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Proposition 7.30. Let\n\n\[ \n\begin{matrix} \left( {X,\mathcal{B},\mu, T}\right) \\ \downarrow \\ \left( {Y,\mathcal{A},\nu, S}\right) \end{matrix} \n\]\n\nbe a relatively weak-mixing extension of measure-preserving transformations on Borel probability spaces. Then for \( f, g \in {L}^{\infty }\left( {X,\mathcal{B},\m... | Proof of Proposition 7.30. For \( {f}_{1} \in {L}^{\infty }\left( \mathcal{A}\right) \), \n\n\[ \nE\left( {{f}_{1}{U}_{T}^{n}g \mid \mathcal{A}}\right) = {f}_{1}E\left( {{U}_{T}^{n}g \mid \mathcal{A}}\right) = E\left( {{f}_{1} \mid \mathcal{A}}\right) {U}_{T}^{n}E\left( {g \mid \mathcal{A}}\right) , \n\]\n\nso in this ... | Yes |
Theorem 8.4. Let \( G \) be a \( \sigma \) -locally compact metrizable group acting continuously on a \( \sigma \) -locally compact metrizable space \( X \) . Then the space \( {\mathcal{M}}^{G}\left( X\right) \) of \( G \) -invariant measures is a closed convex subset of \( \mathcal{M}\left( X\right) \) . A measure \(... | Proof. For \( g \in G \) write \( {\mathcal{M}}^{g}\left( X\right) \) for the space of \( g \) -invariant measures (that is, invariant measures for the transformation \( x \mapsto g \cdot x \) for \( x \in X \) ). Then we know that \( {\mathcal{M}}^{g}\left( X\right) \) is a closed convex subset of \( M\left( X\right) ... | Yes |
Proposition 8.5. With respect to Haar measure \( m = {m}_{X} \) ., the \( {\mathbb{Z}}^{2} \) -action \( T \) is mixing but not mixing of all orders. | Proof. The proof that \( T \) is mixing uses the same ideas as were used in the proof of Theorem 2.19. As in the case of a single transformation, mixing is equivalent to the property that\n\n\[ \int {f}_{0}\left( x\right) {f}_{1}\left( {{T}_{\mathbf{n}}x}\right) \mathrm{d}m \rightarrow \int {f}_{0}\mathrm{\;d}m\int {f}... | Yes |
Lemma 8.7. Let \( G \) be a \( \sigma \) -locally compact metrizable group acting continuously on a locally compact, \( \sigma \) -compact, metrizable space \( X \) . Let \( \mu \) be a locally finite measure on \( X \) which is invariant under \( G \) . Then, for \( p \in \lbrack 1,\infty ) \) and any \( f \in {L}_{\m... | Proof. First notice that for \( f \in {L}_{\mu }^{p} \) , \n\n\[ \n{\int }_{X}{\left| f\left( {g}^{-1} \cdot x\right) \right| }^{p}\mathrm{\;d}\mu \left( x\right) = {\int }_{X}{\left| f\left( x\right) \right| }^{p}\mathrm{\;d}\mu \left( x\right) , \n\] \n\nsince by assumption \( \mu \) is invariant under \( g \in G \) ... | Yes |
Theorem 8.10. If a locally compact group \( G \) is amenable, then every continuous \( G \) -action \( G \rightarrow \operatorname{Homeo}\left( {X,\mathrm{\;d}}\right) \) on a compact metric space has an invariant probability measure. | Proof. Recall that we write \( \mathcal{M}\left( X\right) \) for the space of probability measures on \( \left( {X,\mathrm{\;d}}\right) \) with the weak*-topology (see p. 97); in this argument we will make use of the theory of integration for functions taking values in the space of measures (see Chap. 5 and Sect. A.3).... | Yes |
Theorem 8.11. If \( G \) is an abelian locally compact group, then every continuous \( G \) -action \( G \rightarrow \operatorname{Homeo}\left( {X,\mathrm{\;d}}\right) \) on a compact metric space has an invariant probability measure. | Proof. The proof uses averaging and compactness very much like the proof of Theorem 4.1. For each \( g \in G \) and \( n \geq 0 \) define a map \( {A}_{n, g} : \mathcal{M}\left( X\right) \rightarrow \mathcal{M}\left( X\right) \) by\n\n\[ \n{A}_{n, g}\left( \mu \right) = \frac{1}{n}\mathop{\sum }\limits_{{j = 0}}^{{n - ... | Yes |
Lemma 8.12. If \( G \) is compact, then any continuous action of \( G \) has an invariant probability measure. | Proof. Let \( G \) act continuously on a compact metric space \( \left( {X,\mathrm{\;d}}\right) \), and let \( x \) be a point in \( X \) . Define \( \phi : G \rightarrow X \) by \( \phi \left( g\right) = g \cdot x \) . Writing \( {m}_{G} \) for the Haar measure on \( G \), we see that \( {\phi }_{ * }\left( {m}_{G}\ri... | Yes |
Theorem 8.13. Let \( G \) be a \( \sigma \) -locally compact amenable group with left Haar measure \( {m}_{G} \) acting continuously on \( X \), and let \( \mu \) be a \( G \) -invariant Borel probability measure on \( X \) . Let \( {P}_{G} \) be the orthogonal projection onto the closed subspace\n\n\[ I = \\left\\{ {f... | Proof of Theorem 8.13. Let \( u \) be a function of the form\n\n\[ u\\left( x\\right) = v\\left( {h \\cdot x}\\right) - v\\left( x\\right)\n\]\n\nfor some \( v \\in {L}_{\\mu }^{2}\\left( X\\right) \) and \( h \\in G \), that is \( u = {U}_{{h}^{-1}}v - v \) . Then\n\n\[ {\\int }_{{F}_{n}}{U}_{{g}^{-1}}{U}_{{h}^{-1}}v\... | Yes |
Corollary 8.14. Let \( G \) be a locally compact amenable group with left Haar measure \( {m}_{G} \) acting continuously on \( X \), and let \( \mu \) be a \( G \) -invariant Borel probability measure on \( X \) . Then, for any Følner sequence \( \left( {F}_{n}\right) \) and \( f \in {L}_{\mu }^{1}\left( X\right) \) , | \[ \frac{1}{{m}_{G}\left( {F}_{n}\right) }{\int }_{{F}_{n}}f \circ g\mathrm{\;d}{m}_{G}\left( g\right) \rightarrow E\left( {f \mid \mathcal{E}}\right) \] in \( {L}_{\mu }^{1} \), where \( \mathcal{E} \) is the \( \sigma \) -algebra of \( G \) -invariant sets. | Yes |
Corollary 8.15. Let \( T \) be a measurable and measure-preserving (semi-)flow on the probability space \( \left( {X,\mathcal{B},\mu }\right) \). Then, for any \( f \in {L}_{\mu }^{1} \), there is a measurable set of full measure on which\n\n\[ \frac{1}{s}{\int }_{0}^{s}f\left( {{T}_{s}x}\right) \mathrm{d}s \rightarrow... | Proof. The function \( \left( {x, s}\right) \mapsto f\left( {{T}_{s}\left( x\right) }\right) \) is integrable on \( X \times \left\lbrack {0, s}\right\rbrack \) for any non-negative \( s \) by Fubini’s theorem (Theorem A.13). Thus the integral \( {\int }_{0}^{s}f\left( {{T}_{t}x}\right) \mathrm{d}t \) is well-defined f... | Yes |
Theorem 8.19. Let \( G \) be a unimodular locally compact group with a right-invariant metric satisfying properties \( \left( \mathrm{P}\right) \) ,(D) and \( \left( \mathrm{F}\right) \) . Let \( G \) act continuously on a locally compact \( \sigma \) -compact metric space \( X \), preserving a Borel probability measur... | Proof of Theorem 8.19. Assume first that \( {f}_{0} \in {\mathcal{L}}^{\infty } \), so that\n\n\[ \n{\mathrm{A}}_{n}\left( {f}_{0}\right) \rightarrow {F}_{0} = E\left( {{f}_{0} \mid \mathcal{E}}\right) \n\]\n\nas \( n \rightarrow \infty \) in \( {L}_{\mu }^{1} \) by the mean ergodic theorem for \( {L}^{1} \) (Corollary... | Yes |
Lemma 8.21. Let \( {P}_{1},{P}_{2},\ldots \) be orthogonal projections, all defined on a separable Hilbert space \( \mathcal{H} \). Define the operators\n\n\[ \n{Q}_{1} = {P}_{1},{Q}_{2} = {Q}_{1}{P}_{2}{Q}_{1},\ldots ,{Q}_{n + 1} = {Q}_{n}{P}_{n + 1}{Q}_{n} \n\]\n\nfor any \( n \geq 0 \). Then the sequence \( \left( {... | Proof of Lemma 8.21. Fix some \( v \in \mathcal{H} \), and let \( w \) be a weak*-limit of a subsequence \( \left( {{Q}_{{n}_{k}}v}\right) \). That is, \( w \in \mathcal{H} \) satisfies\n\n\[ \n\left\langle {{Q}_{{n}_{k}}v,\xi }\right\rangle \rightarrow \langle w,\xi \rangle \n\]\n\nas \( k \rightarrow \infty \), for a... | Yes |
