Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Corollary 6.3.12 Let \( \mathcal{M} \vDash T \), and let \( \phi \left( \bar{v}\right) \) be an \( {\mathcal{L}}_{M} \) -formula with \( \operatorname{RM}\left( \phi \right) = \alpha \) and \( {\deg }_{\mathrm{M}}\left( \phi \right) = d \) . i) There is an \( {\mathcal{L}}_{M} \) -formula \( \theta \left( \bar{v}\right... | i) In our argument above, we had \( \psi \left( {\bar{v},\bar{w}}\right) \) and \( \bar{b} \in \mathbb{M} \) and an \( M \) -definable equivalence relation \( E \) with finitely many classes such that if \( \bar{a}E\bar{b} \), then \( \psi \left( {\mathbb{M},\bar{a}}\right) \) is a Morley rank \( \alpha \), Morley degr... | Yes |
Lemma 6.3.14 Suppose that \( \mathcal{M} \vDash T, p \in {S}_{n}\left( M\right), M \subseteq B, q \in {S}_{n}\left( B\right) \) , \( p \subseteq q \), and \( q \) is definable over \( M \) . Then, \( q \) is a nonforking extension of \( p \) . | Proof There is \( r \in {S}_{n}\left( B\right) \) a nonforking extension of \( p \) that is definable over \( M \) . Because \( q \) and \( r \) both extend \( p \) ,\n\n\[ \mathcal{M} \vDash \forall \bar{w}{d}_{r}\phi \left( \bar{w}\right) \leftrightarrow {d}_{q}\phi \left( \bar{w}\right) \]\n\nfor all formulas \( \ph... | No |
Lemma 6.3.16 (Monotonicity) If \( \bar{a}{ \downarrow }_{A}B \) and \( C \subseteq B \), then \( \bar{a}{ \downarrow }_{A}C \) . | Proof Because \( \operatorname{RM}\left( {\bar{a}/A}\right) \geq \operatorname{RM}\left( {\bar{a}/A \cup C}\right) \geq \operatorname{RM}\left( {\bar{a}/A \cup B}\right) \), if \( \operatorname{RM}\left( {\bar{a}/A}\right) = \) \( \operatorname{RM}\left( {\bar{a}/A \cup B}\right) \), then \( \operatorname{RM}\left( {\b... | Yes |
Lemma 6.3.17 (Transitivity) \( \bar{a}{ \bot }_{A}\bar{b},\bar{c} \) if and only if \( \bar{a}{ \bot }_{A}\bar{b} \) and \( \bar{a}{ \bot }_{A,\bar{b}}\bar{c} \) | Proof Because \( \operatorname{RM}\left( {\bar{a}/A,\bar{b},\bar{c}}\right) \leq \operatorname{RM}\left( {\bar{a}/A,\bar{b}}\right) \leq \operatorname{RM}\left( {\bar{a}/A}\right) ,\operatorname{RM}\left( {\bar{a}/A}\right) = \) \( \operatorname{RM}\left( {\bar{a}/A,\bar{b},\bar{c}}\right) \) if and only if \( \operato... | Yes |
Lemma 6.3.18 (Finite Basis) \( \bar{a}{ \downarrow }_{A}B \) if and only if \( \bar{a}{ \downarrow }_{A}{B}_{0} \) for all \( f \) i-nite \( {B}_{0} \subseteq B \) . | Proof\n\n\( \left( \Rightarrow \right) \) This is clear because for any \( {B}_{0} \subseteq B,\operatorname{RM}\left( {\bar{a}/A}\right) \leq \operatorname{RM}\left( {\bar{a}/A \cup {B}_{0}}\right) \leq \) \( \operatorname{RM}\left( {\bar{a}/A \cup B}\right) \).\n\n\( \left( \Leftarrow \right) \) Suppose that \( \bar{... | Yes |
Corollary 6.3.20 \( \bar{a},\bar{b}{ \downarrow }_{A}C \) if and only if \( \bar{a}{ \downarrow }_{A}C \) and \( \bar{b}{ \downarrow }_{A,\bar{a}}C \) . | Proof Because forking occurs over a finite subset, it suffices to assume that \( C \) is a finite sequence \( \bar{c} \) .\n\n\[ \bar{a},\bar{b}{ \downarrow }_{A}\bar{c} \Leftrightarrow \bar{c}{ \downarrow }_{A}\bar{a},\bar{b}\;\text{ by symmetry } \]\n\n\[ \Leftrightarrow \bar{c}{ \downarrow }_{A}\bar{a}\text{ and }\b... | Yes |
Corollary 6.3.21 For any \( \bar{a},\bar{a}{ \downarrow }_{A}\operatorname{acl}\left( A\right) \) . | Proof Suppose that \( \bar{b} \in \operatorname{acl}\left( A\right) \) . By Lemma 6.2.7 iii), \( \operatorname{RM}\left( {\bar{b}/A,\bar{a}}\right) = \) \( \operatorname{RM}\left( {\bar{b}/A}\right) = 0 \) . Thus, \( \bar{b}{ \downarrow }_{A}\bar{a} \) and, by symmetry, \( \bar{a}{ \downarrow }_{A}\bar{b} \) . | Yes |
Theorem 6.4.5 (Uniqueness of Constructible Models) Suppose that \( A \subseteq \mathbb{M},\mathcal{M} \prec \mathbb{M},\mathcal{N} \prec \mathbb{M} \), and \( \mathcal{M} \) and \( \mathcal{N} \) are constructible over \( A \) . The identity map on \( A \) extends to an isomorphism between \( \mathcal{M} \) and \( \mat... | Proof Let \( \left( {{a}_{\alpha } : \alpha < \delta }\right) \) and \( \left( {{b}_{\alpha } : \alpha < \gamma }\right) \) be the constructions of \( \mathcal{M} \) and \( \mathcal{N} \) over \( A \) . Let \( \kappa = \left| M\right| \) . Let \( I = \{ f : X \rightarrow N : f \) is partial elementary, \( A \subset X, ... | Yes |
Lemma 6.4.6 Let \( T \) be \( \omega \) -stable. If \( A \subseteq B, p \in {S}_{n}\left( A\right) ,{p}^{\prime } \in {S}_{n}\left( B\right) ,{p}^{\prime } \) is a nonforking extension of \( p \), and \( {p}^{\prime } \) is isolated, then \( p \) is isolated. | Proof We work in \( {\mathbb{M}}^{\text{eq }} \) . Let \( \phi \left( {\bar{v},\bar{b}}\right) \) isolate \( {p}^{\prime } \) . Let \( q = \operatorname{tp}\left( {\bar{b}/{\operatorname{acl}}^{\text{eq }}\left( A\right) }\right) \) . By Theorem 6.3.9, there is an \( {\mathcal{L}}_{{\operatorname{acl}}^{\text{eq }}\lef... | Yes |
Theorem 6.4.8 (Uniqueness of Prime Models) Suppose that \( T \) is \( \omega \) - stable, \( A \subseteq \mathbb{M},{\mathcal{M}}_{0} \prec \mathbb{M},{\mathcal{M}}_{1} \prec \mathbb{M} \), and \( {\mathcal{M}}_{0} \) and \( {\mathcal{M}}_{1} \) are prime models over \( A \) . The identity map on \( A \) extends to an ... | Proof By Lemma 6.4.2, there is \( \mathcal{N} \prec \mathbb{M} \), a constructible model over \( A \) . Because each \( {\mathcal{M}}_{i} \) is prime over \( A \), we can find an elementary embedding of \( {\mathcal{M}}_{i} \) into \( \mathcal{N} \) . By the previous lemma, each \( {\mathcal{M}}_{i} \) is constructible... | Yes |
Theorem 6.4.10 Let \( k \) be a differential field. There is \( K \supseteq k \) a differential closure of \( k \) . If \( K \) and \( L \) are differential closures of \( k \), then \( K \) and \( L \) are isomorphic over \( k \) . If \( K \) is a differential closure of \( k \), then \( \operatorname{tp}\left( {\bar{... | Proof Because DCF is \( \omega \) -stable and a differential closure of \( k \) is a prime model of DCF over \( k \), this follows from the existence and uniqueness of prime model extensions for \( \omega \) -stable theories (Theorems 4.2.20 and 6.4.8).\n\nBecause the differential closure is a prime model over \( k \),... | Yes |
Theorem 6.5.2 Suppose that \( I = \left( {{a}_{\alpha } : \alpha < \delta }\right) \) is an infinite Morley sequence for \( p \) over \( A \) . Then, \( I \) is an infinite set of indiscernibles. | Proof By Theorem 5.2.13, it suffices to show that \( I \) is a sequence of order indiscernibles. Let \( d = {\deg }_{\mathrm{M}}\left( p\right) \) . We will show by induction on \( n \) that \( \operatorname{tp}\left( {{a}_{{\alpha }_{1}},\ldots ,{a}_{{\alpha }_{n}}/A}\right) = \operatorname{tp}\left( {{a}_{{\beta }_{1... | Yes |
