Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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We show now that \(\left\langle {{f}_{x}v;{sy}}\right\rangle = {\left( -1\right) }^{\deg x\deg y}\langle v;s\left\lbrack {x, y}\right\rbrack \rangle ,\;v \in V, y \in L.\) This formula exhibits \( {f}_{x} \) as the dual (up to sign) of ad \( x \) . In particular, \( {f}_{x} \) is locally conilpotent if and only if ad \... | For the proof of (31.8) note that \( {f}_{x} = {d}_{1}{\eta }_{x} - {\left( -1\right) }^{\deg {\eta }_{x}}{\eta }_{x}{d}_{1} \), where \( {d}_{1} \) is the quadratic part of \( d \) . Then use \( §{21}\left( \mathrm{e}\right) \) to compute \(\left\langle {{f}_{x}v;{sy}}\right\rangle = - {\left( -1\right) }^{\deg {\eta ... | Yes |
Lemma 31.8 If some \( {E}_{i}\left( \sigma \right) \) is locally conilpotent then so is \( H\left( \sigma \right) \) . | proof: If \( H\left( \sigma \right) \) is not locally conilpotent then there is an infinite sequence of non-zero classes \( {\alpha }_{k} \in H\left( M\right) \), such that \( H\left( \sigma \right) {\alpha }_{k + 1} = {\alpha }_{k}, k \geq 0 \) . Let \( p\left( k\right) \) be the greatest integer such that \( {\alpha ... | Yes |
Lemma 31.9 With the notation and hypotheses above suppose ad \( x \) is locally nilpotent and \( {\mathrm{{hl}}}^{\prime }x \) is locally conilpotent. Then \( H\left( \theta \right) \) is locally conilpotent. | proof: First observe that \( \theta \) preserves \( {\Lambda U} \) and also \( {\Lambda }^{ + }U \), because \( \left( {{\Lambda v} \otimes {\Lambda U}, D}\right) \) is a minimal Sullivan algebra. Next, let \( \eta \) be the derivation in \( {\Lambda v} \otimes {\Lambda U} \) defined by \( {\eta w} = \langle w;{sx}\ran... | Yes |
Lemma 31.14 Suppose \( {\left( {x}_{n},{z}_{n}\right) }_{n \geq 0} \) is an infinite sequence of pairs of elements \( {x}_{n},{z}_{n} \in {\Lambda W} \) satisfying:\n\n\[ \bar{d}{z}_{n} = 0\;\text{ and }\;\theta {z}_{n + 1} = {z}_{n} + \bar{d}{x}_{n},\;n \geq 0. \]\n\nThen there is an infinite sequence of elements \( {... | proof: Observe first that each \( \left\lbrack {z}_{n}\right\rbrack \in \mathop{\bigcap }\limits_{k}\operatorname{Im}H{\left( \theta \right) }^{k} = 0 \), since \( \left\lbrack {z}_{n}\right\rbrack = H\left( \theta \right) \left\lbrack {z}_{n + 1}\right\rbrack = \) \( H{\left( \theta \right) }^{2}\left\lbrack {z}_{n + ... | Yes |
The fibration \( p : {S}^{{4m} + 3} \rightarrow \mathbb{H}{P}^{m} \). | The unit sphere \( {S}^{3} \) of the quaternions \( \mathbb{H} \) acts freely by right multiplication on the unit sphere \( {S}^{{4m} + 3} \) of \( {\mathbb{H}}^{m + 1}\left( { \cong {\mathbb{R}}^{{4m} + 4}}\right) \) . This is the action of a classical principal \( {S}^{3} \) bundle ( \( §2\left( \mathrm{a}\right) \) ... | Yes |
The fibration associated with \( {S}^{3} \vee {S}^{3} \rightarrow {S}^{3} \) . | Convert projection from \( {S}^{3} \vee {S}^{3} \) to the first factor to a fibration \( p : X \rightarrow {S}^{3} \) with \( X \simeq {S}^{3} \vee {S}^{3} \) . Since \( p \) is not a rational homology equivalence the fibre, \( F \), has non-trivial rational homology and so \( {\operatorname{cat}}_{0}F \geq 1 = {\opera... | No |
Suppose \( \theta \) is a derivation of even (negative) degree in a commutative cochain algebra \( \left( {A,{d}_{A}}\right) \) such that \( {A}^{0} = \mathbb{R} \) and \( {H}^{1}\left( A\right) = 0 \) . Then we may construct the commutative cochain algebra \( \left( {{\Lambda v} \otimes A, d}\right) \), with \( \deg v... | Now let \( \left( {{\Lambda W},\bar{d}}\right) \) be a minimal model of \( \left( {A, d}\right) \) . Then \( \left( {{\Lambda v} \otimes A, d}\right) \) has a Sullivan model of the form \( \left( {{\Lambda v} \otimes {\Lambda W}, d}\right) \) with \( {d\Phi } = 1 \otimes \bar{d}\Phi + v \otimes {\theta }^{\prime }\Phi ... | Yes |
Example 4 A fibration \( X \rightarrow \mathbb{C}{P}^{m} \) with fibre \( {S}^{3} \) and \( {\operatorname{cat}}_{0}X = 2 \) . | Let \( {S}^{1} \) act on \( {S}^{{2m} + 1} \) by complex multiplication: \( {S}^{1} \) is the unit circle of \( \mathbb{C} \) and \( {S}^{{2m} + 1} \) is the unit sphere in \( {\mathbb{C}}^{m + 1} \) . This is the action of a principal \( {S}^{1} \) -bundle, \( {S}^{{2m} + 1} \rightarrow \mathbb{C}{P}^{m}\left( {§2\lef... | Yes |
Example 1 A space with \( {e}_{0}X = 2 \) and \( {\operatorname{cat}}_{0}X = \infty \) . | In Example 6, \( §{12}\left( \mathrm{\;d}\right) \), we constructed a minimal Sullivan model \( \left( {{\Lambda V}, d}\right) \) such that \( d : V \rightarrow {\Lambda }^{3}V \) and every cocycle in \( {\Lambda }^{ \geq 3}V \) is a coboundary. Moreover \( V = {V}^{ \geq 2} \) and has finite type and it is immediate f... | Yes |
Proposition 32.1 If \( \left( {{\Lambda V}, d}\right) \) is a pure Sullivan algebra then \( H\left( {{\Lambda V}, d}\right) \) is finite dimensional if and only if \( {H}_{0}\left( {{\Lambda V}, d}\right) \) is finite dimensional. | proof: Since \( {\Lambda V} \) is a finitely generated module over the (noetherian) polynomial algebra \( {\Lambda Q} \) any submodule is also finitely generated. Since \( d\left( {\Lambda Q}\right) = 0 \) , \( \ker d \) is a \( {\Lambda Q} \) -submodule of \( {\Lambda V} \) ; hence it is finitely generated. Thus \( H\... | No |
Proposition 32.4 Let \( \left( {{\Lambda V}, d}\right) \) be a minimal Sullivan algebra in which \( V \) is finite dimensional and \( V = {V}^{ \geq 2} \) . Then the following conditions are equivalent:\n\n(i) \( \dim H\left( {{\Lambda V},{d}_{\sigma }}\right) < \infty \) .\n\n(ii) \( \dim H\left( {{\Lambda V}, d}\righ... | proof: Since the odd spectral sequence converges from \( H\left( {{\Lambda V},{d}_{\sigma }}\right) \) to \( H\left( {{\Lambda V}, d}\right) \) the implication (i) \( \Rightarrow \) (ii) is immediate, while (ii) \( \Rightarrow \) (iii) follows from Corollary 1 to Proposition 29.3. To prove (iii) \( \Rightarrow \) (i) l... | Yes |
Example 1 \( \Lambda \left( {{a}_{2},{x}_{3},{u}_{3},{b}_{4},{v}_{5},{w}_{7};{da} = {dx} = 0,{du} = {a}^{2},{db} = {ax},{dv} = }\right. \) \( {ab} - {ux},{dw} = {b}^{2} - {vx}). \) | Here subscripts denote degrees. The differential \( {d}_{\sigma } \) is given by \( {d}_{\sigma }a = {d}_{\sigma }b = \) \( {d}_{\sigma }x = 0,{d}_{\sigma }u = {a}^{2},{d}_{\sigma }v = {ab},{d}_{\sigma }w = {b}^{2} \) . Thus in \( H\left( {{\Lambda V},{d}_{\sigma }}\right) \) we have \( {\left\lbrack a\right\rbrack }^{... | Yes |
