p}_{1235}{p}_{4678} + {p}_{1234}{p}_{5678}. \] This quadric is not in the redu... | Yes |
Proposition 14.22 The semigroup \( \mathcal{G}{\mathcal{T}}_{n} \) has Hilbert basis \( {\mathcal{H}}_{n}^{\prime } \) consisting of partitions with distinct parts of size at most \( n \) . | Proof. Each such partition clearly lies inside \( \mathcal{G}{\mathcal{T}}_{n} \) . Furthermore, given a Gelfand-Tsetlin pattern \( {\left( {\lambda }_{i, j}\right) }_{i + j < n} \), drawn in the manner of (14.9), its set \( \lambda \) of nonzero entries lies in \( {\mathcal{H}}_{n}^{\prime } \) . Subtracting the parti... | Yes |
Theorem 14.23 There is an automorphism of \( {\mathbb{Z}}^{n \times n} \) taking the antidiagonal semigroup to the Gelfand-Tsetlin semigroup. In particular, \( {\mathcal{A}}_{n} \cong \mathcal{G}{\mathcal{T}}_{n} \) . | Proof. It is enough to demonstrate an automorphism of \( {\mathbb{Z}}^{n \times n} \) that induces a bijection between the Hilbert bases of \( {\mathcal{A}}_{n} \) and \( \mathcal{G}{\mathcal{T}}_{n} \) (Exercise 7.2). In our case, the automorphism of \( {\mathbb{Z}}^{n \times n} \) that takes an array \( {\left( {\lam... | Yes |
Corollary 14.24 The initial algebra of the Plücker algebra resulting from either term order in Theorem 14.11 is isomorphic (as a semigroup ring) to the Gelfand-Tsetlin semigroup ring. | Theorem 14.16 specifies a flat (sagbi) degeneration from the Plücker algebra \( \mathbb{k}\left\lbrack \mathbf{p}\right\rbrack /{I}_{n} \) to the Gelfand-Tsetlin semigroup ring \( \mathbb{k}\left\lbrack {\mathcal{G}{\mathcal{T}}_{n}}\right\rbrack = \mathbb{k}\left\lbrack \mathbf{p}\right\rbrack /{J}_{n} \) . The latter... | No |
Example 15.4 Five of the six \( 3 \times 3 \) matrix Schubert varieties for honest permutations are linear subspaces: | \[ {\bar{X}}_{123} = {M}_{33} \] \[ {I}_{213} = \left\langle {x}_{11}\right\rangle \] \[ {\overline{X}}_{213}\; = \;\{ Z \in {M}_{33}\;|\;{x}_{11} = 0\} \] \[ {I}_{231} = \left\langle {{x}_{11},{x}_{12}}\right\rangle \] \[ {\overline{X}}_{231}\; = \;\{ Z \in {M}_{33}\;|\;{x}_{11} = {x}_{12} = 0\} \] \[ {I}_{231} = \lef... | Yes |
Example 15.6 Let \( w = {13865742} \), so that the matrix for \( w \) is given by replacing each \( \times \) by 1 in the left matrix below. | Each \( 8 \times 8 \) matrix in \( {\bar{X}}_{w} \) has the property that every rectangular submatrix contained in the region filled with 1 ’s has rank \( \leq 1 \), and every rectangular submatrix contained in the region filled with 2’s has rank \( \leq 2 \), and so on. The ideal \( {I}_{w} \) contains the 21 minors o... | Yes |
Example 15.7 Let \( w \) be the \( 3 \times 3 \) partial permutation matrix \( \frac{1}{1}\frac{1}{1} \). The matrix Schubert variety \( {\bar{X}}_{w} \) is the set of \( 3 \times 3 \) matrices whose upper left entry is 0, and whose determinant vanishes. The ideal \( {I}_{w} \) is | \[ \left\langle {{x}_{11},\det \left\lbrack \begin{array}{lll} {x}_{11} & {x}_{12} & {x}_{13} \\ {x}_{21} & {x}_{22} & {x}_{23} \\ {x}_{31} & {x}_{32} & {x}_{33} \end{array}\right\rbrack }\right\rangle \] | Yes |
Proposition 15.8 Every partial permutation matrix \( w \) can be extended canonically to a square permutation matrix \( \widetilde{w} \) whose Schubert determinantal ideal \( {I}_{\widetilde{w}} \) has the same minimal generating minors as \( {I}_{w} \) . | Proof. Suppose that \( w \) is not already a permutation and that by symmetry there is a row (as opposed to a column) of \( w \) that has no 1 entries. Define \( {w}^{\prime } \) by adding a new column and placing a 1 entry in its highest possible row. Define \( \widetilde{w} \) by continuing until there is a 1 entry i... | Yes |