Lemma 6.5.3 Suppose that \( I \) is an infinite set of indiscernibles over \( A \subset \) \( \mathbb{M} \) . For any \( \bar{b} \in \mathbb{M} \), there is a finite \( J \subset I \) such that \( I \smallsetminus J \) is a set of indiscernibles over \( A \cup \{ \bar{b}\} \) . | Proof Let \( p = \operatorname{tp}\left( {\bar{b}/A \cup I}\right) \) . There is a finite \( J \subset I \) such that \( p \mid A \cup J \) has the same Morley rank and degree as \( p \) . Let \( {x}_{1},\ldots ,{x}_{n},{y}_{1},\ldots ,{y}_{n} \subset I \smallsetminus J \) with \( {x}_{i} \neq {x}_{j} \) and \( {y}_{i}... | Yes |
Theorem 6.6.37 If \( T \) is superstable, then\n\n\[ \mathrm{{RU}}\left( {\bar{a}/A,\bar{b}}\right) + \mathrm{{RU}}\left( {\bar{b}/A}\right) \leq \mathrm{{RU}}\left( {\bar{a},\bar{b}/A}\right) \leq \mathrm{{RU}}\left( {\bar{a}/A,\bar{b}}\right) \oplus \mathrm{{RU}}\left( {\bar{b}/A}\right) . \] | For a proof, see [18] 6.1.1. | No |
Theorem 7.1.2 (Descending Chain Condition) If \( G \) is an \( \omega \) -stable group, then there is no infinite descending chain of definable subgroups \( G > {G}_{1} > {G}_{2} > \ldots \) | Proof Let \( \alpha = \operatorname{RM}\left( G\right) \) . Let \( {\eta }_{i} = \left( {\operatorname{RM}\left( {G}_{i}\right) ,{\deg }_{\mathrm{M}}\left( {G}_{i}\right) }\right) \) . If \( G > {G}_{1} > \) \( {G}_{2} > \ldots \) is a descending chain, then the remarks above show that \( {\eta }_{1}{ > }_{\text{lex }}... | Yes |
Corollary 7.1.3 Suppose that \( G \) is an \( \omega \) -stable group and \( \sigma : G \rightarrow G \) is a definable injective group homomorphism. Then, \( \sigma \) is surjective. | Proof If not then, because \( {\sigma G} \cong G, G \supset {\sigma G} \supset {\sigma }^{2}G \supset \ldots \), contradicting the Descending Chain Condition. | Yes |
Corollary 7.1.4 If \( G \) is an \( \omega \) -stable group and \( \left\{ {{H}_{i} : i \in I}\right\} \) is a collection of definable subgroups, then there is \( {I}_{0} \subseteq I \) finite such that | Proof If not we can find \( {i}_{0},{i}_{1},\ldots \) such that if \( {G}_{m} = {H}_{{i}_{0}} \cap \ldots \cap {H}_{{i}_{m}} \), then \( {G}_{0} > {G}_{1} > {G}_{2} > \ldots \) | No |
Corollary 7.1.5 If \( G \) is an \( \omega \) -stable group and \( A \subseteq G \), then the centralizer \( C\left( A\right) \) is definable. | Proof Because\n\n\[ C\left( A\right) = \mathop{\bigcap }\limits_{{a \in A}}C\left( {\{ a\} }\right) \]\n\nthere are \( {a}_{1},\ldots ,{a}_{m} \in A \) such that \( C\left( A\right) = \left\{ {g \in G : g{a}_{i} = {a}_{i}g}\right. \) for \( i = \) \( 1,2,\ldots, m\} \). | Yes |
If \( G \) is an \( \omega \) -stable group, there is \( {G}^{0} \leq G \) the smallest definable finite index subgroup of \( G \) . Moreover, \( {G}^{0} \) is a normal subgroup of \( G \) and definable over \( \varnothing \) . | Let \( \mathcal{H} = \left\{ {H \leq G : H\text{definable,}\left\lbrack {G : H}\right\rbrack < {\aleph }_{0}}\right\} \) . By Corollary 7.1.4, there are \( {H}_{1},\ldots ,{H}_{m} \in \mathcal{H} \) such that\n\n\[ \mathop{\bigcap }\limits_{{H \in \mathcal{H}}}H = {H}_{1} \cap \ldots \cap {H}_{m} \]\n\nLet \( {G}^{0} =... | Yes |
Lemma 7.1.11 \( \operatorname{RM}\left( {\operatorname{Stab}\left( p\right) }\right) \leq \operatorname{RM}\left( p\right) \) . | Proof Let \( G \prec {G}_{1} \) with \( a, b \in {G}_{1} \) such that \( a \) realizes \( p, b \in \operatorname{Stab}\left( p\right) \) such that \( \operatorname{RM}\left( {b/G}\right) = \operatorname{RM}\left( {\operatorname{Stab}\left( p\right) }\right) \), and \( a \) and \( b \) are independent over \( G \) . The... | Yes |
Lemma 7.1.12 \( \operatorname{Stab}\left( p\right) \leq {G}^{0} \) . | Proof Let \( a \in \operatorname{Stab}\left( p\right) \), and let \( \psi \left( v\right) \) define \( {G}^{0} \) . Let \( b \in G \) such that \( \psi \left( {{b}^{-1}v}\right) \in p \) . Thus \( \psi \left( {{b}^{-1}{av}}\right) \in p \) . Let \( G \prec H \) with \( c \in H \) realizing \( p \) . Then \( {b}^{-1}{ac... | Yes |
Lemma 7.2.2 If \( \operatorname{tp}\left( {x/G}\right) \) is generic and \( a \in G \), then \( \operatorname{tp}\left( {{ax}/G}\right) \) and \( \operatorname{tp}\left( {{x}^{-1}/G}\right) \) are generic. | Proof The maps \( x \mapsto {ax} \) and \( x \mapsto {x}^{-1} \) are definable bijections and hence preserve Morley rank. | Yes |
Lemma 7.2.3 \( p \in {S}_{1}\left( G\right) \) is generic if and only if \( \left\lbrack {G : \operatorname{Stab}\left( p\right) }\right\rbrack < {\aleph }_{0} \) . | Proof\n\n\( \left( \Leftarrow \right) \) If \( \operatorname{Stab}\left( p\right) \) has finite index, \( \operatorname{RM}\left( {\operatorname{Stab}\left( p\right) }\right) = \operatorname{RM}\left( G\right) \) . But \( \operatorname{RM}\left( {\operatorname{Stab}\left( p\right) }\right) \leq \operatorname{RM}\left( ... | Yes |
Corollary 7.2.4 \( p \in {S}_{1}\left( G\right) \) is generic if and only if \( \operatorname{Stab}\left( p\right) = {G}^{0} \) . | ## Proof\n\n\( \left( \Leftarrow \right) \) Clear from Lemma 7.2.3.\n\n\( \left( \Rightarrow \right) \) By Lemma 7.2.3, \( {G}^{0} \leq \operatorname{Stab}\left( p\right) \), and by Lemma 7.1.12, \( \operatorname{Stab}\left( p\right) \leq \)\n\n\( {G}^{0} \) . | No |
Lemma 7.2.6 If \( g \in G \), there are \( a, b \in \mathbb{G} \) generic over \( G \) such that \( g = {ab} \) . | Proof Let \( a \in \mathbb{G} \) be generic over \( G \) . Because \( x \mapsto g{x}^{-1} \) is a definable bijection, \( b = g{a}^{-1} \) is also generic over \( G \) and \( g = {ab} \) . | Yes |
Corollary 7.2.7 Suppose that \( G \) is connected and \( A \subseteq G \) is a definable subset with \( \operatorname{RM}\left( A\right) = \operatorname{RM}\left( G\right) \) . Then \( G = A \cdot A = \{ {ab} : a, b \in A\} \) . | Proof Let \( \phi \left( v\right) \) be an \( {\mathcal{L}}_{G} \) -formula defining \( A \) . For any \( g \in G \), we can find \( a, b \in \mathbb{G} \) generic over \( G \) such that \( g = {ab} \) . Because there is a unique generic type, \( \phi \left( a\right) \) and \( \phi \left( b\right) \) . Thus \( \mathbb{... | Yes |