Lemma 8.22. Let \( G = \left\{ {{g}_{1},{g}_{2},\ldots }\right\} \) be a countable group acting continuously on a \( \sigma \) -compact metric space \( \left( {X,\mathrm{\;d}}\right) \), and let \( \mu \) be a \( G \) -invariant probability measure on \( X \) . Then \( \mu \) is ergodic if and only if, for any \( f \in... | Proof of Lemma 8.22. Let \( f \in {C}_{c}\left( X\right) \) and assume that \( \mu \) is an ergodic probability measure. Then \( {Q}_{n}f \) converges in \( {L}_{\mu }^{2} \) to \( \int f\mathrm{\;d}\mu \), so there exists a subsequence \( \left( {n}_{k}\right) \) for which \( {Q}_{{n}_{k}}f \rightarrow \int f\mathrm{\... | Yes |
Proposition 8.24. Let \( G \) be a \( \sigma \) -compact metrizable group, and let \( \nu \) be a probability measure on \( G \) . Suppose that \( G \) acts continuously on a compact metric space \( X \) . Then there exists a \( \nu \) -stationary measure on \( X \) . | Proof. Let \( {\mu }_{0} \in \mathcal{M}\left( X\right) \) be any probability measure on \( X \), and define a new probability measure \( \nu * {\mu }_{0} \) by\n\n\[ \nu * {\mu }_{0} = {\int }_{G}{g}_{ * }{\mu }_{0}\mathrm{\;d}\nu \left( g\right) \]\n\nAlso define the averages\n\n\[ {\mu }_{N} = \frac{1}{N}\mathop{\su... | Yes |
Lemma 9.1. The action of \( {\mathrm{{PSL}}}_{2}\left( \mathbb{R}\right) \) on \( \mathbb{H} \) defined by (9.1) has the following properties.\n\n(1) The action is isometric, meaning that\n\n\[ \mathrm{d}\left( {g\left( {z}_{0}\right), g\left( {z}_{1}\right) }\right) = \mathrm{d}\left( {{z}_{0},{z}_{1}}\right) \]\n\nfo... | Proof of Lemma 9.1 (1): Since the metric is defined in terms of the Riemannian metric, we need to start by proving the second claim. For \( v, w \) in \( {\mathrm{T}}_{z}\mathbb{H} \) we have \( {\left( \mathrm{D}g\right) }_{z}v,{\left( \mathrm{D}g\right) }_{z}w \in {\mathrm{T}}_{g\left( z\right) }\mathbb{H} \) and (9.... | Yes |
Lemma 9.2. The action of \( {\mathrm{{PSL}}}_{2}\left( \mathbb{R}\right) \) on \( {\mathrm{T}}^{1}\mathbb{H} \) is simply transitive. | Proof of Lemma 9.2. Since we already know that the action on \( \mathbb{H} \) is transitive, it is enough to consider vectors \( v \in {\mathrm{T}}_{\mathrm{i}}\mathbb{H} \) with base point \( \mathrm{i} \), and here we compute\n\n\[{\left( \mathrm{D}g\right) }_{\mathrm{i}}\left( v\right) = \frac{1}{{\left( \mathrm{i}\... | Yes |
Lemma 9.3. Let \( {z}_{0} = {y}_{0}\mathrm{i} \) and \( {z}_{1} = {y}_{1}\mathrm{i} \) with \( 0 < {y}_{0} < {y}_{1} \). Then\n\n\[ \mathrm{d}\left( {{z}_{0},{z}_{1}}\right) = \log {y}_{1} - \log {y}_{0} \]\n\nand\n\n\[ \phi \left( t\right) = {y}_{0}{\left( \frac{{y}_{1}}{{y}_{0}}\right) }^{t}\mathrm{i} \]\n\nfor \( t ... | Proof of Lemma 9.3. It is readily checked that the path \( \phi \) defined in the lemma has constant speed equal to \( \log {y}_{1} - \log {y}_{0} = \mathrm{L}\left( \phi \right) \) as claimed. It follows that\n\n\[ \mathrm{d}\left( {{z}_{0},{z}_{1}}\right) \leq \log {y}_{1} - \log {y}_{0} \]\n\nSuppose now that \( \et... | Yes |
For any two points \( {z}_{0},{z}_{1} \in \mathbb{H} \) there is a unique path\n\n\[ \phi : \left\lbrack {0,\mathrm{\;d}\left( {{z}_{0},{z}_{1}}\right) }\right\rbrack \rightarrow \mathbb{H} \]\n\nof unit speed with \( \phi \left( 0\right) = {z}_{0} \) and \( \phi \left( {\mathrm{d}\left( {{z}_{0},{z}_{1}}\right) }\righ... | Proof. We first claim that there exists a \( g \in {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) with \( {g}^{-1}\left( {z}_{0}\right) = \mathrm{i} \) and \( {g}^{-1}\left( {z}_{1}\right) = \mathrm{i}{y}_{1} \) for some \( {y}_{1} > 1 \) . By Lemma 9.1(2) we can certainly find some \( \widetilde{g} \in {\operator... | Yes |
Lemma 9.6. There is a neighborhood \( B \) of \( 0 \in {\operatorname{Mat}}_{dd}\left( \mathbb{R}\right) \) such that for any \( v \in B \), and any sequence \( \left( {v}_{m}\right) \) with \( {v}_{m} \rightarrow v \) as \( m \rightarrow \infty \), we have\n\n\[{\left( I + \frac{1}{m}{v}_{m}\right) }^{m} \rightarrow \... | Proof. If \( v \) is sufficiently small, then for large \( m \) we may use the geometric series to get the estimate\n\n\[m\log \left( {I + \frac{1}{m}{v}_{m}}\right) = m\left( {\frac{1}{m}{v}_{m} - \frac{1}{2{m}^{2}}{v}_{m}^{2} + \cdots }\right)\]\n\n\[= {v}_{m} + \mathrm{O}\left( {1/m}\right) \text{.}\]\n\nThis implie... | Yes |
For any closed linear group \( G \subseteq {\mathrm{{GL}}}_{d}\left( \mathbb{R}\right) \) the Lie algebra \( \mathfrak{g} \) uniquely determines the connected component \( {G}^{0} \) of the identity in \( G \) . Indeed, \( {G}^{0} \) is the group generated by \( \exp \left( \mathfrak{g}\right) \) . Moreover, \( {G}^{0}... | Proof. Define, for \( n \geq 1 \) ,\n\n\[{\left( \exp \left( \mathfrak{g}\right) \right) }^{n} = \left\{ {\exp \left( {v}_{1}\right) \exp \left( {v}_{2}\right) \cdots \exp \left( {v}_{n}\right) \mid {v}_{1},\ldots ,{v}_{n} \in \mathfrak{g}}\right\}\]\n\nand\n\n\[H = \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{\left( \... | Yes |
We claim that \( {\mathrm{{SL}}}_{d}\left( \mathbb{R}\right) \) is connected, so that \( {\mathrm{{SL}}}_{d}\left( \mathbb{R}\right) \) is uniquely determined by \( {\mathfrak{{sl}}}_{d}\left( \mathbb{R}\right) \) in the sense of Corollary 9.7. Let \( {E}_{ij} \) denote the matrix with a single non-zero entry 1 in the ... | Equivalently, it is enough to note that any \( g \in {\mathrm{{SL}}}_{d}\left( \mathbb{R}\right) \) can be reduced to \( I \) by a finite sequence of row operations consisting of adding a multiple of the \( j \) th row to the \( i \) th for any \( i \neq j \) . | Yes |
Proposition 9.10. Let \( G \) be a closed linear group, let \( \phi : \left\lbrack {0,1}\right\rbrack \rightarrow G \) be a continuous curve which is differentiable at \( {t}_{0} \in \left\lbrack {0,1}\right\rbrack \), and let \( g \in G \) . Then the curves \( \left( {g\phi }\right) \left( t\right) = {g\phi }\left( t\... | Proof. By definition,\n\n\[ D\left( {g\phi }\right) \left( {t}_{0}\right) = \left( {{g\phi }\left( {t}_{0}\right) ,{\left( g\phi \left( {t}_{0}\right) \right) }^{-1}g{\phi }^{\prime }\left( {t}_{0}\right) }\right) = \left( {{g\phi }\left( {t}_{0}\right) ,\phi {\left( {t}_{0}\right) }^{-1}{\phi }^{\prime }\left( {t}_{0}... | Yes |
For any closed linear group \( G \), the Riemannian metric defined by equation (9.8) defines a left-invariant metric on \( {G}^{0} \) . That is, if we define the length of a piecewise smooth curve \( \phi : \left\lbrack {0,1}\right\rbrack \rightarrow G \) by\n\n\[ \n\mathrm{L}\left( \phi \right) = {\int }_{0}^{1}\paral... | The first statement in Corollary 9.11 is an immediate consequence of Proposition 9.10 and the definitions, and the rest follows just as in our earlier discussion of the hyperbolic plane \( \mathbb{H} \) . The only difference lies in the fact that we need to restrict to \( {g}_{0},{g}_{1} \in {G}^{0} \) in order that th... | Yes |
The Heisenberg group\n\n\[ G = \left\{ {\left. \left( \begin{array}{rrr} 1 & x & z \\ & 1 & y \\ & & 1 \end{array}\right) \right| \;x, y, z \in \mathbb{R}}\right\} \]\n\nhas the Lie algebra\n\n\[ \mathfrak{g} = \left\{ {\left. \left( \begin{array}{lll} 0 & u & w \\ 0 & 0 & v \\ 0 & 0 & 0 \end{array}\right) \right| \;u,... | The closed subgroup\n\n\[ H = \left\{ {\left. \left( \begin{array}{rrr} 1 & 0 & z \\ & 1 & 0 \\ & & 1 \end{array}\right) \right| \;z \in \mathbb{R}}\right\} \]\n\nhas Lie algebra\n\n\[ \mathfrak{h} = \left\{ {\left. \left( \begin{array}{lll} 0 & 0 & w \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right) \right| \;w \in \mathbb... | Yes |