Lemma 15.9 The ideal generated by all minors of size \( r \) in \( {\mathbf{x}}_{p \times q} \) contains every minor of size \( r + 1 \) in \( {\mathbf{x}}_{p \times \left( {q + 1}\right) } \) or in \( {\mathbf{x}}_{\left( {p + 1}\right) \times q} \) . | Proof. Laplace expand each minor of size \( r + 1 \) along its rightmost column or bottom row, respectively. | No |
Proposition 15.10 If \( w \in {S}_{n} \) and \( \widetilde{w} \) extends \( w \) to an element of \( {S}_{n + {n}^{\prime }} \) fixing \( n + 1,\ldots, n + {n}^{\prime } \), then \( {I}_{w} \) and \( {I}_{\widetilde{w}} \) have the same minimal generators. | Proof. Add an extra column to \( w \) containing no 1 entries to get a partial permutation matrix \( {w}^{\prime } \) . Since every row of \( w \) contains a 1, the \ | No |
Theorem 15.15 The Schubert determinantal ideal \( {I}_{w} \subset \mathbb{k}\left\lbrack \mathbf{x}\right\rbrack \) is generated by minors coming from ranks in the essential set of \( w \) : | Proof. Suppose \( \left( {p, q}\right) \) does not lie in \( \mathcal{E}{ss}\left( w\right) \) . Either \( \left( {p, q}\right) \) lies outside \( D\left( w\right) \) , or one of the two locations \( \left( {p, q + 1}\right) \) and \( \left( {p + 1, q}\right) \) lies in \( D\left( w\right) \) .\n\nIn the former case, w... | Yes |
Lemma 15.19 In Bruhat order, \( v \leq w \) if and only if \( w \) lies in \( {\bar{X}}_{v} \) . In other words, \( v \leq w \) if and only if \( \operatorname{rank}\left( {v}_{p \times q}\right) \geq \operatorname{rank}\left( {w}_{p \times q}\right) \) for all \( p, q \) . | Proof. Clearly \( v \leq w \) implies \( w \in {\bar{X}}_{v} \) . For the other direction, note that \( w \in {\bar{X}}_{v} \) implies \( \operatorname{rank}\left( {Z}_{p \times q}\right) \leq \operatorname{rank}\left( {w}_{p \times q}\right) \leq \operatorname{rank}\left( {v}_{p \times q}\right) \) for all \( Z \in {\... | Yes |
Lemma 15.21 Fix a \( k \times \ell \) partial permutation matrix \( v \) with nonzero entries \( v\left( i\right) = j \) and \( v\left( {i}^{\prime }\right) = {j}^{\prime } \). If \( \left( {i, j}\right) < \left( {{i}^{\prime },{j}^{\prime }}\right) \), then the following hold.\n\n1. \( l\left( {{\tau }_{i,{i}^{\prime ... | Proof. Outside the mentioned rectangle, the diagram stays the same after the row switch. Inside the rectangle, nothing changes except that before the switch, no boxes in the diagram lie across the top edge or down the left edge, whereas after the switch, no boxes in the diagram lie on the bottom or right edges. In the ... | No |
Lemma 15.22 Fix a \( k \times \ell \) partial permutation matrix \( w \) with a nonzero entry \( w\left( i\right) = j \) and a zero row \( {i}^{\prime } \) . If \( i < {i}^{\prime } \) then the following hold.\n\n1. \( l\left( {{\tau }_{i,{i}^{\prime }}w}\right) = l\left( w\right) + 1 + {2e} + f \), where \( e \) is th... | Proof. Essentially the same as that of Lemma 15.21, except that the zero rows have no boxes in column \( \ell + 1 \) to move back to column \( j \) . | No |