Lemma 7.2.8 Let \( A \subseteq G \) be a definable generic subset of \( G \) . There are \( {a}_{1},\ldots ,{a}_{n} \in G \) such that \( G = {a}_{1}A \cup \ldots \cup {a}_{n}A \) . | Proof Because finitely many translates of \( {G}^{0} \) cover \( G \), we may, without loss of generality, assume that \( G \) is connected. Let \( \phi \left( v\right) \) be the \( {\mathcal{L}}_{G} \) -formula defining \( A \) . Let \( p \in {S}_{1}\left( G\right) \) be the unique generic type.\n\nClaim For any \( q ... | Yes |
Theorem 7.2.11 If \( G \) is an infinite \( \omega \) -stable group with no proper definable infinite subgroups, then \( G \) is Abelian. | Proof Suppose not. Then, the center \( Z\left( G\right) \) is finite and for all \( a \in G \smallsetminus Z\left( G\right) \) , the centralizer \( C\left( a\right) = \{ g \in G : {ag} = {ga}\} \) is finite.\n\nLet \( a \in G \smallsetminus Z\left( G\right) \), and let \( b \) be generic over \( a \) .\n\nClaim \( {1b}... | Yes |
Corollary 7.2.12 If \( G \) is an infinite \( \omega \) -stable group, then there is an infinite definable Abelian \( H \leq G \) . | Proof By the Descending Chain Condition, there is an infinite definable \( H \leq G \) with no infinite definable proper subgroups. | No |
Corollary 7.2.13 If \( G \) is a group of Morley rank 1, then \( G \) is Abelian-by-finite (i.e., there is a definable Abelian subgroup of finite index). | Proof If \( \operatorname{RM}\left( G\right) = 1 \), then \( {G}^{0} \) is Abelian. | No |
Corollary 7.2.14 If \( G \) is an infinite \( \omega \) -stable group with no definable infinite proper subgroups, then either \( G \) is a divisible Abelian group or every element of \( G \) has order \( p \) for some prime \( p \) . | Proof For any prime \( p,{G}^{p} = \left\{ {{g}^{p} : g \in G}\right\} \) is a definable subgroup and hence must either be finite or all of \( G \) . If \( {G}^{p} = G \), then every element is divisible by \( p \) . If \( G \) is \( p \) -divisible for all primes \( p \), then \( G \) is divisible. If \( {G}^{p} \) is... | Yes |
Theorem 7.3.2 (Zil’ber’s Indecomposability Theorem) Let \( G \) be a group of finite Morley rank and \( \left( {{X}_{i} : i \in I}\right) \) a collection of definable indecomposable subsets of \( G \) each containing 1 . Then, the subgroup of \( G \) generated by \( \mathop{\bigcup }\limits_{{i \in I}}{X}_{i} \) is def... | Proof For each \( \sigma = \left( {{i}_{1},\ldots ,{i}_{n}}\right) \in {I}^{ < \omega } \), let \( {X}^{\sigma } = \left\{ {{x}_{1}\cdots {x}_{n} : {x}_{1} \in }\right. \) \( \left. {{X}_{{i}_{1}},\ldots ,{x}_{n} \in {X}_{{i}_{n}}}\right\} \) . Because \( \operatorname{RM}\left( G\right) \) is finite, there is a \( \si... | Yes |
Lemma 7.3.3 Suppose that there is an \( \omega \) -stable action of \( \Gamma \) on a group \( G \) as a group of automorphisms, \( X \subseteq G \) is \( \Gamma \) -invariant, and for all definable \( \Gamma \) -invariant subgroups \( H \) of \( G \) either \( \left| {X/H}\right| = 1 \) or \( X/H \) is infinite. Then,... | Proof Suppose that \( H \) is a definable subgroup of \( G \) and \( 1 < \left| {X/H}\right| < {\aleph }_{0} \) . Suppose that \( X \subseteq {x}_{1}H \cup \ldots \cup {x}_{n}H \) . If \( \gamma \in \Gamma \) and \( x \in X \), then \( {\gamma }^{-1}x \in \) \( X \) . Thus, \( {\gamma }^{-1}x = {x}_{i}h \) for some \( ... | Yes |
Corollary 7.3.4 Suppose that \( G \) is an \( \omega \) -stable group. If \( H \) is a definable connected subgroup of \( G \) and \( g \in G \), then \( {g}^{H} \) is indecomposable. | Proof The group \( H \) acts on \( G \) via conjugation, and \( {g}^{H} \) is invariant under this action. Thus, by the preceding lemma, it suffices to show that \( {g}^{H} \) is indecomposable for definable \( N \leq G \) where \( {hN}{h}^{-1} = N \) for all \( h \in H \) . Suppose that \( {g}^{H}/N \) is finite and \... | Yes |
Corollary 7.3.5 If \( G \) is a connected group of finite Morley rank, then the commutator subgroup \( {G}^{\prime } \) is a connected definable subgroup of \( G \) . | Proof By Corollary 7.3.4, \( {g}^{G} \) is indecomposable. Thus, \( {g}^{-1}\left( {g}^{G}\right) \) is indecomposable, \( 1 \in {g}^{-1}\left( {g}^{G}\right) \) and \( {G}^{\prime } \) is the group generated by \( \left\{ {{g}^{-1}\left( {g}^{G}\right) }\right. \) : \( g \in G\} \) . By Zil’ber’s Indecomposability The... | Yes |
Theorem 7.3.6 If \( G \) is an infinite non-Abelian group of finite Morley rank and \( G \) has no nontrivial definable normal subgroups, then \( G \) is simple. | Proof Because \( {G}^{0} \) is a normal subgroup of \( G, G \) is connected. For \( a \in G \) , let \( C\left( a\right) \) be the centralizer \( \{ g \in G : {ga} = {ag}\} \) . For \( g, h \in G,{a}^{g} = {a}^{h} \) if and only if \( g \in {hC}\left( a\right) \) . Suppose \( {a}^{G} \) is finite. Then, \( C\left( a\ri... | Yes |
Theorem 7.3.15 If \( G \) is an infinite connected, solvable, nonnilpotent group of finite Morley rank, then \( G \) interprets an algebraically closed field. | Proof Let \( {Z}_{0}\left( G\right) \trianglelefteq {Z}_{1}\left( G\right) \ldots \) be the upper central series of \( G \) . Because \( G \) has finite Morley rank, there is an \( n \) such that \( \operatorname{RM}\left( {{Z}_{n}\left( G\right) }\right) \) is maximal. Then, \( {Z}_{n + 1}\left( G\right) /{Z}_{n}\left... | "No" |
Lemma 7.4.3 If \( V \) is a variety, then \( V \) is interpretable in the algebraically closed field \( K \) . | Proof Let \( V = {V}_{1} \cup \ldots \cup {V}_{n} \) with charts \( {f}_{i} : {V}_{i} \rightarrow {U}_{i} \), without loss of generality, there is an \( m \) such that each \( {U}_{i} \subseteq {K}^{m} \) . Let \( {a}_{1},\ldots ,{a}_{n} \in K \) be distinct, and let \( X = \left\{ {\left( {x, y}\right) \in {K}^{m + 1}... | Yes |
Lemma 7.4.7 Suppose that \( V \) and \( W \) are varieties, \( {V}_{0} \subseteq V \) is open, and \( f : {V}_{0} \rightarrow W \) is a definable function. There is an affine open \( U \subseteq {V}_{0} \) such that \( f \mid U \) is a quasimorphism. | Proof Without loss of generality, we may assume that \( {V}_{0} \) is an affine open subset of \( V \), the closure of \( {V}_{0} \) is irreducible, \( {W}_{0} \) is an affine open subset of \( W \), and \( f : {V}_{0} \rightarrow {W}_{0} \) . By Proposition 3.2.14, there are quasirational functions \( {f}_{1},\ldots ,... | Yes |
Lemma 7.4.9 A definable subgroup of an algebraic group is closed. | Proof Suppose that \( G \) is an algebraic group and \( H \leq G \) is definable. Let \( V \) be the closure of \( H \) in \( G \) . Suppose, for contradiction, that \( a \in V \smallsetminus H \) . By Exercise 6.6.14, \( \operatorname{RM}\left( {V \smallsetminus H}\right) < \operatorname{RM}\left( H\right) \) . Every ... | No |