Proposition 9.14. Let \( G \) be a closed linear group and \( \Gamma \leq G \) a discrete subgroup. Then for any \( x \in X = \Gamma \smallsetminus G \) there exists some \( r > 0 \) such that the map from\n\n\[ \n{B}_{r}^{G} = \\left\\{ {g \in G \mid {\\mathrm{d}}_{G}\\left( {g, e}\\right) < r}\\right\\} \n\]\n\nto\n\... | Proof of Proposition 9.14. Let \( x = {\Gamma h} \) and fix some \( r > 0 \) . Then, for \( {g}_{1},{g}_{2} \in {B}_{r}^{G} \) ,\n\n\[ \n{\\mathrm{d}}_{X}\\left( {{\\Gamma h}{g}_{1},{\\Gamma h}{g}_{2}}\\right) = \\mathop{\\inf }\\limits_{{\\gamma \in \\Gamma }}{\\mathrm{d}}_{G}\\left( {h{g}_{1},{\\gamma h}{g}_{2}}\\rig... | Yes |
We claim that the quotient space \( X = \Gamma \smallsetminus G \) is compact but is not a group with respect to the canonical multiplication of coset representative inherited from the group structure on \( G \). | The statement that \( X \) is not a group is simply the statement that \( \Gamma \) is not a normal subgroup, which is easily seen:\n\n\[ \left( \begin{array}{rrr} 1 & x & 0 \\ 1 & 0 & 1 \\ & 1 & 1 \end{array}\right) \left( \begin{array}{rrr} 1 & 0 & 0 \\ 1 & 1 & 1 \\ & 1 & 1 \end{array}\right) \left( \begin{array}{rrr... | Yes |
Lemma 9.16. The hyperbolic area form \( \mathrm{d}A = \frac{1}{{y}^{2}}\mathrm{\;d}x\mathrm{\;d}y \) on \( \mathbb{H} \), and the hyperbolic volume form\n\n\[ \mathrm{d}m = \frac{1}{{y}^{2}}\mathrm{\;d}x\mathrm{\;d}y\mathrm{\;d}\theta \]\n\non \( {\mathrm{T}}^{1}\mathbb{H} \), where \( \theta \) gives the angle of the ... | Proof. Recall that the complex derivative of \( z \mapsto g\left( z\right) = \frac{{az} + b}{{cz} + d} \) is \( \frac{1}{{\left( cz + d\right) }^{2}} \) , so the Jacobian is \( \frac{1}{{\left| cz + d\right| }^{4}} \) . Therefore, for any continuous function \( f : \mathbb{H} \rightarrow \mathbb{R} \) with compact supp... | Yes |
Proposition 9.18. The set \( E = \{ z \in \mathbb{H}\left| \right| z\left| { \geq 1,}\right| \Re \left( z\right) | \leq \frac{1}{2}\} \) illustrated in Fig. 9.5 is a fundamental domain for the action of \( {\operatorname{PSL}}_{2}\left( \mathbb{Z}\right) \) on \( \mathbb{H} \) in the following sense:\n\n\[ A\left( {{\g... | Proof of Proposition 9.18. Let \( z \in \mathbb{H} \) . We first show that there is some element \( \gamma \in {\operatorname{PSL}}_{2}\left( \mathbb{Z}\right) \) with \( {\gamma z} \in E \), proving (9.13). Recall that for \( \gamma = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \), \n\n\[ \Im \left( {\g... | Yes |
Proposition 9.19. If \( \Gamma \leq {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) is a lattice, the hyperbolic measure defined by the volume form \( \mathrm{d}m = \frac{1}{{y}^{2}}\mathrm{\;d}x\mathrm{\;d}y\mathrm{\;d}\theta \) in Lemma 9.16 induces a \( {fi} \) - nite \( {\mathrm{{PSL}}}_{2}\left( \mathbb{R}\rig... | \[ \pi : {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \rightarrow X \] is the canonical quotient map \( \pi \left( g\right) = {\Gamma g} \) for \( g \in {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) and \( F \) is a finite volume fundamental domain, then \[ {m}_{X}\left( B\right) = m\left( {F \cap {\pi }^{-1}... | Yes |
Proposition 9.20. Let \( G \) be a closed linear group, and let \( \Gamma \leq G \) be a lattice in the sense that \( \Gamma \) is discrete and that there is a fundamental domain \( F \) for \( X = \Gamma \smallsetminus G \) with finite left Haar measure. Then any fundamental domain has the same measure as \( F, G \) i... | Proof of Proposition 9.20. We first show that any two fundamental domains \( F,{F}^{\prime } \subseteq G \) for \( \Gamma \smallsetminus G \) have the same volume. In fact we claim that if \( B,{B}^{\prime } \subseteq G \) are measurable sets with the property that \( {\left. \pi \right| }_{B} \) and \( {\left. \pi \ri... | Yes |
Theorem 9.21. Let \( \Gamma \leq {\mathrm{{PSL}}}_{2}\left( \mathbb{R}\right) \) be a lattice. Then any non-trivial element of the geodesic flow (that is, the map \( {R}_{{a}_{t}} \) for some \( t \neq 0 \) ) is an ergodic transformation on \( X = \Gamma \smallsetminus {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \... | Proof of Theorem 9.21. Normalize the Haar measure \( {m}_{X} \) to ensure that \( {m}_{X}\left( X\right) = 1 \) and let \( f : X \rightarrow \mathbb{R} \) be a measurable \( {R}_{{a}_{t}} \) -invariant function for some \( t \neq 0 \) . Fix \( \varepsilon > 0 \) and choose a compact set \( K \subseteq X \) of measure \... | Yes |
Lemma 9.22. Let \( x = \left( {\mathrm{i}b,\mathbf{v}}\right) \) be in \( {C}_{ + } \) with natural coordinates \( \left( {y, z}\right) \). The next visit, if there is one, of the geodesic flow to the set \( \pi \left( C\right) \) occurs in \( \pi \left( {C}_{ - }\right) \) and has coordinates\n\n\[ \bar{T}\left( {y, z... | Proof. The isometry \( z \mapsto - \frac{1}{z} \) sends the usual fundamental domain \( E \) to another fundamental domain illustrated in Fig. 9.10; this figure also shows a geodesic that never returns to them.\n\n\n... | Yes |
Proposition 9.25. The Gauss map \( T\left( y\right) = \left\{ \frac{1}{y}\right\} \) on \( \left\lbrack {0,1}\right\rbrack = Y \) is ergodic with respect to the Gauss measure \( \mathrm{d}\mu = \frac{1}{\log 2}\frac{1}{1 + x}\mathrm{\;d}x \) . The return time function \( {r}_{C} \) is integrable. | Proof. We will prove ergodicity for the invertible extension of the Gauss map discussed in Sect. 3.4. Recall that this system is the map \( \bar{T} : \bar{Y} \rightarrow \bar{Y} \) given by\n\n\[ \bar{T}\left( {y, z}\right) = \left( {{Ty}, y\left( {1 - {yz}}\right) }\right) \]\n\non the set\n\n\[ \bar{Y} = \{ \left( {y... | Yes |
Theorem 10.1. For the transformation \( S\left( x\right) = {x\tau } \) on \( X \), the following are equivalent:\n\n- \( S \) is uniquely ergodic.\n\n- \( S \) is ergodic with respect to \( {m}_{X} \).\n\n- \( \tau = \left( \begin{array}{rrr} 1 & \alpha & \delta \\ 1 & 1 & \beta \end{array}\right) \) and \( 1,\alpha ,\... | ## 10.3 First Proof of Theorem 10.1\n\nAssume that \( 1,\alpha ,\beta \) are linearly independent over \( \mathbb{Q} \) (in the notation of Theorem 10.1). Even though Theorem 4.21 does not apply directly, since \( X \) is not topologically the direct product \( {\mathbb{T}}^{2} \times \mathbb{T} \), the argument used t... | No |
Lemma 10.3. Let \( S : X \rightarrow X \) be a continuous map on a compact metric space equipped with an \( S \) -invariant and ergodic Borel probability measure \( \mu \) , and let \( R : X \rightarrow X \) be another continuous map that commutes with \( S \) . If there exists a point \( x \in X \) that is \( \mu \) -... | Proof. The proof proceeds quite directly from the definitions. Let \( f \in C\left( X\right) \) ; it is enough to show that\n\n\[ \int f\mathrm{\;d}{R}_{ * }\mu = \int f \circ R\mathrm{\;d}\mu = \int f\mathrm{\;d}\mu . \]\n\nNotice that by continuity of \( R \), we also have \( f \circ R \in C\left( X\right) \) . Since... | Yes |
Lemma 10.4. Fix \( \ell \geq 1 \) . If \( V \) is a sufficiently small neighborhood of \( e \in G \) and \( z \in \operatorname{Supp}{\left. \mu \right| }_{{X}_{1}} \), then for \( \mu \) -almost every \( x \in {zV} \cap {X}_{1} \) there exists an \( m \geq 1 \) such that \( {x}^{\prime } = {S}^{m}x \in {zV} \cap {X}_{... | Proof. The first claim regarding the existence of such an \( m \) is just Poincaré recurrence (Theorem 2.11) since \( \mu \left( {{zV} \cap {X}_{1}}\right) > 0 \) by assumption. Also, the displacement \( g \) lies in \( {V}^{-1}V \) (since we may choose \( h,{h}^{\prime } \in V \) with \( x = {zh} \) and \( {x}^{\prime... | Yes |