The operators \( {\overrightarrow{\sigma }}_{7} \) and \( {\overleftarrow{\sigma }}_{7} \) move the bottom \( \times \) as follows: | If we had chosen a tall and thin ambient rectangle, then it would be possible to have a blank last row and no blank columns (this would fail to satisfy the first of the two parenthesized \ | No |
Proposition 15.27 In each orbit of \( B \times {B}_{ + } \) on \( {M}_{k\ell } \) lies a unique partial permutation \( w \), and the orbit \( {\mathcal{O}}_{w} \) through \( w \) is contained inside \( {\bar{X}}_{w} \). | Proof. Row and column operations that sweep down and to the right can get us from an arbitrary matrix \( Z \) to a partial permutation matrix \( w \) . Such sweeping preserves the ranks of northwest \( p \times q \) submatrices. This proves uniqueness of the partial permutation \( w \) in its orbit and also shows that ... | Yes |
Lemma 15.28 Set \( {w}^{\prime } = {\tau }_{i,{i}^{\prime }}w \), where \( i < {i}^{\prime } \) . If \( w < {w}^{\prime } \), then the closure \( {\mathcal{O}}_{{w}^{\prime }} \) of the orbit through \( {w}^{\prime } \) in \( {M}_{k\ell } \) is properly contained inside \( {\overline{\mathcal{O}}}_{w} \) . The same pro... | Proof. Let \( t \) be an invertible parameter. View each of the following equations as a possible scenario occurring in the two rows \( \left\{ {i,{i}^{\prime }}\right\} \) and two columns \( \left\{ {j,{j}^{\prime }}\right\} \) of an equation \( b\left( t\right) \cdot w \cdot {b}_{ + }{\left( t\right) }^{-1} = w\left(... | No |
Lemma 15.29 Given a \( k \times \ell \) partial permutation \( w \), there exists a chain \( {v}_{0} < {v}_{1} < {v}_{2} < \cdots < {v}_{k\ell - 1} < {v}_{k\ell } \) of covers in the weak order, in which \( l\left( {v}_{i}\right) = i \) for all \( i \), and \( {v}_{l\left( w\right) } = w \) . | Proof. It is enough to show that if \( 0 < l\left( w\right) \) then there exists a cover \( v < w \) , and if \( l\left( w\right) < k\ell \) then there exists a cover \( w < v \) . In the former case, choose \( v = {\sigma }_{i}w \) for the row index \( i \) on any box in the essential set \( \mathcal{E}{ss}\left( w\ri... | Yes |
Proposition 15.30 If \( w \) is a \( k \times \ell \) partial permutation, then the orbit closure \( {\overline{\mathcal{O}}}_{w} \) is an irreducible variety of dimension \( \dim \left( {\overline{\mathcal{O}}}_{w}\right) = k\ell - l\left( w\right) \) . | Proof. The map \( B \times {B}_{ + } \rightarrow {\mathcal{O}}_{w} \) that expresses \( {\mathcal{O}}_{w} \) as an orbit of \( B \times {B}_{ + } \) takes \( \left( {b,{b}_{ + }}\right) \mapsto {bw}{b}_{ + }^{-1} \) . This map of varieties induces a homomorphism \( \mathbb{k}\left\lbrack {M}_{k\ell }\right\rbrack \righ... | Yes |
Theorem 15.31 Let \( w \) be a \( k \times \ell \) partial permutation. The matrix Schubert variety \( {\bar{X}}_{w} \) is the closure \( {\overline{\mathcal{O}}}_{w} \) of the \( B \times {B}_{ + } \) orbit through \( w \in {M}_{k\ell } \) and is irreducible of dimension \( k\ell - l\left( w\right) \) . The matrix \( ... | Proof. Every point on an orbit \( \mathcal{O} \) of an algebraic group is a smooth point of \( \mathcal{O} \), because \( \mathcal{O} \) has a smooth point [Har77, Theorem I.5.3], and the group action is transitive on \( \mathcal{O} \) . Hence the smoothness at \( w \) follows from the rest.\n\nLet \( \widetilde{w} \) ... | Yes |