Example 3 Algebraic closure of \( \\mathbb{k} \) is necessary in Proposition 32.3. | Indeed if \( \\mathbb{k} = \\mathbb{Q} \) the Sullivan algebra \( \\Lambda \\left( {{a}_{2},{b}_{2},{x}_{3};{dx} = {a}^{2} + {b}^{2}}\\right) \) admits no non-trivial morphism to \( \\mathbb{Q}\\left\\lbrack z\\right\\rbrack \), since we would have \( a \\mapsto {\\alpha z}, b \\mapsto {\\beta z} \) with \( \\alpha ,\\... | Yes |
A finite connected graph is n-colourable if each vertex can be assigned one of n distinct colours so that vertices connected by an edge have different colours. | Indeed we lose no generality in assuming \\(\\mathbb{R}\\) is algebraically closed. If the graph is \\(n\\) -colourable identify the colours with the distinct \\({n}^{\\text{th }}\\) roots of unity \\({w}_{\\alpha }\\) and note by \\({w}_{\\alpha \\left( j\\right) }\\) the colour of the vertex \\({v}_{j}\\). If \\({w}_... | Yes |
Theorem 32.6 (Friedlander-Halperin [61] Suppose \( \left( {{\Lambda V}, d}\right) \) is an elliptic Sullivan algebra with formal dimension \( n \) and even and odd exponents \( {a}_{1},\ldots ,{a}_{q} \) and \( {b}_{1},\ldots ,{b}_{p} \) . Then\n\n(i) \( \mathop{\sum }\limits_{{i = 1}}^{p}\left( {2{b}_{i} - 1}\right) -... | proof: We have \( n = \mathop{\sum }\limits_{{i = 1}}^{p}\left( {2{b}_{i} - 1}\right) - \mathop{\sum }\limits_{{j = 1}}^{q}\left( {2{a}_{j} - 1}\right) \geq \mathop{\sum }\limits_{{i = 1}}^{p}\left( {2{b}_{i} - 1}\right) + q \geq p + q,\n\nwhere \( p = \dim {V}^{\text{odd }} \) and \( q = \dim {V}^{\text{even }} \) . | Yes |
Corollary 2 If \( \left( {{\Lambda V}, d}\right) \) is an elliptic Sullivan algebra of formal dimension \( n \) then \( \dim V \leq n \) . | proof: We have \( n = \mathop{\sum }\limits_{{i = 1}}^{p}\left( {2{b}_{i} - 1}\right) - \mathop{\sum }\limits_{{j = 1}}^{q}\left( {2{a}_{j} - 1}\right) \geq \mathop{\sum }\limits_{{i = 1}}^{p}\left( {2{b}_{i} - 1}\right) + q \geq p + q, \n\nwhere \( p = \dim {V}^{\text{odd }} \) and \( q = \dim {V}^{\text{even }} \) . | Yes |
Lemma 32.7 \( \left( {{\Lambda V},{d}_{\sigma }}\right) \) and \( \left( {{\Lambda V}, d}\right) \) have the same formal dimension. | proof: We argue by induction on \( \dim V \) . Write \( \left( {{\Lambda V}, d}\right) \) as a relative Sullivan algebra \( \left( {{\Lambda v} \otimes {\Lambda W}, d}\right) \) in which \( V = \mathbb{R}v \oplus W \) and \( v \) is an element in \( V \) of minimal degree. The Mapping theorem 29.5 asserts that the quot... | Yes |
Lemma 32.11 If \( 2{a}_{1},\ldots ,2{a}_{q} \) are the degrees of a basis \( \left( {y}_{j}\right) \) of \( Q \) and if \( 2{b}_{1} - 1,\ldots ,2{b}_{p} - 1 \) are the degrees of a basis \( \left( {x}_{i}\right) \) of \( P \) then\n\n\[ \mathcal{U}\left( z\right) = \frac{\mathop{\prod }\limits_{{i = 1}}^{p}\left( {1 - ... | proof: Write \( {\Lambda Q} \otimes {\Lambda P} = {\Lambda V} \) . Clearly \( \mathcal{U} = {\mathcal{U}}_{\Lambda V} \) does not depend on the differential, and \( {\mathcal{U}}_{{\Lambda V} \otimes {\Lambda W}} = {\mathcal{U}}_{\Lambda V} \cdot {\mathcal{U}}_{\Lambda W} \) . Since \( {\Lambda V} = \Lambda {y}_{1} \ot... | Yes |
Example 1 Simply connected finite \( H \) -spaces are rationally elliptic. | If \( G \) is as in the title then its Sullivan model is an exterior algebra on a graded vector space \( {P}_{G} \) of finite dimension concentrated in odd degrees, and has zero differential (Example 3, \( §{12}\left( \mathrm{a}\right) \) ). The dimension of \( {P}_{G} \) is called the rank of \( G.▱ \) | No |
Example 2 Simply connected compact homogeneous spaces \( G/K \) are rationally elliptic. | Proposition 15.16 asserts that these spaces have a Sullivan model of the form \( \left( {\Lambda {V}_{{B}_{K}} \otimes \Lambda {P}_{G}, d}\right) \), where \( d = 0 \) in \( \Lambda {V}_{{B}_{K}},{V}_{{B}_{K}} \) is concentrated in even degrees, \( d\left( {P}_{G}\right) \subset \Lambda {V}_{{B}_{K}} \) and \( {P}_{G} ... | Yes |
Suppose an \( r \) -torus \( T = {S}^{1} \times \cdots \times {S}^{1} \) ( \( r \) factors) acts smoothly and freely on a simply connected compact smooth manifold \( M \) . Then the projection \( M \rightarrow M/T \) onto the orbit space is a smooth principal bundle. Hence there is a classifying map \( M/T \rightarrow ... | Now assume \( M \) is rationally elliptic. Since \( {BT} = \mathbb{C}{P}^{\infty } \times \cdots \times \mathbb{C}{P}^{\infty } \) its homotopy groups are concentrated in degree 2, and since \( M/T \) is compact its homology is finite dimensional. Thus \( M/T \) is rationally elliptic. It is immediate from the long exa... | Yes |
Proposition 33.8 Suppose \( {H}^{i}\left( {X;\mathbb{Q}}\right) = 0, i > {n}_{X} \) . Then the integers \( \dim {H}_{i}\left( {{\Omega X};\mathbb{Q}}\right) ,1 \leq i \leq 3\left( {{n}_{X} - 1}\right) \) determine whether \( X \) is rationally elliptic or rationally hyperbolic. | proof: Theorem 33.3 asserts that \( X \) is rationally elliptic if and only if \( {\pi }_{j}\left( X\right) \otimes \) \( \mathbb{Q} = 0,2{n}_{X} \leq j < 3{n}_{X} - 1 \) . This only requires the calculation of \( {r}_{i},2{n}_{X} - 1 \leq \) \( i \leq 3{n}_{X} - 3 \) . | No |
Proposition 33.10 The formal power series \( {P}_{\Omega X} \) and \( \sum {r}_{n}{z}^{n} \) have the same radius of convergence, \( R \) . Moreover\n\n(i) \( R = 1 \) if \( X \) is rationally elliptic and \( R < 1 \) if \( X \) is rationally hyperbolic.\n\n(ii) If \( X \) is rationally hyperbolic and if \( {H}^{i}\lef... | proof: Write \( \sum {a}_{n}{z}^{n} \ll \sum {b}_{n}{z}^{n} \) if \( {a}_{n} \leq {b}_{n} \) for all \( n \) . Since\n\n\[ \sum {r}_{n}{z}^{n} \ll \frac{\mathop{\prod }\limits_{n}{\left( 1 + {z}^{{2n} + 1}\right) }^{{r}_{{2n} + 1}}}{\mathop{\prod }\limits_{n}{\left( 1 - {z}^{2n}\right) }^{{r}_{2n}}} \ll {e}^{\frac{\mat... | Yes |
Example 2 \( X = {S}^{3} \vee {S}^{3} \) . | As in Example 1, \( {P}_{\Omega X} = \frac{1}{1 - 2{z}^{2}} \) and hence \( {H}_{ * }\left( {{\Omega X};\mathbb{Q}}\right) \) is concentrated in even degrees. Thus formula (33.7) becomes\n\n\[ \frac{1}{1 - 2{z}^{2}} = \frac{1}{\mathop{\prod }\limits_{n}{\left( 1 - {z}^{2n}\right) }^{{r}_{2n}}} \]\n\nwhere \( {r}_{2n} =... | Yes |
Lemma 34.1 If \( M \) or \( N \) is \( {UL} \) -free then \( M \otimes N \) is \( {UL} \) -free. | proof: Suppose \( N \) is \( {UL} \) -free on a basis \( {a}_{\alpha } \) . Since \( M \otimes N = {\bigoplus }_{\alpha }M \otimes \left( {{a}_{\alpha } \cdot {UL}}\right) \) it is sufficient to prove that \( M \otimes {UL} \) is \( {UL} \) -free. Let \( {F}_{p} \subset {UL} \) be the linear span of elements of the for... | Yes |
Example 1 \( {\operatorname{Tor}}_{1}^{UI}\left( {\mathbb{k},\mathbb{k}}\right) \cong s\left( {I/\left\lbrack {I, I}\right\rbrack }\right) . \) | Again let \( I \subset L \) be an ideal. Denote by \( \left\lbrack {I, I}\right\rbrack \) the ideal in \( L \) which is the linear span of the elements \( \left\lbrack {y, z}\right\rbrack, y, z \in I \) . If \( x \in I \) denote by \( \left( x\right) \) the image of \( {sx} \) in the suspension \( s\left( {I/\left\lbra... | Yes |