Lemma 11.2. An infinite subgroup \( \Gamma \subseteq {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) is a Fuchsian group if and only if its action on \( \mathbb{H} \) is properly discontinuous. | Proof. If \( \Gamma \) is not discrete then we may choose a sequence of elements \( \left( {g}_{n}\right) \) with \( {g}_{n} \neq e \) for all \( n \geq 1 \) and \( {g}_{n} \rightarrow e \) as \( n \rightarrow \infty \) . If \( P \) is a compact set containing an open set, then \( {g}_{n}P \cap P \neq \varnothing \) fo... | Yes |
For any \( \gamma \in {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) the open set\n\n\[ \n{D}_{\gamma } = \{ z \in \mathbb{H} \mid \mathrm{d}\left( {z, p}\right) < \mathrm{d}\left( {z,{\gamma p}}\right) \} \n\]\n\nis the connected component of \( \mathbb{H} \smallsetminus {L}_{\gamma } \) containing \( p \), where... | To see that \( {L}_{\gamma } \) is a geodesic and the description of \( {D}_{\gamma } \) is valid, notice that both depend only on the points \( p \) and \( {\gamma p} \) . We may apply an isometry \( g \in {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) to map those two points to \( - r + \mathrm{i} \) and \( r + ... | Yes |
Lemma 11.5. Any Dirichlet region for an infinite Fuchsian group \( \Gamma \) is an open fundamental domain for the action of \( \Gamma \) on \( \mathbb{H} \) . The boundary of a Dirichlet region is made up of geodesic segments contained in geodesics defined by\n\n\[ \n{L}_{\gamma } = \{ z \in \mathbb{H} \mid \mathrm{d}... | Proof. Let \( D = D\left( p\right) \) be a Dirichlet region. Since the action of \( \Gamma \) is properly discontinuous by Lemma 11.2, we have that for any \( z \in \mathbb{H} \) there are only finitely many \( \gamma \in \Gamma \) with\n\n\[ \n\mathrm{d}\left( {{\gamma z}, p}\right) \leq \mathrm{d}\left( {z, p}\right)... | Yes |
Lemma 11.6. The boundary of a Dirichlet region \( \partial D \) is the union of at most countably many connected components. Each connected component of \( \partial D \) is the image of a piecewise geodesic path \( \phi : \mathbb{R} \rightarrow \mathbb{H} \) . Either \( D \) is a convex polygon and this path periodical... | Proof. Fix some \( R > 1 \) and write \( {B}_{R} \) for the hyperbolic ball of radius \( R \) around \( p \), the chosen point defining \( D \) . Then we may find finitely many elements \( {\gamma }_{0} = I,{\gamma }_{1},\ldots ,{\gamma }_{n} \in \Gamma \) such that if \( z \in {B}_{R} \) and \( \gamma \left( z\right) ... | Yes |
Proposition 11.8. [Gauss-Bonnet formula] Let \( P \) be a hyperbolic \( n \) - sided convex polygon in \( \mathbb{H} \) with \( n \geq 3 \) vertices in \( \overline{\mathbb{H}} \), with angles \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) at the \( n \) vertices. Then the hyperbolic area of \( P \) is\n\n\[ \left( {n - 2}\... | Proof of Proposition 11.8. Assume for the purposes of an induction that the formula holds for all polygons with no more than \( \left( {n - 1}\right) \) sides, and let \( P \) be a polygon with \( n \) sides as in the statement of the lemma. By cutting off one triangle \( T \) (see Fig. 11.2) to leave an \( \left( {n -... | Yes |
Lemma 11.9. A lattice \( \Gamma \subseteq {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) is uniform (that is, \( \Gamma \smallsetminus {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) is compact) if and only if every vertex of any Dirichlet region for \( \Gamma \) lies in \( \mathbb{H} \) (that is, has compact ... | Proof. Let \( D \) be a Dirichlet region for \( \Gamma \) . If the boundary of \( D \) lies in \( \mathbb{H} \) , then the closure of \( D \) is a compact subset of \( \mathbb{H} \) . The compact subset\n\n\[ F = \left\{ {g \in {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \mid g\left( \mathrm{i}\right) \in \bar{D}}... | Yes |
The subgroup \( \Gamma \left( 2\right) \) described above has index 6 in \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) | since \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) /\Gamma \left( 2\right) \cong {\mathrm{{SL}}}_{2}\left( {\mathbb{F}}_{2}\right) \) has order 6 | Yes |
Lemma 11.12. There is a uniform lattice in \( {\mathrm{{PSL}}}_{2}\left( \mathbb{R}\right) \) . | SKETCH PROOF OF LEMMA 11.12. First notice that there is a regular fourgon \( D \) for which all of the internal angles are equal to \( \frac{\pi }{3} \) . To see this, draw two geodesics intersecting at \( \mathrm{i} \) in a normal angle, and consider the four points on this pair of geodesics at distance \( t \) from i... | No |
Lemma 11.13. Every \( g \in {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) is conjugate to an element of \( \pm A, \pm {U}^{ - } \) , or \( {\mathrm{{SO}}}_{2}\left( \mathbb{R}\right) \) . | Proof. If \( g \in {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) is diagonalizable over \( \mathbb{R} \), then it is conjugate to an element of \( A \) or \( - A \) ; if it is diagonalizable over \( \mathbb{C} \) but not over \( \mathbb{R} \) then it is conjugate to an element of \( {\mathrm{{SO}}}_{2}\left( \mathbb{R... | Yes |
Lemma 11.14. Let \( \Gamma \leq G \) be a discrete subgroup of a closed linear group. Suppose that \( {g}_{2} = h{g}_{1}{h}^{-1} \) for \( {g}_{1},{g}_{2}, h \in G \) . Then the maps \( {R}_{{g}_{1}} \) and \( {R}_{{g}_{2}} \) on \( X = \Gamma \smallsetminus G \) are conjugate via \( {R}_{h} \) . In particular, if \( \... | Proof. This is clear since \( {R}_{{g}_{2}} = {R}_{h}{R}_{{g}_{1}}{R}_{h}^{-1} \), and if \( \Gamma \) is a lattice then \( {R}_{h} \) preserves the finite measure \( {m}_{X} \) . | Yes |
Theorem 11.15. Let \( \Gamma \leq {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) be a lattice, and write \( X = \Gamma \smallsetminus {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) . Let \( g \in {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) be an element that is not conjugate to an element of \( \mathrm{{SO}}\left( 2... | As discussed above, we have to consider two cases, namely (after replacing \( g \) by \( {g}^{2} \) if necessary) elements of \( A \) and elements of \( {U}^{ - } \) . Even though we already dealt with the former, we will give here a different proof covering both cases using the language of unitary representations. | No |
Lemma 11.17. Let \( X \) be a locally compact metric space, and let \( \mu \) be a probability measure on \( X \) . Assume that \( G \) is a metrizable group that acts continuously on \( X \) (see p. 231) and preserves the measure \( \mu \) . Then the action of \( G \) on \( {L}_{\mu }^{2}\left( X\right) \) defined by ... | Proof of Lemma 11.17. Since every \( g \in G \) preserves \( \mu \), we already know that \( f \mapsto g\left( f\right) \) is unitary on \( {L}_{\mu }^{2}\left( X\right) \) . All that remains is to check the continuity requirement, and this follows from the more general result in Lemma 8.7. \( ▱ \) | Yes |
Corollary 11.21. Let \( X \) be a locally compact metric space with a Borel probability measure \( \mu \), and suppose that \( \mu \) is ergodic for a measure-preserving action of \( {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) . Then any element of \( {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) that is not conjug... | Proof of Proposition 11.20. The proof is virtually the same as the proof of Theorem 11.15. An element \( g \in {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) that is not conjugate to an element of \( \mathrm{{SO}}\left( 2\right) \) is either conjugate to an element of \( A \) or to an element of \( {U}^{ - } \) . In th... | No |