Lemma 7.4.10 A connected algebraic group is irreducible. | Proof Let \( G \) be a connected algebraic group. Let \( {V}_{1},\ldots ,{V}_{m} \) be the irreducible components of \( G \) . If \( a \in G \), then \( x \mapsto {ax} \) is continuous. Thus, each \( a{V}_{i} \) is irreducible and \( G = a{V}_{1} \cup \ldots \cup a{V}_{m} \) . Because the decomposition into irreducible... | Yes |
Lemma 7.4.11 Suppose that \( K \) is an algebraically closed field of characteristic zero. Suppose that \( G \) is a variety, \( \left( {G, \cdot }\right) \) is a group, and \( \cdot \) is a morphism. Then, \( \left( {G, \cdot }\right) \) is an algebraic group. | Proof We must show that \( x \mapsto {x}^{-1} \) is a morphism. Let \( {G}^{0} \) be the connected component of \( G \), and let \( U \subseteq {G}^{0} \) be an affine open subset of \( {G}^{0} \) . Because \( {G}^{0} \) is irreducible, it is the closure of \( U \) . Thus, \( U \) is a generic subset of \( {G}^{0} \) .... | Yes |
Lemma 7.4.12 Suppose that \( K \) is an algebraically closed field of characteristic zero, \( G \) and \( H \) are algebraic groups and \( f : G \rightarrow H \) is a definable group homomorphism, then \( f \) is a morphism. | Proof Let \( a \in {G}^{0} \) be generic. By Lemma 7.4.7, there is an affine open set \( U \) such that \( f \mid U \) is a morphism. Finitely many translates of \( U \) cover \( G \) . If \( x \in {aU} \), then \( f\left( x\right) \) is given by the composition\n\n\[ x \mapsto {a}^{-1}x \mapsto f\left( {{a}^{-1}x}\rig... | Yes |
Lemma 7.4.13 Suppose that \( K \) is an algebraically closed field of characteristic zero, \( G \) is a constructible group, and \( {G}^{0} \) is definably isomorphic to an algebraic group; then, so is \( G \) . | Proof Let \( A \) be a set of representatives for \( G/{G}^{0} \) . For \( g \in G \), let \( i\left( g\right) \in A \) such that \( g \in i\left( g\right) {G}^{0} \) . We can choose \( A \) such that \( i\left( 1\right) = 1 \) and \( {a}^{-1} \in A \) for all \( a \in A \) . Then, \( G \) is the disjoint union \( \mat... | Yes |
Theorem 7.4.14 Let \( K \) be an algebraically closed field of characteristic zero. If \( G \subseteq {K}^{n} \) is a constructible group, then \( G \) is definably isomorphic to an algebraic group. | Proof Without loss of generality, we may assume that \( G \) is connected. By quantifier elimination, \( G = \mathop{\bigcup }\limits_{{i = 1}}^{n}{F}_{i} \cap {O}_{i} \) where \( {F}_{i} \) is Zariski closed and irreducible and \( {O}_{i} \) is open. Let \( {V}_{1} \) be some \( {F}_{i} \cap {O}_{i} \) containing the ... | Yes |
Corollary 7.4.16 If \( K \) is an algebraically closed field of characteristic zero, \( G \) is an algebraic group and \( H \leq G \) is an algebraic subgroup, then \( G/H \) is an algebraic group. | Proof Because \( G/H \) is interpretable, by elimination of imaginaries it is constructible and, by Theorem 7.4.14, it is isomorphic to an algebraic group. | No |
Lemma 7.4.19 Let \( k \) be a differential field with constants \( {C}_{k} \) . Let \( f\left( X\right) \in \) \( k\{ X\} \) be a homogeneous linear differential polynomial of order \( n \) .\ni) The solutions to \( f\left( X\right) = 0 \) in \( k \) form a vector space over \( {C}_{k} \) of dimension at most \( n \) . | Proof i) and ii) are standard facts that can be found in any book on differential equations (for example, [39] or [65]). | No |
Lemma 7.4.21 Let \( k \) be a differential field with algebraically closed constant field \( {C}_{k} \), and let \( f\left( X\right) = 0 \) be a homogeneous linear differential equation over \( k \) . There is \( l/k \) a Picard-Vessiot extension for \( f \) with \( l \) contained in the differential closure of \( k \)... | Proof Let \( K \) be the differential closure of \( k \) . By Exercise 6.6.32, \( {C}_{K} = \) \( {C}_{k} \) . By Lemma 7.4.19, we can find \( {x}_{1},\ldots ,{x}_{n} \in K \) a fundamental system of solutions for \( f\left( X\right) = 0 \) . Thus, \( l = k\left\langle {{x}_{1},\ldots ,{x}_{n}}\right\rangle \) is a Pic... | No |
Theorem 7.4.22 Suppose that \( k \) is a differential field with algebraically closed constant field \( C \) . Let \( f\left( X\right) \in k\{ X\} \) be a homogeneous linear differential equation of order \( n \), and let \( l/k \) be a Picard-Vessiot extension for \( f \) . The differential Galois group \( G\left( {l/... | Proof Suppose that \( l = k\left\langle {{x}_{1},\ldots ,{x}_{n}}\right\rangle \), where \( {x}_{1},\ldots ,{x}_{n} \) is a fundamental system of solutions to linear equation \( f\left( X\right) = 0 \) . Because \( l/k \) is Picard-Vessiot, \( {C}_{l} = C \) .\n\nLet \( V = \{ y \in l : f\left( y\right) = 0\} \) . Then... | Yes |
Lemma 7.4.25 \( \psi \left( \bar{v}\right) \) isolates \( \operatorname{tp}\left( {\bar{a}/\left\langle {k,{C}_{\mathbb{K}}}\right\rangle }\right) \) . | Proof Suppose, for contradiction, that \( \bar{b} \in k,\bar{c} \in {C}_{\mathbb{K}} \), and \( \phi \left( {\bar{v},\bar{b},\bar{c}}\right) \) and \( \neg \phi \left( {\bar{v},\bar{b},\bar{c}}\right) \) split \( \psi \left( \bar{v}\right) \) . Then\n\n\[ \n\mathbb{K} \vDash \exists \bar{c}\left( {\bigwedge {c}_{i}^{\p... | Yes |
Theorem 7.5.3 Suppose that \( \mathbb{M} \) is \( \omega \) -stable and \( G \subseteq \mathbb{M} \) is an \( \bigwedge \) -definable group. Then, \( G \) is definable. | Proof Let \( \mathop{\bigwedge }\limits_{{i \in I}}{\phi }_{i}\left( v\right) \) define \( G \) . Without loss of generality, we may assume that if \( {I}_{0} \subseteq I \) is finite, there is \( j \in I \) such that\n\n\[{\phi }_{j}\left( v\right) \rightarrow \mathop{\bigwedge }\limits_{{i \in {I}_{0}}}{\phi }_{i}\le... | Yes |
Theorem 7.5.4 Suppose that \( T \) is \( \omega \) -stable and \( \mathbb{M} \) is a monster model of \( T \) . Let \( A \subset \mathbb{M} \), and let \( p \in {S}_{1}\left( A\right) \) be a stationary type (for notational simplicity, we will assume that \( A = \varnothing \), but this is no loss of generality). For \... | Proof Suppose that \( f \) and \( g \) are definable functions. We say that \( f \) and \( g \) have the same germ at \( p \) if and only if whenever \( A \) is large enough so that \( f \) and \( g \) are both defined over \( A \), and \( a \) realizes \( {p}_{A} \), then \( f\left( a\right) = g\left( a\right) \) . We... | Yes |