The representation of \( L/I \) in \( {\operatorname{Tor}}_{q}^{UI}\left( {\mathbb{R},\mathbb{R}}\right) \). | As in Example 1 the inclusion of \( {P}_{ * } \) in \( {Q}_{ * } \) is a quasi-isomorphism of \( {UI} \) -free resolutions of \( \mathbb{R} \), and in particular applying \( - { \otimes }_{UI}\mathbb{R} \) gives a quasi-isomorphism \( \left( {{\Lambda sI},\bar{d}}\right) \overset{ \simeq }{ \rightarrow }{Q}_{ * }{ \oti... | Yes |
Lemma 34.4 If \( S \) is a right L-module then \( \varphi \) restricts to an isomorphism\n\n\[ \n{\operatorname{Hom}}_{UL}\left( {M \otimes S, N}\right) \overset{ \cong }{ \rightarrow }{\operatorname{Hom}}_{{UL}/I}\left( {M,{\operatorname{Hom}}_{UI}\left( {S, N}\right) }\right) .\n\] | proof: It is immediate that \( \varphi \) is \( L \) -linear. Thus if \( f \in {\operatorname{Hom}}_{UL}\left( {M \otimes S, N}\right) \) we have \( f \cdot x = 0, x \in L \) and so \( \left( {\varphi f}\right) \cdot x = 0 \) . Thus \( {\varphi f} \) is \( L \) -linear. For \( x \in I \) and \( m \in M \) it follows th... | Yes |
(i) \( W \otimes {\operatorname{Ext}}_{UI}^{q}\left( {\mathbb{k},{UI}}\right) \) is a free \( {UL}/I \) -module on \( 1 \otimes {\operatorname{Ext}}_{UI}^{q}\left( {\mathbb{k},{UI}}\right) \) . | proof: Denote \( {\operatorname{Ext}}_{UI}^{q}\left( {\mathbb{k},{UI}}\right) \) by \( {E}^{q} \) . \n\n(i) An \( L/I \) -linear map \( \theta \) from the free \( L/I \) -module \( {E}^{q} \otimes {UL}/I \) to \( W \otimes {E}^{q} \) is given by \n\n\[ \n\theta \left( {\Phi \otimes a}\right) = \left( {1 \otimes \Phi }\... | Yes |
Proposition 34.9 With the hypotheses and notation above assume the Lie algebra \( L/I \) is finitely generated and that \( \alpha \cdot {UL}/I \) is finite dimensional for each \( \alpha \in {\operatorname{Tor}}_{q}^{UI}\left( {N,\mathbf{k}}\right) \) . Then there is an isomorphism of \( {UL}/I \) -modules\n\n\[{\opera... | proof of Proposition 34.9: Dualize the inclusion \( {UI} \rightarrow {UL} \) to a surjection \( {\left( UL\right) }^{\sharp } \rightarrow {\left( UI\right) }^{\sharp } \) of \( {UI} \) -modules. This induces a linear map\n\n\[f : {\operatorname{Tor}}_{q}^{UI}\left( {N,{\left( UL\right) }^{\sharp }}\right) \rightarrow {... | Yes |
Lemma 34.10 With the hypotheses of Proposition 34.9, \( \beta \cdot {UL}/I \) is finite dimensional for all \( \beta \in {\operatorname{Tor}}_{q}^{UI}\left( {N,{\left( UL\right) }^{\sharp }}\right) \) . | proof: Write \( M = {\left( UL\right) }^{\sharp } \) . Then \( M = {M}_{ < 0} \) is the union of the submodules \( {M}_{ \geq - p} \) . It is thus sufficient to prove the lemma for \( \beta \in {\operatorname{Tor}}_{q}^{UI}\left( {N,{M}_{ \geq - p}}\right) \) . Consider the exact sequence\n\n\[ \n{\operatorname{Tor}}_{... | Yes |
Lemma 35.1 Suppose \( {P}_{ * }\overset{ \simeq }{ \rightarrow }M \) is an \( A \) -projective resolution of an \( A \) -module \( M \) and suppose \( {Q}_{ * } = {\left\{ {Q}_{i}\right\} }_{0 \leq i \leq m} \) is a complex of free \( A \) -modules. Then\n\n\[ \n{H}_{i, * }\left( {{\operatorname{Hom}}_{A}\left( {{P}_{ ... | proof: Set \( {Q}_{ * }^{\prime } = {\left\{ {Q}_{i}\right\} }_{0 \leq i \leq m - 1} \) . Because the \( {P}_{i} \) are \( A \) -projective the sequence\n\n\[ \n0 \rightarrow {\operatorname{Hom}}_{A}\left( {{P}_{ * },{Q}_{ * }^{\prime }}\right) \rightarrow {\operatorname{Hom}}_{A}\left( {{P}_{ * },{Q}_{ * }}\right) \ri... | Yes |
Lemma 35.4 Let \( A \subset X \) be an inclusion of \( {\Omega Y} \) -spaces and give \( \left( {X, A}\right) \times {\Omega Y} \) the diagonal action, where \( {\Omega Y} \) acts by right multiplication on \( {\Omega Y} \) . If \( {H}_{ * }\left( {X, A}\right) \) is \( \mathbb{k} \) -free then \( {H}_{ * }\left( {\lef... | proof: If \( \gamma \in {\Omega Y} \) is a loop of length \( \ell \) let \( {\gamma }^{\prime } \) be the loop of length \( \ell \) given by \( {\gamma }^{\prime }\left( t\right) = \gamma \left( {\ell - t}\right) ,0 \leq t \leq \ell \) . Then \( \gamma \mapsto \gamma {\gamma }^{\prime } \) and \( \gamma \mapsto {\gamma... | Yes |
Proposition 35.8 The sequence (35.7) is an \( {H}_{ * }\left( {\Omega Y}\right) \) -free resolution of \( \mathbf{k} \) ; i.e., the Milnor resolution is an Eilenberg-Moore resolution (§20(d)). | proof: We need only show (35.7) is exact. Filter \( {C}_{ * }\left( {\left( \Omega Y\right) }^{*\infty }\right) \) by the submodules \( {C}_{ * }\left( {\left( \Omega Y\right) }^{*n}\right) \) . Then the quasi-isomorphism \( \varphi : V \otimes {C}_{ * }\left( {\Omega Y}\right) \rightarrow {C}_{ * }\left( {\left( \Omeg... | Yes |
Theorem 35.10 If \( \left( {X,{x}_{0}}\right) \) is a normal path connected topological space and if each \( {H}_{i}\left( {\Omega X}\right) \) is \( \mathbb{k} \) -free on a finite basis then\n\n\[ \n\text{depth}{H}_{ * }\left( {\Omega X}\right) \leq \operatorname{cat}X\text{.\n\]\n\nIf equality holds then also \( \op... | proof: Replace \( X \) by a well based space of the same homotopy type by adjoining an interval to \( X \) at the base point \( {x}_{0} \) . Then apply the Corollary above to Theorem 35.9 to \( f = i{d}_{X} \) . | No |
Proposition 35.11 Let \( L = {\left\{ {L}_{i}\right\} }_{i > 1} \) be a graded Lie algebra with each \( {L}_{i} \) finite dimensional.\n\n(i) \( \operatorname{gl}\dim {UL} \) is the largest integer \( n \) (as \( \infty \) ) such that \( {\operatorname{Ext}}_{UL}^{n}\left( {\mathbb{R},\mathbb{R}}\right) \neq 0 \) .\n\n... | proof: (i) Clearly \( n \leq \operatorname{gl}\dim {UL} \), since \( {\operatorname{Ext}}_{UL}^{n}\left( {\mathbb{k},\mathbb{k}}\right) \neq 0 \) . On the other hand, write \( {P}_{ * } = {C}_{ * }\left( L\right) \otimes {UL} \) . Then\n\n\[ \n{\operatorname{Ext}}_{UL}\left( {\mathbb{R},\mathbb{R}}\right) = H\left( {{\... | Yes |
Proposition 35.12 \( \operatorname{cat}\left( {{\Lambda V}, d}\right) \leq \operatorname{gldim}{UL} \) . | proof: Let \( n = \operatorname{gl}\dim {UL} \) ; according to Proposition 35.11 it is the largest integer such that \( {\operatorname{Ext}}_{UL}^{n}\left( {\mathbb{R},\mathbb{R}}\right) \neq 0 \) . Define an ideal \( I \subset {\Lambda V} \) by setting \( I = \) \( {\Lambda }^{ > n}V \oplus {I}^{n} \), where \( {I}^{n... | Yes |