Theorem 11.22. Let \( \Gamma \leq {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) be a lattice. Then the action of \( {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) on \( X = \Gamma \smallsetminus {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) is mixing. | ## 11.4.1 First Proof of Theorem 11.22\n\nIn this section we will prove a stronger and more general result in the language of unitary representations, which will be related to Proposition 11.18 and which has Theorem 11.22 as a consequence. For this, the following notation will be useful. Let \( G \) be a locally compac... | No |
Proposition 11.23. Let \( G \) be a locally compact group and let \( \mathcal{H} \) be a Hilbert space carrying a unitary representation of \( G \). Let \( \alpha = \left( {a}_{n}\right) \in {G}^{\mathbb{N}} \) be a sequence in \( G \), and suppose for some \( v \in \mathcal{H} \) the sequence \( {a}_{n}\left( v\right)... | Proof. Clearly it is sufficient to show that \( {v}_{0} \) is fixed by each \( g \in S\left( \alpha \right) \). So suppose that \( g \in S\left( \alpha \right) \) and let \( {a}_{{n}_{k}} \) be a subsequence such that\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}{a}_{{n}_{k}}^{-1}g{a}_{{n}_{k}} = e. \]\n\n(11.1... | Yes |
Lemma 11.24. Let \( \alpha = \left( {g}_{n}\right) \) be a sequence in \( {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) converging to \( \infty \) (that is, such that for any compact subset \( K \subseteq {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) there are only finitely many \( n \) with \( {g}_{n} \in K \) ). Th... | Proof. Recall from the discussion of \( {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) as a closed linear group on p. 289 that the homomorphism \( \phi : {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \rightarrow \mathrm{{GL}}\left( {{\operatorname{Mat}}_{22}\left( \mathbb{R}\right) }\right) \) defined by \( \left( {\phi ... | Yes |
Theorem 11.25. Let \( \mathcal{H} \) be a Hilbert space carrying a unitary representation of \( {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) without any invariant vectors. Then for any \( v, w \in \mathcal{H} \) the matrix coefficients \( \langle {gv}, w\rangle \) for \( g \in {\mathrm{{SL}}}_{2}\left( \mathbb{R}\rig... | \[ \left\langle {{g}_{n}v, w}\right\rangle \rightarrow 0 \] as \( {g}_{n} \rightarrow \infty \) . | Yes |
Theorem 11.26. Let \( X \) be a \( \sigma \) -compact metric space equipped with a continuous \( {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) -action. Let \( \mu \) be an \( {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) -invariant ergodic probability measure on \( X \) . Then the \( {\mathrm{{SL}}}_{2}\left( \mathbb... | We start with the special case \( r = 2 \), which is essentially the statement of Theorem 11.22.\n\nSecond Proof of Theorem 11.22. Suppose that \( {g}_{n} = {g}_{n}^{\left( 1\right) } \in {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) eventually leaves any compact subset of \( {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right... | Yes |
Lemma 11.28. Let \( \Gamma \subseteq {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) be a discrete subgroup. Assume that the point \( {x}_{0} \in X = \Gamma \smallsetminus {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) is periodic for \( {U}^{ - } \) . Then \( {R}_{{a}_{t}}\left( {x}_{0}\right) \) diverges to infinity i... | Proof. As discussed in Sect. 9.3.3, the space \( X \) is locally isomorphic to \( {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) . That is, for any \( x \in X \) there is an injectivity radius \( {r}_{x} > 0 \) such that the map\n\n\[ \n{B}_{{r}_{x}}^{{\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) } \rightarrow X \n\]\n\n... | Yes |
Proposition 11.30. Let \( K \subseteq X \) be a compact set. Then there exists some constant \( \eta > 0 \) with the following property for all \( {x}_{0} \in X \) . Suppose that \( \left( {t}_{n}\right) \) is a sequence in \( \mathbb{R} \) with \( {t}_{n} \rightarrow \infty \), and with \( {R}_{{a}_{{t}_{n}}}\left( {x... | The basic idea of the proof-ignoring for the moment the slightly mysterious constant \( \eta \) -is as follows. Since \( f \) is uniformly continuous, we may replace the left-hand side of (11.15) by an integral over a slightly thickened tubular neighborhood \( {B}_{n} \) of the piece\n\n\[ \left\{ {{x}_{0}{u}^{ - }\lef... | No |
Lemma 11.31. Let \( G \) be a \( \sigma \) -compact unimodular group and let \( S, T \subseteq G \) be closed subgroups with the property that \( S \cap T = \{ e\} \) and the product set \( {ST} \) contains a neighborhood of \( e \in G \) . Let \( \phi : S \times T \rightarrow G \) be the product map \( \phi \left( {s,... | Proof of Lemma 11.31. Since \( S \cap T = \{ e\} \), an element \( g \in G \) has at most one decomposition as \( g = {st} \) with \( s \in S \) and \( t \in T \) . Therefore for compact subsets \( {K}_{S} \subseteq S \) and \( {K}_{T} \subseteq T \) the map \( \phi \) restricted to \( {K}_{S} \times {K}_{T} \) is a ho... | Yes |
Proposition 11.34. Let \( \Gamma \subseteq {\mathrm{{PSL}}}_{2}\left( \mathbb{R}\right) \) be a lattice. Then either \( \Gamma \) is uniform and \( \Gamma \smallsetminus {\mathrm{{PSL}}}_{2}\left( \mathbb{R}\right) \) is compact, or there exists a compact subset\n\n\[ \n{\Omega }_{cp} \subseteq X = \Gamma \smallsetminu... | Proof of Proposition 11.34. We will leave some of the details of the proof as an exercise, but indicate how the proof of Lemma 11.29 applies to Proposition 11.34. We defined a cusp of \( X = \Gamma \smallsetminus {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) to be an equivalence class of boundary vertices of a Di... | No |
Theorem 11.35. Let \( \Gamma \subseteq {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) be a lattice, and let \( x \in X = \Gamma \smallsetminus {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \) . Then either \( x \) is periodic for the horocycle flow (that is, \( h\left( t\right) \cdot x = x \) for some \( t > 0 \) ), or t... | Proof. Suppose that the point \( {x}_{0} \in X \) is not periodic for the horocycle flow, let \( {T}_{n} \nearrow \infty \) be any sequence, and define a sequence \( \left( {\mu }_{n}\right) \) of probability measures by\n\n\[ \int f\mathrm{\;d}{\mu }_{n} = \frac{1}{{T}_{n}}{\int }_{0}^{{T}_{n}}f\left( {h\left( t\right... | No |
Theorem 1.1.2. The set \( \mathcal{N} \) of monomials belonging to \( I \) is a \( K \) -basis of \( I \) . | Proof. It is clear that the elements of \( \mathcal{N} \) are linearly independent, as \( \mathcal{N} \) is a subset of \( \operatorname{Mon}\left( S\right) \) . Let \( f \in I \) be an arbitrary polynomial. We will show that \( \operatorname{supp}\left( f\right) \subset \mathcal{N} \) . This then yields that \( \mathc... | Yes |
Corollary 1.1.3. Let \( I \subset S \) be an ideal. The following conditions are equivalent:\n\n(a) \( I \) is a monomial ideal;\n\n(b) for all \( f \in S \) one has: \( f \in I \) if and only if \( \operatorname{supp}\left( f\right) \subset I \) . | Proof. (a) \( \Rightarrow \) (b) follows from Theorem 1.1.2.\n\n(b) \( \Rightarrow \) (a): Let \( {f}_{1},\ldots ,{f}_{m} \) be a set of generators of \( I \) . Since \( \operatorname{supp}\left( {f}_{i}\right) \subset I \) for all \( i \), it follows that \( \mathop{\bigcup }\limits_{{i = 1}}^{m}\operatorname{supp}\le... | Yes |
