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Theorem 2.1. The functor \( \operatorname{Der}\left( {\mathfrak{g}, - }\right) \) is represented by the \( \mathfrak{g} \) -module \( I\mathfrak{g} \) , that is, for any \( \mathrm{g} \) -module \( A \) there is a natural isomorphism between the \( K \) -vector spaces \( \operatorname{Der}\left( {\mathfrak{g}, A}\right... | Proof. Given a derivation \( d : \mathrm{g} \rightarrow A \), we define a \( K \) -linear map \( {f}_{d}^{\prime } : T\mathfrak{g} \rightarrow A \) by sending \( K = {T}^{0}\mathfrak{g} \subseteq T\mathfrak{g} \) into zero and \( {x}_{1} \otimes \cdots \otimes {x}_{n} \) into \( {x}_{1} \circ \left( {{x}_{2} \circ \cdo... | Yes |
Proposition 2.3. The vector space \( \operatorname{Der}\left( {\mathfrak{g}, A}\right) \) is naturally isomorphic to the vector space of Lie algebra homomorphisms \( f : \mathfrak{g} \rightarrow A \times \mathfrak{g} \) for which \( {p}_{\mathrm{g}}f = {1}_{\mathrm{g}} \) . | Proof. First we note that \( A \) may be regarded as an \( A \times \mathfrak{g} \) -module via \( {p}_{\alpha } : A \times \mathfrak{g} \rightarrow \mathfrak{g} \), and that then the canonical projection \( {d}^{\prime } = {p}_{A} : A \times \mathfrak{g} \rightarrow A \) becomes a derivation. A Lie algebra homomorphis... | Yes |
Proposition 2.4. Let \( {TV} \) denote the tensor algebra over the \( K \) -vector space \( V \) . The free Lie algebra \( \mathfrak{f}\left( V\right) \) over \( K \) is the Lie subalgebra of LTV generated by \( V \). | Proof. Suppose given \( f : V \rightarrow \mathfrak{g} \) . By the universal property of the tensor algebra the map if: \( V \rightarrow \mathfrak{g} \rightarrow U\mathfrak{g} \) extends to an algebra homomorphism \( {TV} \rightarrow U\mathrm{\;g} \) . Clearly the Lie subalgebra of \( {LTV} \) generated by \( V \) is m... | No |
Theorem 2.5. The augmentation ideal \( I\mathfrak{f} \) of a free Lie algebra \( \mathfrak{f} \) is a free \( \mathfrak{f} \) -module. | Proof. Let \( \mathfrak{f} = \mathfrak{f}\left( V\right) \) and let \( \{ e\} \) be a \( K \) -basis of \( V \), and let \( f : \{ e\} \rightarrow M \) be a function into the \( \mathfrak{f} \) -module \( M \) . We shall show that \( f \) may be extended uniquely to a \( \mathfrak{f} \) -module homomorphism \( {f}^{\pr... | Yes |
Theorem 3.2. If \( n \mapsto g \rightarrow b \) is an exact sequence of Lie algebras and if \( A \) is an \( \mathfrak{h} \) -module, then the following sequence is exact\n\n\[ 0 \rightarrow \operatorname{Der}\left( {\mathfrak{h}, A}\right) \rightarrow \operatorname{Der}\left( {\mathfrak{g}, A}\right) \rightarrow {\ope... | The proof is analogous to the proof of Theorem VI.8.1 and is left to the reader. | No |
Lemma 4.1. The complex \( {\mathbf{W}}^{p} \) is exact. | Proof of Lemma 4.1. In order to show that \( {\mathbf{W}}^{p} \) is exact, we define a \( K \) -linear contracting homotopy \( \sum \) as follows. \( {\sum }_{-1} : K \rightarrow {W}_{0}^{p} \) is given by \( {\sum }_{-1}\left( {1}_{K}\right) = 1\langle \rangle \), and, for \( n \geqq 0 \), we define \( {\sum }_{n} : {... | Yes |