Theorem 7.5.7 Suppose that there is a group configuration in \( {\mathbb{M}}^{\text{eq }} \) . Then, there is a rank one group definable in \( {\mathbb{M}}^{\text{eq }} \) . | We give an application of the group configuration in Theorem 8.3.1. Proofs of Hrushovski’s Theorem appear in [18] §4.5 and [76] §5.4. | No |
Theorem 7.6.26 If \( G \) is a simple group of Morley rank 3, then\ni) \( G \) is a bad group, or\nii) \( G \) interprets an algebraically closed field \( K \) and \( G \) is definably isomorphic to \( {PS}{L}_{2}\left( K\right) \) . | For proofs of these results and more on groups of finite Morley rank, see [86] or [15]. | No |
Lemma 8.2.3 Suppose that \( X \subseteq \mathbb{M} \) is definable. There is \( \alpha \in {\mathbb{M}}^{\text{eq }} \) such that \( \alpha \) is a canonical base for \( X \) . Indeed, if \( X \) is \( A \) -definable, we can find a canonical base in \( {\operatorname{dcl}}^{\mathrm{{eq}}}\left( A\right) \) . | Proof Suppose that \( X \) is defined by the formula \( \phi \left( {\bar{x},\bar{a}}\right) \) . Let \( E \) be the equivalence relation\n\n\[ \bar{a}E\bar{b} \Leftrightarrow \left( {\phi \left( {\bar{x},\bar{a}}\right) \leftrightarrow \phi \left( {\bar{x},\bar{b}}\right) }\right) .\n\nLet \( \alpha = \bar{a}/E \in {\... | Yes |
Lemma 8.2.4 If \( A \) is a canonical base for \( p \in {S}_{n}\left( \mathbb{M}\right) \), then \( B \) is a canonical base for \( p \) if and only if \( {\operatorname{dcl}}^{\mathrm{{eq}}}\left( A\right) = {\operatorname{dcl}}^{\mathrm{{eq}}}\left( B\right) \) . | Proof Suppose that \( C \subset \mathbb{M} \) and \( \left| C\right| < \left| \mathbb{M}\right| \) ; let \( \operatorname{Aut}\left( {\mathbb{M}/C}\right) \) denote the automorphisms of \( \mathbb{M} \) fixing \( C \) pointwise. The proof of Proposition 4.3.25 generalized to \( {\mathbb{M}}^{\text{eq }} \) shows that\n... | Yes |
Lemma 8.2.6 Suppose that \( \mathbb{M} \) is \( \omega \) -stable and \( p \in {S}_{n}\left( \mathbb{M}\right) \) . Then, \( p \) has a canonical base in \( {\mathbb{M}}^{\text{eq }} \) . | Proof For each \( \mathcal{L} \) -formula \( \phi \left( {\bar{v},\bar{w}}\right) \), let \( {X}_{\phi } = \{ \bar{a} \in \mathbb{M} : \phi \left( {\bar{v},\bar{a}}\right) \in p\} \) . By definability of types, \( {X}_{\phi } \) is definable. If \( \sigma \) is an automorphism of \( \mathbb{M} \), then \( {\sigma p} = ... | Yes |
Theorem 8.2.7 Suppose that \( \mathbb{M} \) is \( \omega \) -stable and \( p \in {S}_{n}\left( \mathbb{M}\right) \) does not fork over \( A \subseteq \mathbb{M} \) . There is \( \alpha \in {\operatorname{acl}}^{\mathrm{{eq}}}\left( A\right) \), a canonical base for \( p \) . If \( p \mid A \) is stationary, then we can... | Proof Suppose that \( \phi \left( {\bar{v},\bar{w}}\right) \) is an \( \mathcal{L} \) -formula such that \( \phi \left( {\bar{v},\bar{a}}\right) \in p \) and \( \operatorname{RM}\left( {\phi \left( {\bar{v},\bar{a}}\right) }\right) = \operatorname{RM}\left( p\right) \) . Let \( X = \{ \bar{b} : \phi \left( {\bar{v},\ba... | Yes |
Lemma 8.2.9 Let \( \mathbb{M} \) be a strongly minimal set and let \( X \subset \mathbb{M} \) be infinite. Suppose that \( E \) is an \( \varnothing \) -definable equivalence relation on \( {\mathbb{M}}^{m} \) . Let \( \overline{a} \in {\mathbb{M}}^{m} \) and \( \alpha = \bar{a}/E \) . There is a finite \( C \subset {\... | In particular, if \( {\mathbb{M}}_{X} \) is \( \mathbb{M} \) viewed as an \( {\mathcal{L}}_{X} \) -structure, then for every \( \alpha \in {\mathbb{M}}_{X}^{\text{eq }} \) there is \( \bar{d} \in \mathbb{M} \) such that \( {\operatorname{acl}}^{\text{eq }}\left( {\alpha, X}\right) = {\operatorname{acl}}^{\text{eq }}\le... | Yes |
Theorem 8.2.11 Let \( D \subseteq {\mathbb{M}}^{n} \) be a strongly minimal set. The following are equivalent:\n\ni) for some small \( B \subset D \), the pregeometry \( {D}_{B} \) is modular;\n\nii) \( D \) is linear;\n\niii) for any \( b \in D \smallsetminus \operatorname{acl}\left( \varnothing \right) ,{D}_{b} \) is... | Proof Often when we want to prove things about arbitrary strongly minimal sets \( D \subseteq {\mathbb{M}}^{n} \), we instead assume that \( \mathbb{M} \) is strongly minimal. This is no great loss of generality. By extending the language, we may assume that \( D \) is \( \varnothing \) -definable. By Corollary 6.3.7, ... | Yes |
Lemma 8.2.13 Suppose that \( T \) is \( \omega \) -stable. The following are equivalent.\n\ni) \( T \) is one-based.\n\nii) For all \( \bar{a} \in {\mathbb{M}}^{\text{eq }} \) and \( B \subseteq {\mathbb{M}}^{\text{eq }} \), if \( \operatorname{tp}\left( {\bar{a}/B}\right) \) is stationary, then \( \operatorname{cb}\le... | i) \( \Rightarrow \) ii) Let \( A = {\operatorname{acl}}^{\text{eq }}\left( \bar{a}\right) \) . Because \( \operatorname{tp}\left( {\bar{a}/{\operatorname{acl}}^{\text{eq }}\left( B\right) }\right) \) does not fork over \( B \) , we may without loss of generality assume that \( B = {\operatorname{acl}}^{\mathrm{{eq}}}\... | Yes |
Theorem 8.2.15 Suppose that \( T \) is uncountably categorical and \( \mathbb{M} \) is the monster model of \( T \) . The following are equivalent.\n\ni) \( T \) is one-based.\n\nii) Every strongly minimal \( D \subseteq {\mathbb{M}}^{n} \) is locally modular.\n\niii) Some strongly minimal \( D \subseteq {\mathbb{M}}^{... | For a proof, see Theorem 4.3.1 in [18]. | No |
Corollary 8.3.4 If \( \mathbb{G} \) is an \( \omega \) -stable, one-based group, then there are at most countably many definable subgroups of \( {\mathbb{G}}^{n} \) . | Proof Any definable subgroup \( H \) has a canonical base in \( {\operatorname{acl}}^{\mathrm{{eq}}}\left( \varnothing \right) \) . Because our language is countable, \( {\operatorname{acl}}^{\mathrm{{eq}}}\left( \varnothing \right) \) is countable and there are only countably many definable subgroups. | Yes |
Lemma 8.3.5 If \( \mathbb{G} \) is a connected one-based \( \omega \) -stable group, then \( \mathbb{G} \) is Abelian. Thus every one-based \( \omega \) -stable group is Abelian-by-finite. | Proof For \( g \in \mathbb{G} \), let \( {H}_{g} = \left\{ {\left( {h,{g}^{-1}{hg}}\right) : h \in \mathbb{G}}\right\} \subseteq \mathbb{G} \times \mathbb{G} \) . Then, \( {H}_{g} = {H}_{h} \) if and only if \( g/Z\left( \mathbb{G}\right) = h/Z\left( \mathbb{G}\right) \) . If \( \mathbb{G} \) is non-Abelian, then \( Z\... | Yes |