Corollary 1.1.4. Let \( I \) be a monomial ideal. The residue classes of the monomials not belonging to \( I \) form a \( K \) -basis of the residue class ring \( S/I \) . | Proof. Let \( \mathcal{W} \) be the set of monomials not belonging to \( I \) . It is clear that \( \overline{\mathcal{W}} \) is a set of generators of the \( K \) -vector space \( S/I \) . Suppose there is a non-trivial linear combination\n\n\[ \mathop{\sum }\limits_{{w \in \mathcal{W}}}{a}_{w}\bar{w} = 0 \]\n\nof zer... | Yes |
Proposition 1.1.5. Let \( \left\{ {{u}_{1},\ldots ,{u}_{m}}\right\} \) be a monomial system of generators of the monomial ideal \( I \) . Then the monomial \( v \) belongs to \( I \) if and only if there exists a monomial \( w \) such that \( v = w{u}_{i} \) for some \( i \) . | Proof. Suppose that \( v \in I \) . Then there exist polynomials \( {f}_{i} \in S \) such that \( v = \mathop{\sum }\limits_{{i = 1}}^{m}{f}_{i}{u}_{i} \) . It follows that \( v \in \mathop{\bigcup }\limits_{{i = 1}}^{m}\operatorname{supp}\left( {{f}_{i}{u}_{i}}\right) \), and hence \( v \in \operatorname{supp}\left( {... | Yes |
Proposition 1.1.6. Each monomial ideal has a unique minimal monomial set of generators. More precisely, let \( G \) denote the set of monomials in \( I \) which are minimal with respect to divisibility. Then \( G \) is the unique minimal set of monomial generators. | Proof. Let \( {G}_{1} = \left\{ {{u}_{1},\ldots ,{u}_{r}}\right\} \) and \( {G}_{2} = \left\{ {{v}_{1},\ldots ,{v}_{s}}\right\} \) be two minimal sets of generators of the monomial ideal \( I \) . Since \( {u}_{i} \in I \), there exists \( {v}_{j} \) such that \( {u}_{i} = {w}_{1}{v}_{j} \) for some monomial \( {w}_{1}... | Yes |
Proposition 1.2.1. Let \( I \) and \( J \) be monomial ideals. Then \( I \cap J \) is a monomial ideal, and \( \{ \operatorname{lcm}\left( {u, v}\right) : u \in G\left( I\right), v \in G\left( J\right) \} \) is a set of generators of \( I \cap J \) . | Proof. Let \( f \in I \cap J \) . By Corollary 1.1.3, \( \operatorname{supp}\left( f\right) \subset I \cap J \) . Again applying Corollary 1.1.3 we see that \( I \cap J \) is a monomial ideal.\n\nLet \( w \in \operatorname{supp}\left( f\right) \) ; then since \( \operatorname{supp}\left( f\right) \subset I \cap J \), t... | Yes |
Proposition 1.2.2. Let \( I \) and \( J \) be monomial ideals. Then \( I : J \) is a monomial ideal, and\n\n\[ I : J = \mathop{\bigcap }\limits_{{v \in G\left( J\right) }}I : \left( v\right) \]\n\nMoreover, \( \{ u/\gcd \left( {u, v}\right) : u \in G\left( I\right) \} \) is a set of generators of \( I : \left( v\right)... | Proof. Let \( f \in I : J \) . Then \( {fv} \in I \) for all \( v \in G\left( J\right) \) . In view of Corollary 1.1.3 we have \( \operatorname{supp}\left( f\right) v = \operatorname{supp}\left( {fv}\right) \subset I \) . This implies that \( \operatorname{supp}\left( f\right) \subset I : J \) . Thus Corollary 1.1.3 yi... | Yes |
Proposition 1.2.3. The saturation and the radical of a monomial ideal are again monomial ideals. | Proof. By Proposition 1.2.2, \( I : {\mathfrak{m}}^{k} \) is a monomial ideal for all \( k \) . Since \( \widetilde{I} \) is the union of these ideals, it is a monomial ideal.\n\nLet \( f = c{\mathbf{x}}^{{\mathbf{a}}_{1}} + \cdots \in \sqrt{I} \) with \( 0 \neq c \in K \) . Then \( {f}^{k} \in I \), and consequently \... | Yes |
Proposition 1.2.4. Let \( I \) be a monomial ideal. Then \( \{ \sqrt{u} : u \in G\left( I\right) \} \) is a set of generators of \( \sqrt{I} \) . | Proof. Obviously \( \{ \sqrt{u} : u \in G\left( I\right) \} \subset \sqrt{I} \) . Since \( \sqrt{I} \) is a monomial ideal it suffices to show that each monomial \( v \in \sqrt{I} \) is a multiple of some \( \sqrt{u} \) with \( u \in G\left( I\right) \) . In fact, if \( v \in \sqrt{I} \) then \( {v}^{k} \in I \) for so... | Yes |
Theorem 1.3.1. Let \( I \subset S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) be a monomial ideal. Then \( I = \mathop{\bigcap }\limits_{{i = 1}}^{m}{Q}_{i} \), where each \( {Q}_{i} \) is generated by pure powers of the variables. In other words, each \( {Q}_{i} \) is of the form \( \left( {{x}_{{i}_{1}}... | Proof. Let \( G\left( I\right) = \left\{ {{u}_{1},\ldots ,{u}_{r}}\right\} \), and suppose some \( {u}_{i} \) is not a pure power, say \( {u}_{1} \). Then we can write \( {u}_{1} = {vw} \) where \( v \) and \( w \) are coprime monomials, that is, \( \gcd \left( {v, w}\right) = 1 \) and \( v \neq 1 \neq w \). We claim t... | Yes |
Corollary 1.3.2. A monomial ideal is irreducible if and only if it is generated by pure powers of the variables. | Proof. Let \( Q = \left( {{x}_{{i}_{1}}^{{a}_{1}},\ldots ,{x}_{{i}_{k}}^{{a}_{k}}}\right) \), and suppose \( Q = I \cap J \) where \( I \) and \( J \) are monomial ideals properly containing \( Q \) . By Theorem 1.3.1 we have \( I = \mathop{\bigcap }\limits_{{i = 1}}^{r}{Q}_{i} \) and \( J = \mathop{\bigcap }\limits_{{... | Yes |
Let\n\n\[ I = \left( {{x}_{1}^{2}{x}_{2},{x}_{1}^{2}{x}_{3}^{2},{x}_{2}^{2},{x}_{2}{x}_{3}^{2}}\right) . \]\n\nThen | \[ I = \left( {{x}_{1}^{2},{x}_{1}^{2}{x}_{3}^{2},{x}_{2}^{2},{x}_{2}{x}_{3}^{2}}\right) \cap \left( {{x}_{2},{x}_{1}^{2}{x}_{3}^{2},{x}_{2}^{2},{x}_{2}{x}_{3}^{2}}\right) = \left( {{x}_{1}^{2},{x}_{2}^{2},{x}_{2}{x}_{3}^{2}}\right) \cap \left( {{x}_{2},{x}_{1}^{2}{x}_{3}^{2}}\right) \]\n\n\[ = \left( {{x}_{1}^{2},{x}_... | Yes |
Lemma 1.3.5. Suppose \( I \) has irredundant presentation \( I = {P}_{1} \cap \cdots \cap {P}_{m} \) as an intersection of prime ideals. Then \( \operatorname{Min}\left( I\right) = \left\{ {{P}_{1},\ldots ,{P}_{m}}\right\} \) | Proof. Suppose \( {P}_{i} \) is not a minimal prime ideal of \( I \) . Then there exists a prime ideal \( P \) with \( I \subset P \), and \( P \) is properly contained in \( {P}_{i} \) . Since \( {P}_{j}{R}_{{P}_{i}} = {R}_{{P}_{i}} \) for \( i \neq j \) and since localization commutes with intersections, it follows t... | Yes |
Proposition 1.3.7. The irreducible ideal \( \left( {{x}_{{i}_{1}}^{{a}_{1}},\ldots ,{x}_{{i}_{k}}^{{a}_{k}}}\right) \) is \( \left( {{x}_{{i}_{1}},\ldots ,{x}_{{i}_{k}}}\right) \) -primary. | Proof. Let \( Q = \left( {{x}_{{i}_{1}}^{{a}_{1}},\ldots ,{x}_{{i}_{k}}^{{a}_{k}}}\right) \) and \( P = \left( {{x}_{{i}_{1}},\ldots ,{x}_{{i}_{k}}}\right) \) . Since \( P \) is a minimal prime ideal of \( Q \), it follows that \( P \in \operatorname{Ass}\left( Q\right) \) . Notice that \( {P}^{m} \subset Q \) for \( m... | Yes |
The ideal \( I = \left( {{x}_{1}^{3},{x}_{2}^{3},{x}_{1}^{2}{x}_{3}^{2},{x}_{1}{x}_{2}{x}_{3}^{2},{x}_{2}^{2}{x}_{3}^{2}}\right) \) has the irredundant presentation as intersection of irreducible ideals | \[ I = \left( {{x}_{1}^{3},{x}_{2}^{3},{x}_{3}^{2}}\right) \cap \left( {{x}_{1}^{2},{x}_{2}}\right) \cap \left( {{x}_{1},{x}_{2}^{2}}\right) . \] We have \( \operatorname{Ass}\left( {{x}_{1}^{2},{x}_{2}}\right) = \operatorname{Ass}\left( {{x}_{1},{x}_{2}^{2}}\right) = \left\{ \left( {{x}_{1},{x}_{2}}\right) \right\} \)... | Yes |
Corollary 1.3.10. Let \( I \subset S \) be a monomial ideal, and let \( P \in \operatorname{Ass}\left( I\right) \) . Then there exists a monomial \( v \) such that \( P = I : v \) . | Proof. Since \( P \in \operatorname{Ass}\left( I\right) \), there exists \( f \in S \) such that \( P = I : f \) . Thus for each \( {x}_{i} \in P \) we have that \( {x}_{i}f \in I \) . Since \( I \) is a monomial ideal, this implies that \( {x}_{i}u \in I \) for all \( u \in \operatorname{supp}\left( f\right) \) . It f... | Yes |