Corollary 15.33 Let \( w \) be a \( k \times \ell \) partial permutation and fix \( i < k \) . If \( {\sigma }_{i}w < w \), then \( {\sigma }_{i}\left( {\bar{X}}_{{\sigma }_{i}w}\right) = {\bar{X}}_{{\sigma }_{i}w} \) . | Proof. Let \( B \times {B}_{ + } \) act on \( {\sigma }_{i}\left( {\bar{X}}_{{\sigma }_{i}w}\right) \) and take the closure. As \( {\sigma }_{i}\left( {\bar{X}}_{{\sigma }_{i}w}\right) \) is irreducible, its image under the morphism \( \mu : B \times {B}_{ + } \times {\sigma }_{i}\left( {\bar{X}}_{{\sigma }_{i}w}\right... | Yes |
Lemma 15.35 Let \( v \) be an \( n \times n \) partial permutation and \( w \) an \( n \times n \) permutation with \( {\sigma }_{i}w < w \) and \( {\sigma }_{i}w < v \) . If \( l\left( v\right) = l\left( w\right) \), then \( {\bar{X}}_{v} \) has codimension 1 inside \( {\bar{X}}_{{\sigma }_{i}w} \), and \( {\bar{X}}_{... | Proof. The codimension statement comes from Theorem 15.31. Using Proposition 15.23, we find that \( v \) is obtained from \( {\sigma }_{i}w \) either by switching a pair of rows or deleting a single nonzero entry from \( {\sigma }_{i}w \) .\n\nAny 1 that we delete from \( {\sigma }_{i}w \) must have no 1’s southeast of... | Yes |
Lemma 15.36 Let \( w \) be an \( n \times n \) permutation with \( {\sigma }_{i}w < w \) . If \( \mathfrak{m} = {\mathfrak{m}}_{{\sigma }_{i}w} \) is the maximal ideal of \( {\sigma }_{i}w \in {\bar{X}}_{{\sigma }_{i}w} \), then the variable \( {x}_{i + 1, w\left( {i + 1}\right) } \) maps to \( \mathfrak{m} \smallsetmi... | Proof. Let \( v \) be the permutation \( {\sigma }_{i}w \), and consider the map \( B \times {B}_{ + } \rightarrow {M}_{nn} \) sending \( \left( {b,{b}^{ + }}\right) \mapsto {bv}{b}^{ + } \) . The image of this map is the orbit \( {\mathcal{O}}_{v} \subset {\bar{X}}_{v} \), and the identity id \( \mathrel{\text{:=}} \l... | Yes |
Proposition 15.37 Assume \( {\sigma }_{i}w < w \), set \( j = w\left( i\right) - 1 \), and define \( \Delta \) as the minor in \( \mathbf{x} \) using all rows and columns in which \( {\left( {\sigma }_{i}w\right) }_{i \times j} \) is nonzero. The images of \( \Delta \) and \( {\sigma }_{i}\Delta \) in \( \mathbb{k}\lef... | Proof. Lemma 15.35 says that \( {\sigma }_{i} \) induces an automorphism of the local ring at the prime ideal of \( {\bar{X}}_{v} \) inside \( {\bar{X}}_{{\sigma }_{i}w} \), for every boundary component \( {\bar{X}}_{v} \) of \( {\bar{X}}_{{\sigma }_{i}w} \) other than \( {\bar{X}}_{w} \) . This automorphism takes \( \... | Yes |
Let \( w \) be the partial permutation matrix in Example 15.2 with \( k \leq \ell \) . In this classical case, the double Schubert polynomial \( {\mathfrak{S}}_{w} \) is the Schur polynomial associated to the partition with rectangular Ferrers shape \( \left( {k - r}\right) \times \left( {\ell - r}\right) \) . The Jaco... | \[ \frac{\mathop{\prod }\limits_{{j = 1}}^{\ell }\left( {1 - {s}_{j}q}\right) }{\mathop{\prod }\limits_{{i = 1}}^{k}\left( {1 - {t}_{i}q}\right) }. \] (15.3) This formula appears in any book on symmetric functions, e.g. [Macd95]. \( \diamond \) | Yes |
The first five of the six \( 3 \times 3 \) matrix Schubert varieties in Example 15.4 have \( {\mathbb{Z}}^{3 + 3} \) -graded multidegrees that are products of expressions having the form \( {t}_{i} - {s}_{j} \) by Proposition 8.49. They are, in the order they appear in Example 15.4: \( 1,{t}_{1} - {s}_{1},\left( {{t}_{... | \[ \mathcal{C}\left( {{\bar{X}}_{132};\mathbf{t},\mathbf{s}}\right) = {t}_{1} + {t}_{2} - {s}_{1} - {s}_{2} \] of \( {\bar{X}}_{132} \), as the reader should check. | No |