Proposition 36.2\n\n(i) If \( L \) is the direct sum of ideals \( I \) and \( J \) then\n\n\[ \text{depth}L = \operatorname{depth}I + \operatorname{depth}J.\]\n\n(ii) If \( L \) is the infinite direct sum of non-zero ideals then \( \operatorname{depth}L = \infty \). | proof: \( \; \) (i) Because of (36.1) we may identify \( {\operatorname{Ext}}_{UL}^{p}\left( {\mathbb{R},{UL}}\right) \) with \( {\operatorname{Tor}}_{p}^{UL}{\left( \mathbb{R}, U{L}^{\sharp }\right) }^{\sharp } \), \( p \geq 0 \) (Lemma 34.3(iii)). Thus depth \( L \) is the least integer \( m \) such that \( {\operato... | Yes |
Proposition 36.3 Let \( I \subset L \) be an ideal.\n\n(i) \( \operatorname{depth}I \leq \operatorname{depth}L \) . | proof: (i) The Hochschild-Serre spectral sequence (§34(b)) converges from \( {E}_{2}^{p, q} = {\operatorname{Ext}}_{{UL}/I}^{p}\left( {\mathbb{k},{\operatorname{Ext}}_{UI}^{q}\left( {\mathbb{k},{UL}}\right) }\right) \) to \( {\operatorname{Ext}}_{UL}^{p + q}\left( {\mathbb{k},{UL}}\right) \) . Since \( {UL} \) is \( {U... | Yes |
Theorem 36.4 The graded Lie algebra \( L \) is solvable and of finite depth if and only if \( L \) is finite dimensional. In this case \( {\operatorname{Ext}}_{UL}^{ * }\left( {\mathbb{k},{UL}}\right) \) is one dimensional, and \[ \text{depth}L = \dim {L}_{\text{even }}\text{.} \] | proof: Suppose \( L \) is solvable and has finite depth. The ideal \( \left\lbrack {L, L}\right\rbrack \) has finite depth (Proposition 36.3(i)), and so by induction on solvlength, \( \left\lbrack {L, L}\right\rbrack \) is finite dimensional. In particular, for some \( k,{L}_{ \geq k} \) is an abelian ideal, also of fi... | Yes |
Theorem 36.5 [54] If L satisfying (36.1) has finite depth then its radical, \( R \) , is finite dimensional and \( \dim {R}_{\text{even }} \leq \operatorname{depth}L \) . | proof: Every solvable ideal \( I \subset L \) satisfies \( \dim {I}_{\text{even }} \leq \operatorname{depth}L \), by Theorem 36.4. Choose \( I \) so that \( \dim {I}_{\text{even }} \) is maximized. For any solvable ideal \( J, I + J \) is solvable, and hence \( {J}_{\text{even }} \subset {I}_{\text{even }} \) . It foll... | Yes |
Theorem 36.8 A graded Lie algebra L satisfying (36.1) and of depth \( m \) contains at most \( m \) linearly independent Engel elements of even degree. | proof: We show that if \( L = I \oplus \mathbb{R}x \), with \( I \) an ideal and \( x \) a non-zero Engel element of even degree, then \( \operatorname{depth}I < \operatorname{depth}L \) . (The theorem follows from this by an obvious argument.)\n\nTo establish this assertion note first that since ad \( x \) is locally ... | Yes |
A graded Lie algebra \( L \) has depth 0 if and only if \( {\operatorname{Hom}}_{UL}\left( {\mathbb{R},{UL}}\right) \neq 0 \) ; i.e., if and only if \( a \cdot U{L}_{ + } = 0 \) for some non-zero \( a \in {UL} \). | It follows at once from the Poincaré Birkoff Witt theorem 21.1 that this occurs if and only if \( L \) is finite dimensional and concentrated in odd degrees. | No |
Example 2 Free products have depth 1. | Let \( E \) and \( L \) be graded Lie algebras and recall the free product \( E \coprod L \) defined in \( §{21}\left( \mathrm{c}\right) \) . We shall show that \( \operatorname{depth}E \coprod L = 1 \) . Indeed, choose free resolutions of the form \[ \overset{d}{ \rightarrow }V\left( 2\right) \otimes {UE}\overset{d}{ ... | Yes |
Example 3 \( X \vee Y \) . | Let \( X \) and \( Y \) be simply connected spaces with rational homotopy of finite type. In Example 2 of \( §{24}\left( \mathrm{f}\right) \) we observed that \( {L}_{X \vee Y} = {L}_{X} \coprod {L}_{Y} \) . Thus depth \( {L}_{X \vee Y} = \) 1. On the other hand, \( {\operatorname{cat}}_{0}\left( {X \vee Y}\right) = \m... | Yes |
If \( E \) and \( L \) are graded Lie algebras then\n\n\[ \operatorname{depth}\left( {E \oplus L}\right) = \operatorname{depth}E + \operatorname{depth}L \] | as observed in Proposition 36.2. Since \( {L}_{X \times Y} = {L}_{X} \oplus {L}_{Y} \) we have that depth \( {L}_{X \times Y} \) \( = \operatorname{depth}{L}_{X} + \operatorname{depth}{L}_{Y} \) in analogy with \( {\operatorname{cat}}_{0}\left( {X \times Y}\right) = {\operatorname{cat}}_{0}X + {\operatorname{cat}}_{0}Y... | Yes |
Example 5 \( X = {S}_{a}^{3} \vee {S}_{b}^{3}{ \cup }_{{\left\lbrack a{\left\lbrack a, b\right\rbrack }_{W}\right\rbrack }_{W}}{D}^{8} \) . | This CW complex was first discussed in Example 2, \( §{13}\left( \mathrm{\;d}\right) \) and subsequently in Example 4, \( §{24}\left( \mathrm{f}\right) \) and in Example 3, \( §{33}\left( \mathrm{c}\right) \) . In \( §{33}\left( \mathrm{c}\right) \) we showed that the homotopy fibre of the retraction \( X \rightarrow {... | Yes |
Example 6 \( \mathbb{C}{P}^{\infty }/\mathbb{C}{P}^{n} \) . | Because the inclusion \( i : \mathbb{C}{P}^{n} \rightarrow \mathbb{C}{P}^{\infty } \) induces the surjection \( {\Lambda x} \rightarrow {\Lambda x}/{x}^{n + 1} \) in cohomology \( \left( {\deg x = 2}\right) \), the cohomology algebra of \( \mathbb{C}{P}^{\infty }/\mathbb{C}{P}^{n} \) is just \( \mathbb{Q} \oplus \) \( ... | Yes |
Proposition 37.2 ([S-T]) With the notation above,\n\n\[ \n{\pi }_{ * }\left( p\right) \gamma = {\left\lbrack {\pi }_{ * }\left( p\right) \beta ,\alpha \right\rbrack }_{W}\;\text{ and }\;\operatorname{hur}\gamma = \operatorname{hur}\beta \cdot \operatorname{hur}{\partial }_{ * }\alpha .\n\] | proof: The first assertion is immediate from the definition of the Whitehead product \( \left( {§{13}\left( \mathrm{e}\right) }\right) \) . For the second, observe that \( c \) factors over the surjection \( \partial \left( {{D}^{m} \times {D}^{n + 1}}\right) \rightarrow \left( {{S}^{m} \times {S}^{n}}\right) \cup {D}^... | Yes |
Proposition 37.8 (Anick [6]) With the notation preceding Proposition 37.6,\n\n\[ U{L}_{Y}{\left( z\right) }^{-1} = \left( {1 + z}\right) {UL}{\left( z\right) }^{-1} - \left( {z - {H}_{ + }\left( Z\right) \left( z\right) + {H}_{ + }\left( X\right) \left( z\right) }\right) . \] | proof of Proposition 37.8: If \( {C}_{0} \leftarrow {C}_{1} \leftarrow \cdots \leftarrow {C}_{n} \) is a finite dimensional chain complex then \( \sum {\left( -1\right) }^{p}\dim {H}_{p}\left( C\right) = \sum {\left( -1\right) }^{p}\dim {C}_{p} \), as follows by a trivial calculation. Hence if \( {C}_{0, * } \leftarrow... | Yes |
Proposition 37.9 With the notation above \( \left( {{s}^{2} = }\right. \) double suspension), (i) \( {sV} \cong {\operatorname{Tor}}_{1}^{UL}\left( {\mathbb{Q},\mathbb{Q}}\right) \) . (ii) \( {s}^{2}R \cong {\operatorname{Tor}}_{2}^{UL}\left( {\mathbb{Q},\mathbb{Q}}\right) \) . | proof: (i) Indeed \( V \cong {\mathbb{L}}_{V}/\left\lbrack {{\mathbb{L}}_{V},{\mathbb{L}}_{V}}\right\rbrack = L/\left\lbrack {L, L}\right\rbrack \) and so \( {sV} \cong {\operatorname{Tor}}_{1}^{UL}\left( {\mathbb{Q},\mathbb{Q}}\right) \) by Example 1, \( §{34}\left( \mathrm{a}\right) \) . (ii) Consider the Hochschild-... | Yes |