Corollary 8.3.7 Suppose that \( p \in {S}_{n}\left( G\right) \) then, there is \( \bar{b} \in {G}^{n} \) such that “ \( v \in \operatorname{Stab}\left( p\right) \bar{b} \) ” \( \in p \) . | Proof As we argued above, the group \( {G}^{n} \) is also one-based, and we can view \( p \) as a 1-type over \( {G}^{n} \) and apply the previous lemma. | No |
Theorem 8.3.9 If \( G \) is an \( \omega \) -stable one-based group and \( X \subseteq {G}^{n} \) is definable, then \( X \) is a finite Boolean combination of cosets of definable subgroups \( H \leq {G}^{n} \) . | Proof This follows from Corollary 8.3.8 and Exercise 4.5.13. | No |
Theorem 8.3.20 Suppose that \( K \) is a differentially closed field and \( A \) is a simple Abelian variety defined over \( K \) that is not isomorphic to an Abelian variety defined over \( C \) . Let \( {A}^{\# } \) be the closure in the Kolchin topology of the torsion points of \( A \) . Then, \( {A}^{\# } \) is a o... | The results of Manin and Buium show that \( {A}^{\# } \) is a finite Morley rank group with no infinite definable subgroups. There are two cases to consider. If all strongly minimal subsets of \( {A}^{\# } \) are locally modular, then \( {A}^{\# } \) is one-based. If there is any nonlocally modular strongly minimal sub... | Yes |
Theorem 8.3.21 Let \( K \) and \( k \) be algebraically closed fields of characteristic zero with \( k \subseteq K \) . Let \( A \) be a simple \( A \) belian variety defined over \( K \) that is not isomorphic to an Abelian variety defined over the algebraic closure of \( k \) . If \( V \subset A \) is a proper subvar... | Proof We can define a derivation \( \delta \) on \( K \) such that the constant field is \( k \) (see [58] X §7). If \( \widehat{K} \) is the differential closure of \( \left( {K,\delta }\right) \), then, by Exercise 6.6.32, the constant field of \( \widehat{K} \) is \( k \) . Thus, replacing \( K \) by \( \widehat{K} ... | Yes |
Theorem 1.2 (Topological Invariance of Dimension). A nonempty n-dimensional topological manifold cannot be homeomorphic to an \( m \) -dimensional manifold unless \( m = n \) . | For the proof, see Theorem 17.26. In Chapter 2, we will also prove a related but weaker theorem (diffeomorphism invariance of dimension, Theorem 2.17). See also [LeeTM, Chap. 13] for a different proof of Theorem 1.2 using singular homology theory. | No |
Example 1.3 (Graphs of Continuous Functions). Let \( U \subseteq {\mathbb{R}}^{n} \) be an open subset, and let \( f : U \rightarrow {\mathbb{R}}^{k} \) be a continuous function. The graph of \( f \) is the subset of \( {\mathbb{R}}^{n} \times {\mathbb{R}}^{k} \) defined by\n\n\[ \Gamma \left( f\right) = \left\{ {\left... | Because \( \varphi \) is the restriction of a continuous map, it is continuous; and it is a homeomorphism because it has a continuous inverse given by \( {\varphi }^{-1}\left( x\right) = \left( {x, f\left( x\right) }\right) \) . Thus \( \Gamma \left( f\right) \) is a topological manifold of dimension \( n \) . In fact,... | Yes |
For each integer \( n \geq 0 \), the unit \( n \)-sphere \( {\mathbb{S}}^{n} \) is Hausdorff and second-countable because it is a topological subspace of \( {\mathbb{R}}^{n + 1} \). To show that it is locally Euclidean, for each index \( i = 1,\ldots, n + 1 \) let \( {U}_{i}^{ + } \) denote the subset of \( {\mathbb{R}... | Let \( f : {\mathbb{B}}^{n} \rightarrow \mathbb{R} \) be the continuous function\n\n\[ f\left( u\right) = \sqrt{1 - {\left| u\right| }^{2}} \]\n\nThen for each \( i = 1,\ldots, n + 1 \), it is easy to check that \( {U}_{i}^{ + } \cap {\mathbb{S}}^{n} \) is the graph of the function\n\n\[ {x}^{i} = f\left( {{x}^{1},\ldo... | Yes |
Example 1.5 (Projective Spaces). The \( n \) -dimensional real projective space, denoted by \( {\mathbb{{RP}}}^{n} \) (or sometimes just \( {\mathbb{P}}^{n} \) ), is defined as the set of 1-dimensional linear subspaces of \( {\mathbb{R}}^{n + 1} \), with the quotient topology determined by the natural map \( \pi : {\ma... | For each \( i = 1,\ldots, n + 1 \), let \( {\widetilde{U}}_{i} \subseteq {\mathbb{R}}^{n + 1} \smallsetminus \{ 0\} \) be the set where \( {x}^{i} \neq 0 \) , and let \( {U}_{i} = \pi \left( {\widetilde{U}}_{i}\right) \subseteq {\mathbb{{RP}}}^{n} \) . Since \( {\widetilde{U}}_{i} \) is a saturated open subset, \( {U}_... | Yes |
Example 1.8 (Product Manifolds). Suppose \( {M}_{1},\ldots ,{M}_{k} \) are topological manifolds of dimensions \( {n}_{1},\ldots ,{n}_{k} \), respectively. The product space \( {M}_{1} \times \cdots \times {M}_{k} \) is shown to be a topological manifold of dimension \( {n}_{1} + \cdots + {n}_{k} \) as follows. | It is Hausdorff and second-countable by Propositions A. 17 and A.23, so only the locally Euclidean property needs to be checked. Given any point \( \left( {{p}_{1},\ldots ,{p}_{k}}\right) \in \) \( {M}_{1} \times \cdots \times {M}_{k} \), we can choose a coordinate chart \( \left( {{U}_{i},{\varphi }_{i}}\right) \) for... | Yes |
Lemma 1.10. Every topological manifold has a countable basis of precompact coordinate balls. | Proof. Let \( M \) be a topological \( n \) -manifold. First we consider the special case in which \( M \) can be covered by a single chart. Suppose \( \varphi : M \rightarrow \widehat{U} \subseteq {\mathbb{R}}^{n} \) is a global coordinate map, and let \( \mathcal{B} \) be the collection of all open balls \( {B}_{r}\l... | Yes |
Proposition 1.11. Let \( M \) be a topological manifold.\n\n(a) \( M \) is locally path-connected.\n\n(b) \( M \) is connected if and only if it is path-connected.\n\n(c) The components of \( M \) are the same as its path components.\n\n(d) \( M \) has countably many components, each of which is an open subset of \( M ... | Proof. Since each coordinate ball is path-connected, (a) follows from the fact that \( M \) has a basis of coordinate balls. Parts (b) and (c) are immediate consequences of (a) and Proposition A.43. To prove (d), note that each component is open in \( M \) by Proposition A.43, so the collection of components is an open... | Yes |
Proposition 1.12 (Manifolds Are Locally Compact). Every topological manifold is locally compact. | Proof. Lemma 1.10 showed that every manifold has a basis of precompact open subsets. | No |