Theorem 1.4.2. Let \( I \subset S \) be a monomial ideal. Then \( \bar{I} \) is a monomial ideal generated by all monomials \( u \in S \) for which there exists an integer \( k \) such that \( {u}^{k} \in {I}^{k} \) . | Proof. We first show that if \( J \) is monomial ideal and \( u \in \bar{J} \) is monomial, then there exists an integer \( k \) such that \( {u}^{k} \in {J}^{k} \) . Indeed, let \( {u}^{m} + {c}_{1}{u}^{m - 1} + \cdots + \) \( {c}_{m - 1}u + {c}_{m} = 0 \) be an equation of integral dependence of \( u \) over \( J \),... | Yes |
Corollary 1.4.3. Let \( I \subset S \) be a monomial ideal. Then \( \bar{I} \) is generated by the monomials \( {\mathbf{x}}^{\mathbf{a}} \) with \( \mathbf{a} \in \mathcal{C}\left( I\right) \) . | Proof. By Theorem 1.4.2, \( {\mathbf{x}}^{\mathbf{a}} \in \bar{I} \) if and only if there exists an integer \( k > 0 \) such that \( {\left( {\mathbf{x}}^{\mathbf{a}}\right) }^{k} \in {I}^{k} \) . It follows from Proposition 1.1.5 that this is the case if and only if there exist \( {\mathbf{x}}^{{\mathbf{a}}_{1}},\ldot... | Yes |
Proposition 1.4.4. Let \( I \subset S \) be a squarefree monomial ideal. Then\n\n\[ \n{I}^{\left( k\right) } = \mathop{\bigcap }\limits_{{P \in \operatorname{Min}\left( I\right) }}{P}^{k} \n\] | Proof. Because of Corollary 1.3.6 we have \( I{S}_{P} = P{S}_{P} \) for \( P \in \operatorname{Min}\left( I\right) \) . It follows that \( {I}^{k}{S}_{P} = {P}^{k}{S}_{P} \) . Thus it is clear that \( {P}^{k} \subset \operatorname{Ker}\left( {S \rightarrow {\left( S/{I}^{k}\right) }_{P}}\right) \) .\n\nConversely, if \... | Yes |
Theorem 1.4.6. Let \( I \subset S \) be a squarefree monomial ideal. Then the following conditions are equivalent:\n\n(a) \( I \) is normally torsionfree;\n\n(b) \( {I}^{\left( k\right) } = {I}^{k} \) for all \( k \) .\n\nIf the equivalent conditions hold, then \( I \) is a normal ideal. | Proof. Let \( {I}^{k} = \mathop{\bigcap }\limits_{{P \in \operatorname{Ass}\left( {I}^{k}\right) }}Q\left( P\right) \) be an irredundant primary decomposition of \( {I}^{k} \) . Then \( {I}^{\left( k\right) } = {I}^{k} \), if and only if \( \mathop{\bigcap }\limits_{{P \in \operatorname{Ass}\left( {I}^{k}\right) }}Q\le... | Yes |
Proposition 1.5.1. The set of all monomials \( {x}_{1}^{{a}_{1}}\cdots {x}_{n}^{{a}_{n}} \) of \( S \) with \( \{ i \in \left\lbrack n\right\rbrack \) : \( \left. {{a}_{i} \neq 0}\right\} \in \Delta \) is a \( K \) -basis of \( S/{I}_{\Delta } \) . | Proof. Let \( u = {x}_{1}^{{a}_{1}}\cdots {x}_{n}^{{a}_{n}} \) be a monomial of \( S \) . If \( \left\{ {i \in \left\lbrack n\right\rbrack : {a}_{i} \neq 0}\right\} \notin \Delta \) then by definition \( \sqrt{u} \in {I}_{\Delta } \) . Thus \( u \in {I}_{\Delta } \) . On the other hand, if \( u \in {I}_{\Delta } \) , t... | Yes |
Lemma 1.5.2. The collection of sets \( {\Delta }^{ \vee } \) is a simplicial complex and\n\n\[ \n{\left( {\Delta }^{ \vee }\right) }^{ \vee } = \Delta \text{.} \n\] | Proof. Let \( F \in {\Delta }^{ \vee } \) and \( {F}^{\prime } \subset F \) . Then \( \left\lbrack n\right\rbrack \smallsetminus F \notin \Delta \) . Since \( \left\lbrack n\right\rbrack \smallsetminus F \subset \left\lbrack n\right\rbrack \smallsetminus {F}^{\prime } \), it follows that \( \left\lbrack n\right\rbrack ... | Yes |
Lemma 1.5.3. One has\n\n\[ \n{I}_{{\Delta }^{ \vee }} = I\left( \bar{\Delta }\right) \n\] | Proof. A squarefree monomial \( {\mathbf{x}}_{F} \) belongs to \( G\left( {I}_{{\Delta }^{ \vee }}\right) \) if and only if \( F \) is a minimal nonface of \( {\Delta }^{ \vee } \) . In other words, \( F \) is a nonface of \( {\Delta }^{ \vee } \) and all proper subsets of \( F \) are faces of \( {\Delta }^{ \vee } \) ... | Yes |
Lemma 1.5.4. The standard primary decomposition of \( {I}_{\Delta } \) is\n\n\[ \n{I}_{\Delta } = \mathop{\bigcap }\limits_{{F \in \mathcal{F}\left( \Delta \right) }}{P}_{\bar{F}} \n\] | Proof. Let \( u = {x}_{1}^{{a}_{1}}\cdots {x}_{n}^{{a}_{n}} \) be a monomial of \( S \) and \( {F}_{u} = \left\{ {i \in \left\lbrack n\right\rbrack : {a}_{i} \neq 0}\right\} \) . If \( u \in {I}_{\Delta } \), then by Proposition 1.5.1, \( {F}_{u} \notin \Delta \) . Thus no facet of \( \Delta \) contains \( {F}_{u} \) .... | Yes |
Let \( \Delta \) be the simplicial complex of Figure 1.1. Since\n\n\[ {I}_{\Delta } = \left( {{x}_{3},{x}_{4}}\right) \cap \left( {{x}_{3},{x}_{5}}\right) \cap \left( {{x}_{1},{x}_{4},{x}_{5}}\right) \cap \left( {{x}_{1},{x}_{2},{x}_{5}}\right) ,\]\nthe ideal \( {I}_{{\Delta }^{ \vee }} \) is generated by \( {x}_{3}{x}... | Let \( I \subset S \) be an arbitrary squarefree monomial ideal. Then there is a unique simplicial complex \( \Delta \) such that \( I = {I}_{\Delta } \) . For simplicity, we often write \( {I}^{ \vee } \) to denote the ideal \( {I}_{{\Delta }^{ \vee }} \) . | No |
Lemma 1.6.1. Let \( I \subset S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) be a monomial ideal with \( G\left( I\right) = \) \( \left\{ {{u}_{1},\ldots ,{u}_{m}}\right\} \) where \( {u}_{i} = \mathop{\prod }\limits_{{j = 1}}^{n}{x}_{j}^{{a}_{ij}} \) for \( i = 1,\ldots, m \) . Fix an integer \( j \in \le... | Proof. Suppose \( y - {x}_{j} \) is a zero divisor modulo \( J \) . Then \( y - {x}_{j} \in P \) for some \( P \in \operatorname{Ass}\left( J\right) \) . Since by Corollary 1.3.9, \( P \) is a monomial prime ideal, it follows that \( y,{x}_{j} \in P \) . Hence there exists \( w \in S \smallsetminus J \) such that \( {y... | Yes |
Corollary 1.6.3. Let \( I \subset S \) be a monomial ideal and \( J \subset T \) its polarization. Then\n\n(a) \( {\beta }_{ij}\left( I\right) = {\beta }_{ij}\left( J\right) \) for all \( i \) and \( j \) ;\n\n(b) \( {H}_{S/I}\left( t\right) = {\left( 1 - t\right) }^{\delta }{H}_{T/J}\left( t\right) \) where \( \delta ... | Proof. (a) Since \( \mathbf{z} \) is a \( T/J \) -sequence, Corollary A.3.5 and Theorem A.3.4 imply that\n\n\[ \n{\operatorname{Tor}}_{i}^{T}\left( {T/\left( \mathbf{z}\right), T/J}\right) = {H}_{i}\left( {\mathbf{z};T/J}\right) = 0.\n\]\n\nHence if \( \mathbb{F} \) is a graded minimal free \( T \) -resolution of \( T/... | Yes |
(a) Let \( \mathbf{a} = \left( {{a}_{1},{a}_{2},\ldots ,{a}_{n}}\right) \) and \( \mathbf{b} = \left( {{b}_{1},{b}_{2},\ldots ,{b}_{n}}\right) \) be vectors belonging to \( {\mathbb{Z}}_{ + }^{n} \) . We define the total order \( { < }_{\text{lex }} \) on \( \operatorname{Mon}\left( S\right) \) by setting \( {\mathbf{x... | It follows that \( { < }_{\text{lex }} \) is a monomial order on \( S \), which is called the lexicographic order on \( S \) induced by the ordering \( {x}_{1} > {x}_{2} > \cdots > {x}_{n} \). | Yes |
Lemma 2.1.4. Let \( u, v \) be monomials of \( S \) and \( f, g \) nonzero polynomials of S. Then one has\n\n(i) if \( u \) divides \( v \), then \( u \leq v \) ;\n\n(ii) \( {\operatorname{in}}_{ < }\left( {uf}\right) = u{\operatorname{in}}_{ < }\left( f\right) \) ;\n\n(iii) \( {\operatorname{in}}_{ < }\left( {fg}\righ... | Proof. (i) In fact, if \( u \) divides \( v \) and if \( v = {uw} \) with \( w \in \operatorname{Mon}\left( S\right) \), then since \( 1 \leq w \) one has \( 1 \cdot u \leq w \cdot u \). Thus \( u \leq v \), as desired.\n\n(ii) Let \( w \in \operatorname{supp}\left( f\right) \) with \( w < {\operatorname{in}}_{ < }\lef... | Yes |