Proposition 38.3 If \( \left( {{\Lambda V}, d}\right) \) is an elliptic Sullivan algebra (introduction to §32) then \( H\left( {{\Lambda V}, d}\right) \) is a Poincaré duality algebra. Its formal dimension is \( n = \) \( \mathop{\sum }\limits_{i}\deg {x}_{i} - \mathop{\sum }\limits_{j}\left( {\deg {y}_{j} - 1}\right) ... | proof: Recall the odd spectral sequence defined in \( §{32}\left( \mathrm{\;b}\right) \) . Its first term is \( \left( {{\Lambda V},{d}_{\sigma }}\right) \) with \( {d}_{\sigma }\left( {V}^{\text{even }}\right) = 0 \) and \( {d}_{\sigma }\left( {V}^{\text{odd }}\right) \subset {V}^{\text{even }} \) . Proposition 32.4 a... | Yes |
Theorem 38.4 Let \( X \) be a simply connected topological space and let \( \left( {{\Lambda V}, d}\right) \) be a minimal simply connected Sullivan algebra such that \( H\left( X\right) \) and \( H\left( {{\Lambda V}, d}\right) \) are Poincaré duality algebras. Then \[ {\mathrm{e}}_{0}\left( X\right) = {\operatorname{... | proof of Theorem 38.4: Choose \( z \in {\left( \Lambda V\right) }^{\sharp } \) so that \( {d}^{\sharp }z = 0 \) and \( z \) represents a fundamental class of \( \left( {{\Lambda V}, d}\right) \) . Define \( \theta : \left( {{\Lambda V}, d}\right) \rightarrow \left( {{\left( \Lambda V\right) }^{\sharp },{d}^{\sharp }}\r... | Yes |
Theorem 2.1.1 Suppose \( {\left\{ {a}_{n}\right\} }_{n = 1}^{\infty } \) is a sequence of complex numbers and \( f\left( t\right) \) is a continuously differentiable function on \( \left\lbrack {1, x}\right\rbrack \) . Set\n\n\[ \nA\left( t\right) = \mathop{\sum }\limits_{{n \leq t}}{a}_{n} \n\]\n\nThen\n\n\[ \n\mathop... | Proof. First, suppose \( x \) is a natural number. We write the left-hand side as\n\n\[ \n\mathop{\sum }\limits_{{n \leq x}}{a}_{n}f\left( n\right) = \mathop{\sum }\limits_{{n \leq x}}\{ A\left( n\right) - A\left( {n - 1}\right) \} f\left( n\right) \n\]\n\n\[ \n= \mathop{\sum }\limits_{{n \leq x}}A\left( n\right) f\lef... | Yes |
Theorem 2.1.9 (Euler-Maclaurin summation formula) Let \( k \) be a nonnegative integer and \( f \) be \( \left( {k + 1}\right) \) times differentiable on \( \left\lbrack {a, b}\right\rbrack \) with \( a, b \in \mathbb{Z} \) . Then\n\n\[ \mathop{\sum }\limits_{{a < n \leq b}}f\left( n\right) = {\int }_{a}^{b}f\left( t\r... | Example 2.1.10 For integers \( x \geq 1 \) , \n\n\[ \mathop{\sum }\limits_{{n \leq x}}\frac{1}{n} = \log x + \gamma + \frac{1}{2x} + \frac{1}{{12}{x}^{2}} + O\left( \frac{1}{{x}^{3}}\right) . \]\n\nSolution. Put \( f\left( t\right) = 1/t \) in Theorem 2.1.9, \( a = 1 \) , \( b = x \), and \( k = 2 \) . Then\n\n\[ \math... | Yes |
For integers \( x \geq 1 \) , \n\n\[ \mathop{\sum }\limits_{{n \leq x}}\frac{1}{n} = \log x + \gamma + \frac{1}{2x} + \frac{1}{{12}{x}^{2}} + O\left( \frac{1}{{x}^{3}}\right) . \] | Solution. Put \( f\left( t\right) = 1/t \) in Theorem 2.1.9, \( a = 1 \) , \( b = x \), and \( k = 2 \) . Then \n\n\[ \mathop{\sum }\limits_{{2 \leq n \leq x}}\frac{1}{n} = \log x + \frac{1}{2}\left( {\frac{1}{x} - 1}\right) + \frac{1}{12}\left( {\frac{1}{{x}^{2}} - 1}\right) - {\int }_{1}^{x}\frac{{B}_{3}\left( t\righ... | Yes |
Theorem 2.4.1 For any \( y > 0 \) ,\n\n\[ \mathop{\sum }\limits_{{n \leq x}}f\left( n\right) = \mathop{\sum }\limits_{{d \leq y}}g\left( d\right) H\left( \frac{x}{d}\right) + \mathop{\sum }\limits_{{d \leq \frac{x}{y}}}h\left( d\right) G\left( \frac{x}{d}\right) - G\left( y\right) H\left( \frac{x}{y}\right) . \] | Proof. We have\n\n\[ \mathop{\sum }\limits_{{n \leq x}}f\left( n\right) = \mathop{\sum }\limits_{{{de} \leq x}}g\left( d\right) h\left( e\right) \]\n\n\[ = \mathop{\sum }\limits_{{{de} \leq x}}g\left( d\right) h\left( e\right) + \mathop{\sum }\limits_{{{de} \leq x}}g\left( d\right) h\left( e\right) \]\n\n\[ = \mathop{\... | Yes |
Theorem 3.1.9 (Bertrand’s postulate) For \( n \) sufficiently large, there is a prime between \( n \) and \( {2n} \) . | Proof: (S. Ramanujan) Observe that if\n\n\[ \n{a}_{0} \geq {a}_{1} \geq {a}_{2} \geq \cdots \n\] \n\nis a decreasing sequence of real numbers tending to zero, then\n\n\[ \n{a}_{0} - {a}_{1} \leq \mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( -1\right) }^{n}{a}_{n} \leq {a}_{0} - {a}_{1} + {a}_{2} \n\] \n\nThis is the... | No |
Theorem 4.1.4 Let \( \delta \left( x\right) \) be defined as above. Let\n\n\[ I\left( {x, R}\right) = \frac{1}{2\pi i}{\int }_{c - {iR}}^{c + {iR}}\frac{{x}^{s}}{s}{ds}. \]\n\nThen, for \( x > 0, c > 0, R > 0 \), we have\n\n\[ \left| {I\left( {x, R}\right) - \delta \left( x\right) }\right| < \left\{ \begin{array}{ll} {... | Proof. Suppose first \( 0 < x < 1 \) . Consider the rectangular contour \( {K}_{U} \) oriented counterclockwise with vertices \( c - {iR}, c + {iR}, U + {iR} \) , \( U - {iR}, U > 0 \) . By Cauchy’s theorem\n\n\[ \frac{1}{2\pi i}{\int }_{{K}_{U}}\frac{{x}^{s}}{s}{ds} = 0 = \delta \left( x\right) \]\n\nTo prove the theo... | Yes |
Theorem 4.2.7 Let \( s = \sigma + {it} \). There are positive constants \( {c}_{1} \) and \( {c}_{2} \) such that\n\n\[ 1 - \frac{{c}_{1}}{{\left( \log T\right) }^{9}} \leq \sigma \leq 2 \]\n\n\[ \left| {\zeta \left( s\right) }\right| > \frac{{c}_{2}}{{\left( \log T\right) }^{7}} \]\n\nwhere \( 1 \leq \left| {\operator... | Proof. In Exercise 3.2.5, we proved\n\n\[ \left| {\zeta {\left( \sigma \right) }^{3}\zeta {\left( \sigma + it\right) }^{4}\zeta \left( {\sigma + {2it}}\right) }\right| \geq 1 \]\n\nfor \( \sigma > 1 \). Thus,\n\n\[ {\left| \zeta \left( \sigma + it\right) \right| }^{4} \geq {\left| \zeta \left( \sigma + 2it\right) \righ... | No |
Theorem 5.1.3 (Poisson summation formula) Let \( F \in {L}^{1}\left( \mathbb{R}\right) \) . Suppose that the series\n\n\[ \mathop{\sum }\limits_{{n \in \mathbb{Z}}}F\left( {n + v}\right) \]\n\nconverges absolutely and uniformly in \( v \), and that\n\n\[ \mathop{\sum }\limits_{{m \in \mathbb{Z}}}\left| {\widehat{F}\lef... | Proof. The function\n\n\[ G\left( v\right) = \mathop{\sum }\limits_{{n \in \mathbb{Z}}}F\left( {n + v}\right) \]\n\nis a continuous function of \( v \) of period 1 . The Fourier coefficients of \( G \) are given by\n\n\[ {c}_{m} = {\int }_{0}^{1}G\left( v\right) {e}^{-{2\pi imv}}{dv} \]\n\n\[ = \mathop{\sum }\limits_{{... | Yes |
Corollary 5.1.4 With \( F \) as above,\n\n\[ \mathop{\sum }\limits_{{n \in \mathbb{Z}}}F\left( n\right) = \mathop{\sum }\limits_{{n \in \mathbb{Z}}}\widehat{F}\left( n\right) \] | Proof. Set \( v = 0 \) in the theorem. | No |