The ideal \( {I}_{2143} \) from Example 15.7 equals \( I\left( {\bar{X}}_{2143}\right) \) | since it has a squarefree initial ideal \( \left\langle {{x}_{11},{x}_{13}{x}_{22}{x}_{31}}\right\rangle \) and is therefore a radical ideal. The multidegree of \( {\bar{X}}_{2143} \) is the double Schubert polynomial\n\n\[ \n{\mathfrak{S}}_{2143}\left( {\mathbf{t} - \mathbf{s}}\right) \n\]\n\n\[ \n= {\partial }_{2}{\p... | Yes |
Corollary 16.1 Schubert polynomials have nonnegative coefficients. | Proof. Write \( {\mathfrak{S}}_{w}\left( \mathbf{t}\right) = \mathcal{C}\left( {{\bar{X}}_{w};\mathbf{t}}\right) \) as in Theorem 15.40. Choosing a term order on \( \mathbb{k}\left\lbrack \mathbf{x}\right\rbrack \), Corollary 8.47 implies that \( {\mathfrak{S}}_{w}\left( \mathbf{t}\right) = \mathcal{C}\left( {\mathbb{k... | Yes |
The long permutation \( {w}_{0} = n\ldots {321} \) in \( {S}_{n} \) has a unique \( n \times n \) reduced pipe dream \( {D}_{0} \), whose + tiles fill the region strictly above the main antidiagonal, in spots \( \left( {i, j}\right) \) with \( i + j \leq n \) . | The right-hand pipe dream displayed before Definition 16.2 is \( {D}_{0} \) for \( n = 5 \) . | No |
The permutation \( w = {2143} \) has three reduced pipe dreams: | \[ \mathcal{{RP}}\left( {2143}\right) = \left\{ \begin{array}{lllll} 1 & 1 & 2 & 3 & 4 \\ 1 & 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 4 & 1 & 1 & 1 & 1 \\ 3 & 1 & 1 & 1 & 1 \\ 3 & 1 & 1 & 1 & 1 \end{array}\right\} . \] | No |
Lemma 16.5 Suppose that \( D \in \mathcal{R}\mathcal{P}\left( w\right) \), and let \( j \) be a fixed column index with \( \left( {i + 1, j}\right) \notin D \), but \( \left( {i, p}\right) \in D \) for all \( p \leq j \), and \( \left( {i + 1, p}\right) \in D \) for all \( p < j \) . Then \( l\left( {{\sigma }_{i}w}\ri... | Proof. Removing \( \left( {i, j}\right) \) only switches the exit points of the two pipes starting in rows \( i \) and \( i + 1 \) . Thus the pipe starting in row \( p \) of \( {D}^{\prime } \) exits out of column \( {\sigma }_{i}w\left( p\right) \) for every row index \( p \) . No pair of pipes can cross twice in \( {... | No |
Lemma 16.7 Chuting \( D \in \mathcal{R}\mathcal{P}\left( w\right) \) yields another reduced pipe dream for \( w \) . | Proof. If two pipes intersect at the - in the northeast corner of a chutable rectangle \( C \), then chuting that + only relocates the crossing point of those two pipes to the southwest corner of \( C \) . No other pipes are affected. | No |
Example 16.9 The pipe dream \( D \) at left is a reduced pipe dream for \( w = {13865742} \) . Applying mitosis \( {}_{3} \) yields the indicated set of pipe dreams: | The three offspring on the right are listed in the order they are produced by successive chute moves. | No |
Lemma 16.10 Fix a \( k \times \ell \) partial permutation \( w \), and suppose that \( i < k \) satisfies \( {\sigma }_{i}w < w \) . Then every pipe dream \( {D}^{\prime } \in \mathcal{R}\mathcal{P}\left( {{\sigma }_{i}w}\right) \) lies in \( {\operatorname{mitosis}}_{i}\left( D\right) \) for some pipe dream \( D \in \... | Proof. In column start \( {}_{i + 1}\left( {D}^{\prime }\right) \), rows \( i \) and \( i + 1 \) in \( {D}^{\prime } \) look like \( + \), because otherwise one of two illegal things must happen: the pipes passing through the row \( i \) of column start \( {}_{i + 1} \) in \( {D}^{\prime } \) intersect again at the clo... | Yes |