Theorem 1.15 (Manifolds Are Paracompact). Every topological manifold is paracompact. In fact, given a topological manifold \( M \), an open cover \( X \) of \( M \) , and any basis \( \mathcal{B} \) for the topology of \( M \), there exists a countable, locally finite open refinement of \( X \) consisting of elements o... | Proof. Given \( M, X \), and \( \mathcal{B} \) as in the hypothesis of the theorem, let \( {\left( {K}_{j}\right) }_{j = 1}^{\infty } \) be an exhaustion of \( M \) by compact sets (Proposition A.60). For each \( j \), let \( {V}_{j} \doteq {\bar{K}}_{j + 1} \smallsetminus \) Int \( {K}_{j} \) and \( {W}_{j} = \operato... | Yes |
Proposition 1.17. Let \( M \) be a topological manifold.\n\n(a) Every smooth atlas \( \mathcal{A} \) for \( M \) is contained in a unique maximal smooth atlas, called the smooth structure determined by \( \mathcal{A} \) . | Proof. Let \( \mathcal{A} \) be a smooth atlas for \( M \), and let \( \overline{\mathcal{A}} \) denote the set of all charts that are smoothly compatible with every chart in \( \mathcal{A} \) . To show that \( \overline{\mathcal{A}} \) is a smooth atlas, we need to show that any two charts of \( \overline{\mathcal{A}}... | Yes |
For each nonnegative integer \( n \), the Euclidean space \( {\mathbb{R}}^{n} \) is a smooth \( n \)-manifold with the smooth structure determined by the atlas consisting of the single chart \( \left( {{\mathbb{R}}^{n},{\operatorname{Id}}_{{\mathbb{R}}^{n}}}\right) \). | We call this the standard smooth structure on \( {\mathbb{R}}^{n} \) and the resulting coordinate map standard coordinates. Unless we explicitly specify otherwise, we always use this smooth structure on \( {\mathbb{R}}^{n} \). With respect to this smooth structure, the smooth coordinate charts for \( {\mathbb{R}}^{n} \... | Yes |
Consider the homeomorphism \( \psi : \mathbb{R} \rightarrow \mathbb{R} \) given by\n\n\[ \psi \left( x\right) = {x}^{3}. \]\n\nThe atlas consisting of the single chart \( \left( {\mathbb{R},\psi }\right) \) defines a smooth structure on \( \mathbb{R} \). | This chart is not smoothly compatible with the standard smooth structure, because the transition map \( {\operatorname{Id}}_{\mathbb{R}} \circ {\psi }^{-1}\left( y\right) = {y}^{1/3} \) is not smooth at the origin. Therefore, the smooth structure defined on \( \mathbb{R} \) by \( \psi \) is not the same as the standard... | Yes |
Let \( M \) be a smooth \( n \) -manifold and let \( U \subseteq M \) be any open subset. Define an atlas on \( U \) by \[ {\mathcal{A}}_{U} = \{ \text{ smooth charts }\left( {V,\varphi }\right) \text{ for }M\text{ such that }V \subseteq U\} . \] | Every point \( p \in U \) is contained in the domain of some chart \( \left( {W,\varphi }\right) \) for \( M \) ; if we set \( V = W \cap U \), then \( \left( {V,{\left. \varphi \right| }_{V}}\right) \) is a chart in \( {\mathcal{A}}_{U} \) whose domain contains \( p \) . Therefore, \( U \) is covered by the domains of... | Yes |
Example 1.27 (The General Linear Group). The general linear group \( \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \) is the set of invertible \( n \times n \) matrices with real entries. | It is a smooth \( {n}^{2} \) -dimensional manifold because it is an open subset of the \( {n}^{2} \) -dimensional vector space \( \mathrm{M}\left( {n,\mathbb{R}}\right) \) , namely the set where the (continuous) determinant function is nonzero. | Yes |
If \( U \subseteq {\mathbb{R}}^{n} \) is an open subset and \( f : U \rightarrow {\mathbb{R}}^{k} \) is a smooth function, we have already observed above (Example 1.3) that the graph of \( f \) is a topological \( n \) -manifold in the subspace topology. | Since \( \Gamma \left( f\right) \) is covered by the single graph coordinate chart \( \varphi : \Gamma \left( f\right) \rightarrow U \) (the restriction of \( {\pi }_{1} \) ), we can put a canonical smooth structure on \( \Gamma \left( f\right) \) by declaring the graph coordinate chart \( \left( {\Gamma \left( f\right... | Yes |
We showed in Example 1.4 that the \( n \)-sphere \( {\mathbb{S}}^{n} \subseteq {\mathbb{R}}^{n + 1} \) is a topological \( n \)-manifold. We put a smooth structure on \( {\mathbb{S}}^{n} \) as follows. For each \( i = 1,\ldots, n + 1 \), let \( \left( {{U}_{i}^{ \pm },{\varphi }_{i}^{ \pm }}\right) \) denote the graph ... | In the case \( i < j \), we get \[ {\varphi }_{i}^{ \pm } \circ {\left( {\varphi }_{j}^{ \pm }\right) }^{-1}\left( {{u}^{1},\ldots ,{u}^{n}}\right) = \left( {{u}^{1},\ldots ,\widehat{{u}^{i}},\ldots , \pm \sqrt{1 - {\left| u\right| }^{2}},\ldots ,{u}^{n}}\right) \] (with the square root in the \( j \) th position), and... | Yes |
Example 1.32 (Level Sets). The preceding example can be generalized as follows. Suppose \( U \subseteq {\mathbb{R}}^{n} \) is an open subset and \( \Phi : U \rightarrow \mathbb{R} \) is a smooth function. For any \( c \in \mathbb{R} \), the set \( {\Phi }^{-1}\left( c\right) \) is called a level set of \( \Phi \) . Cho... | Because \( {D\Phi }\left( a\right) \) is a row matrix whose entries are the partial derivatives \( \left( {\partial \Phi /\partial {x}^{1}\left( a\right) ,\ldots ,\partial \Phi /\partial {x}^{n}\left( a\right) }\right) \), for each \( a \in M \) there is some \( i \) such that \( \partial \Phi /\partial {x}^{i}\left( a... | Yes |
The \( n \)-dimensional real projective space \( {\mathbb{{RP}}}^{n} \) is a topological \( n \)-manifold by Example 1.5. Let us check that the coordinate charts \( \left( {{U}_{i},{\varphi }_{i}}\right) \) constructed in that example are all smoothly compatible. | Assuming for convenience that \( i > j \), it is straightforward to compute that\n\n\[ \n{\varphi }_{j} \circ {\varphi }_{i}^{-1}\left( {{u}^{1},\ldots ,{u}^{n}}\right) = \left( {\frac{{u}^{1}}{{u}^{j}},\ldots ,\frac{{u}^{j - 1}}{{u}^{j}},\frac{{u}^{j + 1}}{{u}^{j}},\ldots ,\frac{{u}^{i - 1}}{{u}^{j}},\frac{1}{{u}^{j}}... | Yes |
Example 1.34 (Smooth Product Manifolds). If \( {M}_{1},\ldots ,{M}_{k} \) are smooth manifolds of dimensions \( {n}_{1},\ldots ,{n}_{k} \), respectively, we showed in Example 1.8 that the product space \( {M}_{1} \times \cdots \times {M}_{k} \) is a topological manifold of dimension \( {n}_{1} + \cdots + {n}_{k} \), wi... | \[ \left( {{\psi }_{1} \times \cdots \times {\psi }_{k}}\right) \circ {\left( {\varphi }_{1} \times \cdots \times {\varphi }_{k}\right) }^{-1} = \left( {{\psi }_{1} \circ {\varphi }_{1}^{-1}}\right) \times \cdots \times \left( {{\psi }_{k} \circ {\varphi }_{k}^{-1}}\right) , \] which is a smooth map. This defines a nat... | Yes |