Example 5.3.1 If \( \\left( {n, q}\\right) = 1 \\), then\n\n\\[ \n\\chi \\left( n\\right) \\tau \\left( \\bar{\\chi }\\right) = \\mathop{\\sum }\\limits_{{m = 1}}^{q}\\bar{\\chi }\\left( m\\right) e\\left( \\frac{mn}{q}\\right) .\n\\] | Solution. We have\n\n\\[ \n\\chi \\left( n\\right) \\tau \\left( \\bar{\\chi }\\right) = \\mathop{\\sum }\\limits_{{m = 1}}^{q}\\bar{\\chi }\\left( m\\right) \\chi \\left( n\\right) e\\left( \\frac{m}{q}\\right)\n\\]\n\n\\[ \n= \\mathop{\\sum }\\limits_{{h = 1}}^{q}\\bar{\\chi }\\left( h\\right) e\\left( \\frac{nh}{q}\... | Yes |
Theorem 5.3.3 If \( \chi \) is a primitive character \( \left( {\;\operatorname{mod}\;q}\right) \), then \( \left| {\tau \left( \chi \right) }\right| = {q}^{1/2} \) . | Proof. By Exercise 5.3.2,\n\n\[ \chi \left( n\right) \tau \left( \bar{\chi }\right) = \mathop{\sum }\limits_{{m = 1}}^{q}\bar{\chi }\left( m\right) e\left( \frac{mn}{q}\right) . \]\n\nThus\n\n\[ {\left| \chi \left( n\right) \right| }^{2}{\left| \tau \left( \chi \right) \right| }^{2} = \mathop{\sum }\limits_{{{m}_{1} = ... | No |
Show that an entire function \( f\left( z\right) \) of finite order \( \beta \) without any zeros must be of the form \( f\left( z\right) = {e}^{g\left( z\right) } \), where \( g\left( z\right) \) is a polynomial and \( \beta = \deg g \) . | ## Solution.\n\nLet \( h\left( z\right) = \log f\left( z\right) - \log f\left( 0\right) \) . Then \( h\left( z\right) \) is entire, since \( f\left( z\right) \) has no zeros. Also, for any \( \epsilon > 0 \) ,\n\n\[ \n\operatorname{Re}h\left( z\right) = \log \left| {f\left( z\right) }\right| \ll {R}^{\beta + \epsilon }... | Yes |
Theorem 6.1.2 (Jensen’s theorem) Let \( f\left( z\right) \) be an entire function of order \( \beta \) such that \( f\left( 0\right) \neq 0 \) . If \( {z}_{1},{z}_{2},\ldots ,{z}_{n} \) are the zeros of \( f\left( z\right) \) in \( \left| z\right| < R \), counted with multiplicity, then | Proof. We may assume, without loss of generality, that \( f\left( 0\right) = 1 \) . Also, it is clear that if the theorem is true for functions \( g \) and \( h \), that it is also true for the product \( {gh} \) . Thus, it suffices to prove it for functions with either no zero or one zero in \( \left| z\right| < R \) ... | Yes |
Corollary 6.1.3 Let \( f \) be as in Theorem 6.1.2. Then\n\n\[ \log \left( \frac{{R}^{n}}{\left| {z}_{1}\right| \cdots \left| {z}_{n}\right| }\right) \leq \mathop{\max }\limits_{{\left| z\right| = R}}\log \left| {f\left( z\right) }\right| - \log \left| {f\left( 0\right) }\right| .\n\] | Proof. This is clear from Jensen's theorem. | No |
Theorem 6.5.6 There exists a constant \( c > 0 \) such that \( \zeta \left( s\right) \) has no zero in the region\n\n\[ \sigma \geq 1 - \frac{c}{\log \left| t\right| },\;\left| t\right| \geq 2 \] | Proof. By Exercise 6.5.5,\n\n\[ - \operatorname{Re}\left( \frac{{\zeta }^{\prime }\left( {\sigma + {it}}\right) }{\zeta \left( {\sigma + {it}}\right) }\right) < {A}_{1}\log \left| t\right| - \frac{1}{\sigma - \beta }.\]\n\nWe also know, by Exercises 6.5.2 and 6.5.4, that\n\n\[ - \frac{{\zeta }^{\prime }\left( \sigma \r... | Yes |
Corollary 6.5.7 There exists a constant \( c > 0 \) such that \( \zeta \left( s\right) \) has no zero in the region\n\n\[ \sigma \geq 1 - \frac{c}{\log \left( {\left| t\right| + 2}\right) } \] | Proof. The region \( \sigma \geq 1,\left| t\right| \leq 2 \) contains no zeros of \( \zeta \left( s\right) \) . Thus, there must be a constant \( {c}_{1} > 0 \) such that \( \zeta \left( s\right) \) has no zeros in \( \sigma \geq \) \( 1 - {c}_{1} \) and \( \left| t\right| \leq 2 \) . Combining such a region with the z... | No |
Theorem 6.5.12 There exists a positive absolute constant \( c \) such that if \( 0 < \delta < c \), then \( L\left( {s,\chi }\right) \) has no zeros in the region\n\n\[ \delta > 1 - \frac{c}{\log q\left( {\left| t\right| + 2}\right) } \]\n\nexcept possibly if \( \chi \) is real and nonprincipal, in which case there is ... | Proof. We need only consider the case where \( \chi \) is real and nonprin-cipal and \( \left| \gamma \right| < \delta /\log q \) . First suppose there are two complex zeros in the region. We have\n\n\[ - \frac{{L}^{\prime }\left( {\sigma ,\chi }\right) }{L\left( {\sigma ,\chi }\right) } < {c}_{1}\log q - \mathop{\sum ... | Yes |
Theorem 7.1.7 Let \( N\left( T\right) \) be the number of zeros of \( \zeta \left( s\right) \) in the rectangle \( 0 < \sigma < 1,0 < t < T \) . Then\n\n\[ N\left( T\right) = \frac{T}{2\pi }\log \frac{T}{2\pi } - \frac{T}{2\pi } + \frac{7}{8} + S\left( T\right) + O\left( \frac{1}{T}\right) ,\] \n\nwhere \n\n\[ {\pi S}\... | Proof. Let \( R \) be the rectangle with vertices \( 2,2 + {iT}, - 1 + {iT} \), and -1 , traversed in the counterclockwise direction. Then\n\n\[ {2\pi N}\left( T\right) = {\Delta }_{R}\arg \xi \left( s\right) \]\n\nThere is no change in the argument as \( s \) goes from -1 to 2 . Also, the change when \( s \) moves fro... | Yes |
Theorem 7.2.8 For some constant \( {c}_{1} > 0 \) ,\n\n\[ \psi \left( x\right) = x + O\left( {x\exp \left( {-{c}_{1}\sqrt{\log x}}\right) }\right) \] | Proof. By the solution to Exercise 7.2.7, we know that\n\n\[ \psi \left( x\right) = x - \mathop{\sum }\limits_{{\left| \rho \right| < R}}\frac{{x}^{\rho }}{\rho } - \frac{{\zeta }^{\prime }\left( 0\right) }{\zeta \left( 0\right) } + \frac{1}{2}\log \left( {1 - {x}^{-2}}\right) + O\left( {\frac{x{\log }^{2}x}{R} + \frac... | Yes |
Theorem 7.3.2 (Weil’s explicit formula) Assume that \( h\left( s\right) \) satisfies the conditions of Lemma 7.3.1. In addition, assume that \( h\left( {it}\right) = {h}_{0}\left( {t/{2\pi }}\right) \) is a real-valued function for \( t \in \mathbb{R} \) whose Fourier transform\n\n\[ \n{\widehat{h}}_{0}\left( y\right) ... | Proof. Recall that\n\n\[ \n\frac{{\xi }^{\prime }\left( {\frac{1}{2} + s}\right) }{\xi \left( {\frac{1}{2} + s}\right) } \n\]\n\n\[ \n= \frac{1}{s + 1/2} + \frac{1}{s - 1/2} - \frac{1}{2}\log \pi + \frac{{\Gamma }^{\prime }\left( {1/4 + s/2}\right) }{\Gamma \left( {1/4 + s/2}\right) } - \mathop{\sum }\limits_{{n = 1}}^... | Yes |
Theorem 8.1.3 (Phragmén - Lindelöf) Suppose that \( f\left( s\right) \) is entire in the region\n\n\[ \nS\left( {a, b}\right) = \{ s \in \mathbb{C} : a \leq \operatorname{Re}\left( s\right) \leq b\} \]\n\nand that as \( \left| t\right| \rightarrow \infty \),\n\n\[ \n\left| {f\left( s\right) }\right| = O\left( {e}^{{\le... | Proof. We first select an integer \( m > \alpha, m \equiv 2\left( {\;\operatorname{mod}\;4}\right) \) . Since arg \( s \rightarrow \) \( \pi /2 \) as \( t \rightarrow \infty \), we can choose \( {T}_{1} \) sufficiently large so that\n\n\[ \n\left| {\arg s - \pi /2}\right| < \pi /{4m} \]\n\nThen for \( \left| {\operator... | Yes |