Corollary 4.3. Let \( \mathfrak{g} \) be a Lie algebra of dimension \( n \) over \( K \) . Then for any \( \mathfrak{g} \) -module \( A,{H}^{k}\left( {\mathfrak{g}, A}\right) = 0 \) for \( k \geqq n + 1 \) . | Proof. For \( k \geqq n + 1 \) we have \( {E}_{k}V = 0 \) . | No |
Lemma 5.1.\n\n\\[ \n\\beta \\left( {\\left\\lbrack {x, y}\\right\\rbrack, z}\\right) = \\beta \\left( {x,\\left\\lbrack {y, z}\\right\\rbrack }\\right), x, y, z \\in \\mathfrak{g}.\n\\] | Proof. Since the trace function is additive and \\( \\operatorname{Tr}\\left( {\\varphi \\psi }\\right) = \\operatorname{Tr}\\left( {\\psi \\varphi }\\right) \\) , for \\( \\varphi ,\\psi \\in {\\operatorname{End}}_{K}A \\), we have\n\n\\[ \n\\beta \\left( {\\left\\lbrack {x, y}\\right\\rbrack, z}\\right) = \\operatorn... | Yes |
Corollary 5.3. The Killing form of a semi-simple Lie algebra is nondegenerate. | Proof. The structure map ad : \( \mathfrak{g} \rightarrow L\left( {{\operatorname{End}}_{K}\mathfrak{g}}\right) \) of the \( \mathfrak{g} \) -module \( \mathfrak{g} \) has the center of \( g \) as kernel (see Exercise 1.2). Since the center is an abelian ideal, it is trivial. Hence ad is injective. | No |
Corollary 5.4. Let \( \mathfrak{a} \) be an ideal in the semi-simple Lie algebra \( \mathfrak{g} \). Then there exists an ideal \( \mathfrak{b} \) of \( \mathfrak{g} \) such that \( \mathfrak{g} = \mathfrak{a} \oplus \mathfrak{b} \), as Lie algebras. | Proof. Define \( \mathfrak{b} \) to be the orthogonal complement of \( \mathfrak{a} \) with respect to the Killing form \( \beta \). Clearly it is sufficient to show (i) that \( \mathfrak{b} \) is an ideal and (ii) that \( \mathfrak{a} \cap \mathfrak{b} = \{ 0\} \). To prove (i) let \( x \in \mathfrak{g}, b \in \mathfr... | Yes |
Corollary 5.5. If \( \mathfrak{g} \) is semi-simple, then every ideal \( \mathfrak{a} \) in \( \mathfrak{g} \) is semi-simple also. | Proof. Since \( \mathfrak{g} = \mathfrak{a} \oplus \mathfrak{b} \) by Corollary 5.4, every ideal \( {\mathfrak{a}}^{\prime } \) in \( \mathfrak{a} \) is also an ideal in \( \mathfrak{g} \) . In particular if \( {\mathfrak{a}}^{\prime } \) is an abelian ideal, it follows that \( {\mathfrak{a}}^{\prime } = 0 \) . | No |
Proposition 6.1. Let \( \mathfrak{g} \) be semi-simple, then \( {H}^{1}\left( {\mathfrak{g}, A}\right) = 0 \) . | Proof. Suppose there is a \( \mathfrak{g} \) -module \( A \) with \( {H}^{1}\left( {\mathfrak{g}, A}\right) \neq 0 \) . Then there is such a g-module \( A \) with minimal \( K \) -dimension. If \( A \) is not simple, then there is a proper submodule \( 0 \neq {A}^{\prime } \subset A \) . Consider \( 0 \rightarrow {A}^{... | Yes |
Theorem 6.2 (Weyl). Every (finite-dimensional) module \( A \) over a semisimple Lie algebra \( \mathfrak{g} \) is a direct sum of simple \( \mathfrak{g} \) -modules. | Proof. Using induction on the \( K \) -dimension of \( A \), we have only to show that every non-trivial submodule \( 0 \neq {A}^{\prime } \subset A \) is a direct summand in \( A \) . To that end we consider the short exact sequence\n\n\[ \n{A}^{\prime } \rightarrowtail A \rightarrow {A}^{\prime \prime } \n\]\n\n(6.1)... | Yes |