Lemma 1.35 (Smooth Manifold Chart Lemma). Let \( M \) be a set, and suppose we are given a collection \( \left\{ {U}_{\alpha }\right\} \) of subsets of \( M \) together with maps \( {\varphi }_{\alpha } : {U}_{\alpha } \rightarrow {\mathbb{R}}^{n} \), such that the following properties are satisfied:\n\n(i) For each \(... | Proof. We define the topology by taking all sets of the form \( {\varphi }_{\alpha }^{-1}\left( V\right) \), with \( V \) an open subset of \( {\mathbb{R}}^{n} \), as a basis. To prove that this is a basis for a topology, we need to show that for any point \( p \) in the intersection of two basis sets \( {\varphi }_{\a... | Yes |
Theorem 1.37 (Topological Invariance of the Boundary). If \( M \) is a topological manifold with boundary, then each point of \( M \) is either a boundary point or an interior point, but not both. Thus \( \partial M \) and \( \operatorname{Int}M \) are disjoint sets whose union is \( M \) . | For the proof, see Problem 17-9. | No |
Proposition 1.45. Suppose \( {M}_{1},\ldots ,{M}_{k} \) are smooth manifolds and \( N \) is a smooth manifold with boundary. Then \( {M}_{1} \times \cdots \times {M}_{k} \times N \) is a smooth manifold with boundary, and \( \partial \left( {{M}_{1} \times \cdots \times {M}_{k} \times N}\right) = {M}_{1} \times \cdots ... | Proof. Problem 1-12. | No |
Theorem 1.46 (Smooth Invariance of the Boundary). Suppose \( M \) is a smooth manifold with boundary and \( p \in M \) . If there is some smooth chart \( \left( {U,\varphi }\right) \) for \( M \) such that \( \varphi \left( U\right) \subseteq {\mathbb{H}}^{n} \) and \( \varphi \left( p\right) \in \partial {\mathbb{H}}^... | Proof. Suppose on the contrary that \( p \) is in the domain of a smooth interior chart \( \left( {U,\psi }\right) \) and also in the domain of a smooth boundary chart \( \left( {V,\varphi }\right) \) such that \( \varphi \left( p\right) \in \) \( \partial {\mathbb{H}}^{n} \) . Let \( \tau = \varphi \circ {\psi }^{-1} ... | Yes |
Proposition 2.4. Every smooth map is continuous. | Proof. Suppose \( M \) and \( N \) are smooth manifolds with or without boundary, and \( F : M \rightarrow N \) is smooth. Given \( p \in M \), smoothness of \( F \) means there are smooth charts \( \left( {U,\varphi }\right) \) containing \( p \) and \( \left( {V,\psi }\right) \) containing \( F\left( p\right) \), suc... | Yes |
Proposition 2.10. Let \( M, N \), and \( P \) be smooth manifolds with or without boundary.\n\n(d) If \( F : M \rightarrow N \) and \( G : N \rightarrow P \) are smooth, then so is \( G \circ F : M \rightarrow P \) . | Proof. We prove (d) and leave the rest as exercises. Let \( F : M \rightarrow N \) and \( G : N \rightarrow \) \( P \) be smooth maps, and let \( p \in M \) . By definition of smoothness of \( G \), there exist smooth charts \( \left( {V,\theta }\right) \) containing \( F\left( p\right) \) and \( \left( {W,\psi }\right... | No |
Proposition 2.12. Suppose \( {M}_{1},\ldots ,{M}_{k} \) and \( N \) are smooth manifolds with or without boundary, such that at most one of \( {M}_{1},\ldots ,{M}_{k} \) has nonempty boundary. For each \( i \), let \( {\pi }_{i} : {M}_{1} \times \cdots \times {M}_{k} \rightarrow {M}_{i} \) denote the projection onto th... | ## Proof. Problem 2-2. | No |
Theorem 2.17 (Diffeomorphism Invariance of Dimension). A nonempty smooth manifold of dimension \( m \) cannot be diffeomorphic to an \( n \) -dimensional smooth manifold unless \( m = n \). | Proof. Suppose \( M \) is a nonempty smooth \( m \) -manifold, \( N \) is a nonempty smooth \( n \) - manifold, and \( F : M \rightarrow N \) is a diffeomorphism. Choose any point \( p \in M \), and let \( \left( {U,\varphi }\right) \) and \( \left( {V,\psi }\right) \) be smooth coordinate charts containing \( p \) and... | Yes |
Theorem 2.18 (Diffeomorphism Invariance of the Boundary). Suppose \( M \) and \( N \) are smooth manifolds with boundary and \( F : M \rightarrow N \) is a diffeomorphism. Then \( F\left( {\partial M}\right) = \partial N \), and \( F \) restricts to a diffeomorphism from \( \operatorname{Int}M \) to \( \operatorname{In... | ## - Exercise 2.19. Use Theorem 1.46 to prove the preceding theorem. | No |
Lemma 2.20. The function \( f : \mathbb{R} \rightarrow \mathbb{R} \) defined by\n\n\[ f\left( t\right) = \left\{ \begin{array}{ll} {e}^{-1/t}, & t > 0 \\ 0, & t \leq 0 \end{array}\right. \]\n\nis smooth. | Proof. The function in question is pictured in Fig. 2.4. It is smooth on \( \mathbb{R} \smallsetminus \{ 0\} \) by composition, so we need only show \( f \) has continuous derivatives of all orders at the origin. Because existence of the \( \left( {k + 1}\right) \) st derivative implies continuity of the \( k \) th, it... | Yes |
Lemma 2.21. Given any real numbers \( {r}_{1} \) and \( {r}_{2} \) such that \( {r}_{1} < {r}_{2} \), there exists a smooth function \( h : \mathbb{R} \rightarrow \mathbb{R} \) such that \( h\left( t\right) \equiv 1 \) for \( t \leq {r}_{1},0 < h\left( t\right) < 1 \) for \( {r}_{1} < \) \( t < {r}_{2} \), and \( h\lef... | Proof. Let \( f \) be the function of the previous lemma, and set\n\n\[ h\left( t\right) = \frac{f\left( {{r}_{2} - t}\right) }{f\left( {{r}_{2} - t}\right) + f\left( {t - {r}_{1}}\right) }.\]\n\n(See Fig. 2.5.) Note that the denominator is positive for all \( t \), because at least one of the expressions \( {r}_{2} - ... | Yes |
Lemma 2.22. Given any positive real numbers \( {r}_{1} < {r}_{2} \), there is a smooth function \( H : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) such that \( H \equiv 1 \) on \( {\bar{B}}_{{r}_{1}}\left( 0\right) ,0 < H\left( x\right) < 1 \) for all \( x \in {B}_{{r}_{2}}\left( 0\right) \smallsetminus {\bar{B}}_{{r}_{... | Proof. Just set \( H\left( x\right) = h\left( \left| x\right| \right) \), where \( h \) is the function of the preceding lemma. Clearly, \( H \) is smooth on \( {\mathbb{R}}^{n} \smallsetminus \{ 0\} \), because it is a composition of smooth functions there. Since it is identically equal to 1 on \( {B}_{{r}_{1}}\left( ... | Yes |
Proposition 2.25 (Existence of Smooth Bump Functions). Let \( M \) be a smooth manifold with or without boundary. For any closed subset \( A \subseteq M \) and any open subset \( U \) containing \( A \), there exists a smooth bump function for \( A \) supported in \( U \) . | Proof. Let \( {U}_{0} = U \) and \( {U}_{1} = M \smallsetminus A \), and let \( \left\{ {{\psi }_{0},{\psi }_{1}}\right\} \) be a smooth partition of unity subordinate to the open cover \( \left\{ {{U}_{0},{U}_{1}}\right\} \) . Because \( {\psi }_{1} \equiv 0 \) on \( A \) and thus \( {\psi }_{0} = \) \( \mathop{\sum }... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.