Corollary 8.1.4 Suppose that \( f\left( s\right) \) is entire in \( S\left( {a, b}\right) \) and that \( \left| {f\left( s\right) }\right| \) \( = O\left( {e}^{{\left| t\right| }^{\alpha }}\right) \) for some \( \alpha \geq 1 \) as \( \left| t\right| \rightarrow \infty \) . If \( f\left( s\right) \) is \( O\left( {\lef... | Proof. We apply the theorem to the function \( g\left( s\right) = f\left( s\right) /{\left( s - u\right) }^{A} \) , where \( u > b \) . Then \( g \) is bounded on the two vertical strips, and the result follows. | Yes |
Theorem 8.2.1 (Selberg) For any \( F \in \mathcal{S} \), let \( {N}_{F}\left( T\right) \) be the number of zeros \( \rho \) of \( F\left( s\right) \) satisfying \( 0 \leq \operatorname{Im}\left( \rho \right) \leq T \), counted with multiplicity. Then\n\n\[ \n{N}_{F}\left( T\right) \sim \left( {2\mathop{\sum }\limits_{{... | Proof. This is easily derived by the method used to count zeros of \( \zeta \left( s\right) \) and \( L\left( {s,\chi }\right) \) as in Theorem 7.1.7 and Exercise 7.4.4. | No |
Lemma 8.2.2 (Conrey and Ghosh) If \( F \in S \) and \( \deg F = 0 \), then \( F = 1 \) . | Proof. We follow [CG]. A Dirichlet series can be viewed as a power series in the infinitely many variables \( {p}^{-s} \) as we range over primes \( p \) . Thus, if \( \deg F = 0 \), we can write our functional equation as\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{\left( \frac{{Q}^{2}}{n}\right) }^{s} = {w... | Yes |
Theorem 8.2.3 (Selberg) If \( F \in \mathcal{S} \) and \( F \) is of positive degree, then \( \deg F \geq 1 \) . | Proof. We follow [CG]. Consider the identity\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{e}^{-{nx}} = \frac{1}{2\pi i}{\int }_{\left( 2\right) }F\left( s\right) {x}^{-s}\Gamma \left( s\right) {ds}. \]\n\nBecause of the Phragmen - Lindelöf principle and the functional equation, we find that \( F\left( s\right... | Yes |
Example 9.1.1 (Eratosthenes-Legendre) Let \( {P}_{z} \) be the product of the primes \( p \leq z \), and \( \pi \left( {x, z}\right) \) the number of \( n \leq x \) that are not divisible by any prime \( p \leq z \). Then\n\n\[ \pi \left( {x, z}\right) = \mathop{\sum }\limits_{{d \mid {P}_{z}}}\mu \left( d\right) \left... | Solution. Clearly,\n\n\[ \pi \left( {x, z}\right) = \mathop{\sum }\limits_{{n \leq x}}\mathop{\sum }\limits_{{d \mid \left( {n,{P}_{z}}\right) }}\mu \left( d\right) \]\n\n\[ = \mathop{\sum }\limits_{{d \mid {P}_{z}}}\mu \left( d\right) \mathop{\sum }\limits_{\substack{{n \leq x} \\ {d \mid n} }}1 = \mathop{\sum }\limit... | Yes |
Theorem 10.1.5 (Ostrowski) Every nontrivial norm \( \parallel \cdot \parallel \) on \( \mathbb{Q} \) is equivalent to \( {\left| \cdot \right| }_{p} \) for some prime \( {\left. p\text{or}\left| \cdot \right| \right| }_{\infty } \) . | Proof. Case (i): Suppose there is a natural number \( n \) such that \( \left| \right| n\left| \right| > \) 1. Let \( {n}_{0} \) be the least such \( n \) . We know that \( {n}_{0} > 1 \), so we can write \( \begin{Vmatrix}{n}_{0}\end{Vmatrix} = {n}_{0}^{\alpha } \) for some positive \( \alpha \) . Write any natural nu... | Yes |
Theorem 10.1.8 \( {\mathbb{Q}}_{p} \) is complete with respect to \( {\left| \cdot \right| }_{p} \) . | Proof. Let \( {\left\{ {a}^{\left( j\right) }\right\} }_{j = 1}^{\infty } \) be a Cauchy sequence of equivalence classes in \( {\mathbb{Q}}_{p} \) . We must show that there is a Cauchy sequence to which it converges. We write \( {a}^{\left( j\right) } = {\left\{ {a}_{n}^{\left( j\right) }\right\} }_{n = 1}^{\infty } \)... | No |
Theorem 10.1.11 Every equivalence class \( s \) in \( {\mathbb{Q}}_{p} \) for which \( {\left| s\right| }_{p} \leq 1 \) has exactly one representative Cauchy sequence \( {\left\{ {a}_{i}\right\} }_{i = 1}^{\infty } \) satisfying \( 0 \leq {a}_{i} < {p}^{i} \) and \( {a}_{i} \equiv {a}_{i + 1}\left( {\;\operatorname{mod... | Proof. The uniqueness is clear, for if \( {\left\{ {a}_{i}^{\prime }\right\} }_{i = 1}^{\infty } \) is another such sequence, we have \( {a}_{i} \equiv {a}_{i}^{\prime }\left( {\;\operatorname{mod}\;{p}^{i}}\right) \), which forces \( {a}_{i} = {a}_{i}^{\prime } \) . Now let \( {\left\{ {c}_{i}\right\} }_{i = 1}^{\inft... | Yes |
Show that \( {x}^{2} = 6 \) has a solution in \( {\mathbb{Q}}_{5} \) . | The equation \( {x}^{2} \equiv 6\left( {\;\operatorname{mod}\;5}\right) \) has a solution (namely \( x \equiv \) 1 (mod 5)). We will show inductively that \( {x}^{2} \equiv 6\left( {\;\operatorname{mod}\;{5}^{n}}\right) \) has a solution for every \( n \geq 1 \) . Suppose\n\n\[ \n{x}_{n}^{2} \equiv 6\left( {\;\operator... | Yes |
Theorem 10.2.8 \( {\left| \cdot \right| }_{p} \) is a nonarchimedean norm on \( K \) . | Proof. It is clear that \( {\left| x\right| }_{p} = 0 \) if and only if \( x = 0 \) . It is also clear that \( {\left| xy\right| }_{p} = {\left| x\right| }_{p}{\left| y\right| }_{p} \), since the norm is multiplicative. To prove that\n\n\[ \n{\left| x + y\right| }_{p} \leq \max \left( {{\left| x\right| }_{p},{\left| y\... | No |
Theorem 10.3.12 (Mahler,1961) Suppose \( f : {\mathbb{Z}}_{p} \rightarrow {\mathbb{Q}}_{p} \) is continuous. Let\n\n\[ \n{a}_{n}\left( f\right) = \mathop{\sum }\limits_{{k = 0}}^{n}{\left( -1\right) }^{n - k}\left( \begin{array}{l} n \\ k \end{array}\right) f\left( k\right) .\n\]\n\nThen the series\n\n\[ \n\mathop{\sum... | Proof. We know that given any positive integer \( s \), there exists a positive integer \( t \) such that for \( x, y \in {\mathbb{Z}}_{p} \), \n\n\[ \n{\left| x - y\right| }_{p} \leq {p}^{-t} \Rightarrow {\left| f\left( x\right) - f\left( y\right) \right| }_{p} \leq {p}^{-s}.\n\]\n\nIn particular,\n\n\[ \n{\left| f\le... | Yes |
Theorem 10.4.7 (Mazur, 1972)\n\n\\[ \n- \\left( {1 - {p}^{k - 1}}\\right) {B}_{k}/k = \\frac{1}{{\\alpha }^{-k} - 1}\\int _{{\\mathbb{Z}}_{p}^{ * }}{x}^{k - 1}d{\\mu }_{1,\\alpha }.\n\\] | By Exercise 8.2.12, we can interpret the left hand side of the equation in Theorem 10.4.7 as\n\n\\[ \n\\left( {1 - {p}^{k - 1}}\\right) \\zeta \\left( {1 - k}\\right)\n\\] | No |
Theorem 11.1.5 [Weyl,1916] A sequence \( {\left\{ {x}_{n}\right\} }_{n = 1}^{\infty } \) is u.d. if and only if\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{N}{e}^{{2\pi im}{x}_{n}} = o\left( N\right) ,\;m = \pm 1, \pm 2,\ldots \] | Proof. As observed earlier, the necessity is clear. For sufficiency, let \( \epsilon > 0 \) and \( f \) a continuous function \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{C} \) . By the Weierstrass approximation theorem, there is a trigonometric polynomial \( \phi \left( x\right) \) such that \( \deg \phi... | Yes |