Theorem 16.11 If \( w \) is a \( k \times \ell \) partial permutation and \( i < k \) is a row index that satisfies \( {\sigma }_{i}w < w \), then the set of reduced pipe dreams for \( {\sigma }_{i}w \) is the disjoint union \( {\biguplus }_{D \in \mathcal{R}\mathcal{P}\left( w\right) }{\operatorname{mitosis}}_{i}\left... | Proof. Lemmas 16.5 and 16.7 imply that mitosis \( {}_{i}\left( D\right) \subseteq \mathcal{R}\mathcal{P}\left( {{\sigma }_{i}w}\right) \) whenever \( D \in \mathcal{R}\mathcal{P}\left( w\right) \), and Lemma 16.10 gives the reverse containment. That the union is disjoint (i.e., that mitosis \( {}_{i}\left( D\right) \ca... | Yes |
Theorem 16.13 \( {\mathfrak{S}}_{w}\left( \mathbf{t}\right) = \mathop{\sum }\limits_{{D \in \mathcal{{RP}}\left( w\right) }}{\mathbf{t}}^{D} \), where \( {\mathbf{t}}^{D} = \mathop{\prod }\limits_{{\left( {i, j}\right) \in D}}{t}_{i} \) . | The proof, at the end of this section, comes down to an attempt at calculating \( {\partial }_{i}\left( {\mathbf{t}}^{D}\right) \) directly. Fixing the loose ends in this method requires the involution in Proposition 16.16, to gather terms together in pairs. The involution is defined by first partitioning rows \( i \) ... | No |
Lemma 16.15 Let \( C \) be an intron in a reduced pipe dream. There is a unique intron \( \tau \left( C\right) \) satisfying the following two conditions.\n\n1. The sets of \( \boxplus \) columns are the same in \( C \) and \( \tau \left( C\right) \) .\n\n2. The number \( {c}_{i} \) of + tiles in row \( i \) of \( C \)... | Proof. First assume \( {c}_{i} > {c}_{i + 1} \) and work by induction on \( c = {c}_{i} - {c}_{i + 1} \) . If \( c = 0 \), then \( \tau \left( C\right) = C \) and the lemma is obvious. If \( c > 0 \), then consider the leftmost \( \boxplus \) column. Moving to the left from this column, there must be a column not equal... | Yes |
Proposition 16.16 For each \( i \) there is an involution \( {\tau }_{i} : \mathcal{R}\mathcal{P}\left( w\right) \rightarrow \mathcal{R}\mathcal{P}\left( w\right) \) such that \( {\tau }_{i}^{2} = 1 \), and for all \( D \in \mathcal{R}\mathcal{P}\left( w\right) \), the following hold:\n\n1. \( {\tau }_{i}D \) agrees wi... | Proof. Let \( D \in \mathcal{R}\mathcal{P}\left( w\right) \) . Consider the union of all columns in rows \( i \) and \( i + 1 \) of \( D \) that are east of or coincide with column start \( {}_{i}\left( D\right) \) . Since the first and last tiles in this region (numbered as in Definition 16.14) are elbows, this region... | Yes |
Theorem 16.18 The facets of the antidiagonal complex \( {\mathcal{L}}_{w} \) are the complements of the reduced pipe dreams for \( w \), yielding the prime decomposition\n\n\[ \n{J}_{w} = \mathop{\bigcap }\limits_{{D \in \mathcal{{RP}}\left( w\right) }}\left\langle {{x}_{ij} \mid \left( {i, j}\right) \text{ is a crossi... | It is convenient to identify each antidiagonal \( \underline{a} \in \mathbb{k}\left\lbrack \mathbf{x}\right\rbrack \) with the subset of the \( k \times \ell \) array of variables dividing \( \underline{a} \), just as we identify pipe dreams with their sets of + tiles. Then Theorem 16.18 can be equivalently rephrased a... | No |
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