Example 2.1.6. (a) Let \( S = K\left\lbrack {{x}_{1},{x}_{2},\ldots ,{x}_{7}}\right\rbrack \) and \( { < }_{\text{lex }} \) the lexicographic order on \( S \) induced by \( {x}_{1} > {x}_{2} > \cdots > {x}_{7} \) . Let \( f = {x}_{1}{x}_{4} - {x}_{2}{x}_{3} \) and \( g = {x}_{4}{x}_{7} - {x}_{5}{x}_{6} \) with their in... | In fact, the polynomial \( h = {x}_{7}f - {x}_{1}g = {x}_{1}{x}_{5}{x}_{6} - {x}_{2}{x}_{3}{x}_{7} \) belongs to \( I \), but its initial monomial in \( {}_{{ < }_{\mathrm{{lex}}}}\left( h\right) = {x}_{1}{x}_{5}{x}_{6} \) can be divided by neither in \( {}_{{ < }_{\mathrm{{lex}}}}\left( f\right) \) nor in \( {}_{{ < }... | Yes |
Lemma 2.1.7. Let \( < \) be a monomial order on \( S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) . Then, for any monomial \( u \) of \( S \), there is no infinite descending sequence of the form\n\n\[ \cdots < {u}_{2} < {u}_{1} < {u}_{0} = u. \] | Proof. Suppose, on the contrary, that one has an infinite descending sequence (2.1) and write \( \mathcal{M} \) for the set of monomials \( \left\{ {{u}_{0},{u}_{1},{u}_{2},\ldots }\right\} \) . It follows from Dickson’s Lemma that \( {\mathcal{M}}^{\min } \) is a finite set, say \( {\mathcal{M}}^{\min } = \left\{ {{u}... | Yes |
Theorem 2.1.8. Let \( I \) be a nonzero ideal of \( S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) and \( \mathcal{G} = \) \( \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \) a Gröbner basis of \( I \) with respect to a monomial order \( < \) on \( S \) . Then \( I = \left( {{g}_{1},\ldots ,{g}_{s}}\right) \) ... | Proof. (Gordan) Let \( 0 \neq f \in I \) . Since \( {\operatorname{in}}_{ < }\left( f\right) \in {\operatorname{in}}_{ < }\left( I\right) \), it follows that there is \( {g}_{{i}_{0}} \) such that \( {\operatorname{in}}_{ < }\left( {g}_{{i}_{0}}\right) \) divides \( {\operatorname{in}}_{ < }\left( f\right) \) . Let \( ... | Yes |
Example 2.1.10. Let \( S = K\left\lbrack {{x}_{1},{x}_{2},\ldots ,{x}_{10}}\right\rbrack \) and \( I \) the ideal of \( S \) generated by\n\n\[ \n{f}_{1} = {x}_{1}{x}_{8} - {x}_{2}{x}_{6},\;{f}_{2} = {x}_{2}{x}_{9} - {x}_{3}{x}_{7},\;{f}_{3} = {x}_{3}{x}_{10} - {x}_{4}{x}_{8},\n\]\n\n\[ \n{f}_{4} = {x}_{4}{x}_{6} - {x}... | Suppose, on the contrary, that there exists a monomial order \( < \) on \( S \) such that \( \mathcal{G} = \left\{ {{f}_{1},\ldots ,{f}_{5}}\right\} \) is a Gröbner basis of \( I \) with respect to \( < \) . First, note that each of the five polynomials\n\n\[ \n{x}_{1}{x}_{8}{x}_{9} - {x}_{3}{x}_{6}{x}_{7},{x}_{2}{x}_{... | Yes |
Theorem 2.2.1 (The division algorithm). Let \( S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) denote the polynomial ring in \( n \) variables over a field \( K \) and fix a monomial order \( < \) on \( S \) . Let \( {g}_{1},{g}_{2},\ldots ,{g}_{s} \) be nonzero polynomials of \( S \) . Then, given a polyno... | Proof (of Theorem 2.2.1). Let \( I = \left( {{\operatorname{in}}_{ < }\left( {g}_{1}\right) ,\ldots ,{\operatorname{in}}_{ < }\left( {g}_{s}\right) }\right) \) . If none of the monomials \( u \in \operatorname{supp}\left( f\right) \) belongs to \( I \), then the desired expression can be obtained by setting \( {f}^{\pr... | Yes |
Example 2.2.2. Let \( { < }_{\text{lex }} \) denote the lexicographic order on \( S = K\left\lbrack {x, y, z}\right\rbrack \) induced by \( x > y > z \) . Let \( {g}_{1} = {x}^{2} - z,{g}_{2} = {xy} - 1 \) and \( f = {x}^{3} - {x}^{2}y - {x}^{2} - 1 \) . Each of | \[ f = {x}^{3} - {x}^{2}y - {x}^{2} - 1 = x\left( {{g}_{1} + z}\right) - {x}^{2}y - {x}^{2} - 1 \] \[ = x{g}_{1} - {x}^{2}y - {x}^{2} + {xz} - 1 = x{g}_{1} - \left( {{g}_{1} + z}\right) y - {x}^{2} + {xz} - 1 \] \[ = x{g}_{1} - y{g}_{1} - {x}^{2} + {xz} - {yz} - 1 = x{g}_{1} - y{g}_{1} - \left( {{g}_{1} + z}\right) + {... | Yes |
Lemma 2.2.3. If \( \mathcal{G} = \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \) is a Gröbner basis of \( I = \left( {{g}_{1},\ldots ,{g}_{s}}\right) \) , then for any nonzero polynomial \( f \) of \( S \), there is a unique remainder of \( f \) with respect to \( {g}_{1},\ldots ,{g}_{s} \) . | Proof. Suppose there exist remainders \( {f}^{\prime } \) and \( {f}^{\prime \prime } \) with respect to \( {g}_{1},\ldots ,{g}_{s} \) with \( {f}^{\prime } \neq {f}^{\prime \prime } \) . Since \( 0 \neq {f}^{\prime } - {f}^{\prime \prime } \in I \), the initial monomial \( w = {\operatorname{in}}_{ < }\left( {{f}^{\pr... | Yes |
Corollary 2.2.4. If \( \mathcal{G} = \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \) is a Gröbner basis of \( I = \left( {{g}_{1},\ldots ,{g}_{s}}\right) \) , then a nonzero polynomial \( f \) of \( S \) belongs to \( I \) if and only if the unique remainder of \( f \) with respect to \( {g}_{1},\ldots ,{g}_{s} \) is 0 . | Proof. First, in general, if a remainder of a nonzero polynomial \( f \) of \( S \) with respect to \( {g}_{1},{g}_{2},\ldots ,{g}_{s} \) is 0, then \( f \) belongs to \( I = \left( {{g}_{1},{g}_{2},\ldots ,{g}_{s}}\right) \) . Second, suppose that a nonzero polynomial \( f \) belongs to \( I \) and \( f = {f}_{1}{g}_{... | Yes |
Proposition 2.2.5. Let \( I \) be a nonzero ideal of \( S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \), and \( < a \) monomial order on \( S \) . Then (a) the set of monomials which do not belong to \( {\operatorname{in}}_{ < }\left( I\right) \) form a \( K \) -basis of \( S/I \) . (b) \( {\dim }_{K}{I}_{j... | Proof. (a) Let \( {g}_{1},\ldots ,{g}_{m} \) be a Gröbner basis of \( I \), let \( f \in S \) and \( {f}^{\prime } \) the remainder of \( f \) with respect to \( {g}_{1},\ldots ,{g}_{m} \) . Then \( f + I = {f}^{\prime } + I \) and \( \operatorname{supp}\left( {f}^{\prime }\right) \cap \) \( {\operatorname{in}}_{ < }\l... | Yes |
Proposition 2.2.6. Let \( I \subset J \) be nonzero ideals of \( S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) with \( I \neq J \), and let \( < \) and \( { < }^{\prime } \) be monomial orders on \( S \) . Then\n\n(a) \( {\operatorname{in}}_{ < }\left( I\right) \subset {\operatorname{in}}_{ < }\left( J\ri... | Proof. (a) \( {\operatorname{in}}_{ < }\left( I\right) \) is generated by all monomials \( {\operatorname{in}}_{ < }\left( f\right) \) with \( f \in I \) . Since \( I \subset J \), each \( f \in I \) belongs to \( J \) . Therefore \( {\operatorname{in}}_{ < }\left( I\right) \subset {\operatorname{in}}_{ < }\left( J\rig... | Yes |
Lemma 2.3.1. Let \( f \) and \( g \) be nonzero polynomials and suppose that \( {\operatorname{in}}_{ < }\left( f\right) \) and \( {\operatorname{in}}_{ < }\left( g\right) \) are relatively prime, i.e. \( \operatorname{lcm}\left( {{\operatorname{in}}_{ < }\left( f\right) ,{\operatorname{in}}_{ < }\left( g\right) }\righ... | Proof. To simplify notation we will assume that each of the coefficients of \( {\operatorname{in}}_{ < }\left( f\right) \) in \( f \) and \( {\operatorname{in}}_{ < }\left( g\right) \) in \( g \) is equal to 1 . Let \( f = {\operatorname{in}}_{ < }\left( f\right) + {f}_{1} \) and \( g = \) \( {\operatorname{in}}_{ < }\... | Yes |
Lemma 2.3.3. Let \( w \) be a monomial of \( S \) and \( {f}_{1},{f}_{2},\ldots ,{f}_{s} \) polynomials of \( S \) with \( {\operatorname{in}}_{ < }\left( {f}_{i}\right) = w \) for all \( 1 \leq i \leq s \) . Let \( g = \sum {b}_{i}{f}_{i}\left( { \neq 0}\right) \) be a linear combination of \( {f}_{1},{f}_{2},\ldots ,... | Proof. Let \( {c}_{i} \) denote the coefficient of \( w = {\operatorname{in}}_{ < }\left( {f}_{i}\right) \) in \( {f}_{i} \) . Then\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{s}{b}_{i}{c}_{i} = 0 \]\n\nLet \( {g}_{i} = \left( {1/{c}_{i}}\right) {f}_{i} \) . Then\n\n\[ S\left( {{f}_{j},{f}_{k}}\right) = {g}_{j} - {g}_{k},\;... | Yes |
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