Theorem 11.1.16 (van der Corput,1931) Let \( {y}_{1},\ldots ,{y}_{N} \) be complex numbers. Let \( H \) be an integer with \( 1 \leq H \leq N \). Then\n\n\[ \n{\left| \mathop{\sum }\limits_{{n = 1}}^{N}{y}_{n}\right| }^{2} \leq \n\]\n\n\[ \n\frac{N + H}{H + 1}\mathop{\sum }\limits_{{n = 1}}^{N}{\left| {y}_{n}\right| }^... | Proof. It is convenient to set \( {y}_{n} = 0 \) for \( n \leq 0 \) and \( n > N \). Clearly,\n\n\[ \n{\left( H + 1\right) }^{2}{\left| \mathop{\sum }\limits_{n}{y}_{n}\right| }^{2} = {\left| \mathop{\sum }\limits_{{h = 0}}^{H}\mathop{\sum }\limits_{n}{y}_{n + h}\right| }^{2} = {\left| \mathop{\sum }\limits_{n}\mathop{... | Yes |
Corollary 11.1.17 (van der Corput,1931) If for each positive integer \( r \) , the sequence \( {x}_{n + r} - {x}_{n} \) is u.d. mod 1, then the sequence \( {x}_{n} \) is u.d. mod 1 . | Proof. We apply Theorem 11.1.16 with \( {y}_{n} = {e}^{{2\pi im}{x}_{n}} \) to get\n\n\[ \n{\left| \frac{1}{N}\mathop{\sum }\limits_{{n = 1}}^{N}{e}^{{2\pi im}{x}_{n}}\right| }^{2} \n\]\n\n\[ \n\leq \frac{1 + H/N}{H + 1} + \frac{2\left( {N + H}\right) }{{N}^{2}\left( {H + 1}\right) }\mathop{\sum }\limits_{{r = 1}}^{H}\... | Yes |
Theorem 11.2.2 The number \( x \) is normal to the base \( b \) if and only if the sequence \( \left( {x{b}^{n}}\right) \) is u.d. mod 1 . | Proof. Let \( {B}_{k} = {b}_{1}{b}_{2}\cdots {b}_{k} \) be a block of \( k \) digits. The block\n\n\[ \n{a}_{m}{a}_{m + 1}\cdots {a}_{m + k - 1} \n\]\n\nin the \( b \) -adic expansion of \( x \) is identical with \( {B}_{k} \) if and only if\n\n\[ \n\frac{{B}_{k}}{{b}^{k}} \leq \left( {x{b}^{m - 1}}\right) < \frac{{B}_... | Yes |
Theorem 11.3.3 [Wiener - Schoenberg,1928] The sequence \( {\left\{ {x}_{n}\right\} }_{n = 1}^{\infty } \) has a continuous a.d.f. if and only if for every integer \( m \), the limit\n\n\[ \n{a}_{m} \mathrel{\text{:=}} \mathop{\lim }\limits_{{N \rightarrow \infty }}\frac{1}{N}\mathop{\sum }\limits_{{n = 1}}^{N}{e}^{{2\p... | Proof. Suppose the sequence has a continuous a.d.f. \( g\left( x\right) \) . The existence of the limits is clear. Now, by Exercise 11.3.2, we have\n\n\[ \n{a}_{m} = {\int }_{0}^{1}{e}^{2\pi imx}{dg}\left( x\right) \]\n\nThus,\n\n\[ \n\mathop{\lim }\limits_{{N \rightarrow \infty }}\frac{1}{N}\mathop{\sum }\limits_{{m =... | Yes |
Theorem 11.4.7 Let \( M \) be a natural number. For any interval \( I = \\left\\lbrack {a, b}\\right\\rbrack \) with length \( b - a < 1 \), write\n\n\[ \n{\\Xi }_{I}\\left( x\\right) = \\mathop{\\sum }\\limits_{{n \\in \\mathbb{Z}}}{\\chi }_{I}\\left( {n + x}\\right) \n\]\n\nThen, there are trigonometric polynomials\n... | Proof. Take \( \\delta = M + 1 \) in Exercise 11.4.5 and let \( {H}_{ \\pm } \) be the functions obtained by that exercise. Put\n\n\[ \n{V}_{ \\pm }\\left( x\\right) = \\mathop{\\sum }\\limits_{{n \\in \\mathbb{Z}}}{H}_{ \\pm }\\left( {n + x}\\right) \n\]\n\nBy Exercise 11.4.6, \( {V}_{ \\pm }\\left( x\\right) \\in {L}... | No |
Theorem 11.4.8 (Erdös-Turán,1948) For any integer \( M \geq 1 \) ,\n\n\[ \n{D}_{N} \leq \frac{1}{M + 1} + 3\mathop{\sum }\limits_{{m = 1}}^{M}\frac{1}{Nm}\left| {\mathop{\sum }\limits_{{n = 1}}^{N}{e}^{{2\pi im}{x}_{n}}}\right| .\n\] | Proof. Let \( {\chi }_{I} \) be the characteristic function of the interval \( I = \left\lbrack {a, b}\right\rbrack \) . Using Theorem 11.4.7, we have\n\n\[ \n\mathop{\sum }\limits_{{n = 1}}^{N}{\Xi }_{I}\left( {x}_{n}\right) \leq \mathop{\sum }\limits_{{n = 1}}^{N}{S}_{M}^{ + }\left( {x}_{n}\right)\n\]\n\n\[ \n\leq N\... | Yes |
Lemma 1.3.1 If \( x \) is a vertex of the graph \( X \) and \( g \) is an automorphism of \( X \), then the vertex \( y = {x}^{g} \) has the same valency as \( x \) . | Proof. Let \( N\left( x\right) \) denote the subgraph of \( X \) induced by the neighbours of \( x \) in \( X \) . Then\n\n\[ N{\left( x\right) }^{g} = N\left( {x}^{g}\right) = N\left( y\right) \]\n\nand therefore \( N\left( x\right) \) and \( N\left( y\right) \) are isomorphic subgraphs of \( X \) . Consequently they ... | Yes |
Lemma 1.4.1 The chromatic number of a graph \( X \) is the least integer \( r \) such that there is a homomorphism from \( X \) to \( {K}_{r} \) . | Proof. Suppose \( f \) is a homomorphism from the graph \( X \) to the graph \( Y \) . If \( y \in V\left( Y\right) \), define \( {f}^{-1}\left( y\right) \) by\n\n\[ \n{f}^{-1}\left( y\right) \mathrel{\text{:=}} \{ x \in V\left( X\right) : f\left( x\right) = y\} .\n\] \n\nBecause \( y \) is not adjacent to itself, the ... | Yes |
Lemma 1.6.1 If \( v \geq k \geq i \), then \( J\left( {v, k, i}\right) \cong J\left( {v, v - k, v - {2k} + i}\right) \) . | Proof. The function that maps a \( k \) -set to its complement in \( \Omega \) is an isomorphism from \( J\left( {v, k, i}\right) \) to \( J\left( {v, v - k, v - {2k} + i}\right) \) ; you are invited to check the details. | No |
Lemma 1.6.2 If \( v \geq k \geq i \), then \( \operatorname{Aut}\left( {J\left( {v, k, i}\right) }\right) \) contains a subgroup isomorphic to \( \operatorname{Sym}\left( v\right) \) . | Note that \( \operatorname{Aut}\left( {J\left( {v, k, i}\right) }\right) \) is a permutation group acting on a set of size \( \left( \begin{array}{l} v \\ k \end{array}\right) \) , and so when \( k \neq 1 \) or \( v - 1 \), it is not actually equal to \( \operatorname{Sym}\left( v\right) \) . Nevertheless, it is true t... | No |
Lemma 1.7.1 If \( X \) is regular with valency \( k \), then \( L\left( X\right) \) is regular with valency \( {2k} - 2 \) . | Each vertex in \( X \) determines a clique in \( L\left( X\right) \) : If \( x \) is a vertex in \( X \) with valency \( k \), then the \( k \) edges containing \( x \) form a \( k \) -clique in \( L\left( X\right) \) . Thus if \( X \) has \( n \) vertices, there is a set of \( n \) cliques in \( L\left( X\right) \) wi... | No |
Theorem 1.7.2 A nonempty graph is a line graph if and only if its edge set can be partitioned into a set of cliques with the property that any vertex lies in at most two cliques. | Proof. Let \( C \) be a clique in \( L\left( X\right) \) containing exactly \( c \) vertices. If \( c > 3 \) , then the vertices of \( C \) correspond to a set of \( c \) edges in \( X \), meeting at a common vertex. Consequently, there is a bijection between the vertices of \( X \) and the maximal cliques of \( L\left... | No |
Lemma 1.7.3 Suppose that \( X \) and \( Y \) are graphs with minimum valency four. Then \( X \cong Y \) if and only if \( L\left( X\right) \cong L\left( Y\right) \). | Proof. Let \( C \) be a clique in \( L\left( X\right) \) containing exactly \( c \) vertices. If \( c > 3 \) , then the vertices of \( C \) correspond to a set of \( c \) edges in \( X \), meeting at a common vertex. Consequently, there is a bijection between the vertices of \( X \) and the maximal cliques of \( L\left... | No |
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