Proposition 6.3. Let \( \mathfrak{g} \) be a semi-simple Lie algebra and let \( A \) be a (finite-dimensional) \( \mathfrak{g} \) -module. Then \( {H}^{2}\left( {\mathfrak{g}, A}\right) = 0 \) . | Proof. We begin as in the proof of Proposition 6.1. Suppose there is a \( \mathfrak{g} \) -module \( A \) with \( {H}^{2}\left( {\mathfrak{g}, A}\right) \neq 0 \) . Then there is such a \( \mathfrak{g} \) -module \( A \) with minimal \( K \) -dimension. If \( A \) is not simple, then there is a proper submodule \( 0 \n... | Yes |
Proposition 6.6. \( \mathrm{g}/\mathrm{r} \) is semi-simple. | Proof. Let \( \mathfrak{a}/\mathfrak{r} \) be an abelian ideal of \( \mathfrak{g}/\mathfrak{r} \) ; then the sequence \( \mathfrak{r} \rightarrowtail \mathfrak{a} \rightarrow \mathfrak{a}/\mathfrak{r} \) has both ends solvable, hence a is solvable by Lemma 6.4. By the maxi-mality of \( r \), it follows that \( a = r \)... | Yes |
Theorem 6.7 (Levi-Malcev). Every (finite-dimensional) Lie algebra \( \mathfrak{g} \) is the split extension of a semi-simple Lie algebra by the radical \( \mathfrak{r} \) of \( \mathfrak{g} \) . | Proof. We proceed by induction on the derived length of \( \mathfrak{r} \) . If \( \mathfrak{r} \) is abelian, then it is a \( \mathrm{g}/\mathrm{r} \) -module and \( {H}^{2}\left( {\mathrm{\;g}/\mathrm{r},\mathrm{r}}\right) = 0 \) by Proposition 6.3. Since \( {H}^{2} \) classifies extensions with abelian kernel the ex... | Yes |
Theorem 7.1. The global dimension of \( P = K\left\lbrack {{x}_{1},\ldots ,{x}_{m}}\right\rbrack \) is \( m \) . Moreover, every projective graded P-module is free. | The proof of this theorem will be executed in several steps; not all these steps require that \( K \) be a field, so we will specify the assumptions at each stage. Now \( K \) can be considered as a graded \( P \) -module through the augmentation \( \varepsilon : P \rightarrow K \), which associates with each polynomia... | No |
Proposition 7.2 (Koszul). Let \( {D}_{n} = P{ \otimes }_{K}{E}_{n}\mathfrak{a} \), and let \( {d}_{n} : {D}_{n} \rightarrow {D}_{n - 1} \) be defined by \[ {d}_{n}\left( {p \otimes \left\langle {{x}_{{j}_{1}},\ldots ,{x}_{{j}_{n}}}\right\rangle }\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{\left( -1\right) }^{i + 1}p{... | We note in passing that Proposition 7.2 admits a fairly easy direct proof (see Exercise 7.1). | No |
Proposition 7.3. Let \( M \) be a graded P-module. If \( K \) is a commutative ring and if \( M{ \otimes }_{P}K = 0 \) then \( M = 0 \) . | Proof. Let \( I \) be the augmentation ideal in \( P \), that is,\n\n\[ I\overset{\mu }{ \rightarrow }P\overset{\varepsilon }{ \rightarrow }K \]\n\nis exact. Then the homogeneous non-zero elements of \( I \) all have positive degree. Now the sequence\n\n\[ M{ \otimes }_{P}I\overset{{\mu }_{ * }}{ \rightarrow }M{ \otime... | Yes |
Theorem 7.4. Let \( B \) be a graded \( P \) -module. If \( K \) is a field and if \( {\operatorname{Tor}}_{1}^{P}\left( {B, K}\right) = 0 \) then \( B \) is free. | Proof. It is plain that every element of \( B{ \otimes }_{P}K \) can be expressed as \( b \otimes 1, b \in B \), where 1 is the unity element of \( K \) . Thus we may select a basis \( \left\{ {{b}_{i} \otimes 1}\right\}, i \in I \), for \( B{ \otimes }_{P}K \) as vector space over \( K \) . Let \( F \) be the free gra... | Yes |
Theorem 1.1. The couple is exact. | Proof. Exactness at top left \( {D}_{1} : {\alpha }_{1}{\gamma }_{1}\left\lbrack z\right\rbrack = {\alpha \gamma }\left( z\right) = 0 \) . Conversely, if \( x \in {D}_{1} = {\alpha D} \) and \( {\alpha x} = 0 \), then \( x \in \ker \beta \) and \( x = {\gamma y}, y \in E \) . Thus \( {d}_{0}y = {\beta \gamma y} = 0, y ... | Yes |
Theorem 1.2. \( {E}_{n} = {\gamma }^{-1}\left( {{\alpha }^{n}D}\right) /\beta \left( {{\alpha }^{-n}\left( 0\right) }\right) \), and \( {d}_{n} : {E}_{n} \rightarrow {E}_{n} \) is induced by \( \beta {\alpha }^{-n}\gamma \) . | Indeed we have, for each \( n \), the exact sequence\n\n\[ \n{\alpha }^{n}D\overset{{\alpha }_{n}}{ \rightarrow }{\alpha }^{n}D\overset{{\beta }_{n}}{ \rightarrow }{E}_{n}\overset{{\gamma }_{n}}{ \rightarrow }{\alpha }^{n}D\overset{{\alpha }_{n}}{ \rightarrow }{\alpha }^{n}D, \n\]\n\n(1.5)\n\nwhere \( {\alpha }_{n} \) ... | Yes |
Theorem 3.1. If \( \alpha \) is stationary, the spectral sequence associated with the exact couple (2.9) converges finitely; that is, given \( \left( {p, q}\right) \), there exists \( r \) such that \( {E}_{r}^{p, q} = {E}_{r + 1}^{p, q} = \cdots = {E}_{\infty }^{p, q} \) . | Proof. Consider the exact sequence\n\n\[ \n{D}^{p - 1, q}\overset{\alpha }{ \rightarrow }{D}^{p, q}\overset{\beta }{ \rightarrow }{E}^{p, q}\overset{\gamma }{ \rightarrow }{D}^{p - 1, q - 1}\overset{\alpha }{ \rightarrow }{D}^{p, q - 1}. \n\]\n\n(3.1)\n\nFix \( q \) . Since \( \alpha \) is positively stationary, it fol... | Yes |
Proposition 3.4. If the filtration of the chain complex \( C \) is finite, it is homologically finite. | Proof. Plainly (3.8) (i) implies (3.9) (i). Also (3.8) (ii) implies that, given \( q \) ,\n\n\[ \n{C}_{q}^{\left( p\right) } = {C}_{q},{C}_{q + 1}^{\left( p\right) } = {C}_{q + 1},\text{ for }p\text{ large }.\n\]\n\nThus \( {H}_{q}\left( {C}^{\left( p\right) }\right) = {H}_{q}\left( C\right) \) for \( p \) large. | No |
Theorem 3.5. If the filtration of the chain complex \( C \) is homologically finite, then\n\n(i) the associated spectral sequence converges finitely;\n\n(ii) the induced filtration of \( H\left( C\right) \) is finite;\n\n(iii) \( {E}_{\infty } \cong {Gr} \circ H\left( C\right) \) ; precisely,\n\n\[ \n{E}_{\infty }^{p, ... | Proof. (i) Plainly (3.9) (i) asserts that \( D \) is positively graded; and (3.9) (ii) is equivalent to the statement that \( \bar{D} \) is negatively graded, as is seen immediately by applying homology to the sequence\n\n\[ \n{C}^{\left( p\right) } \rightarrowtail C \rightarrow C/{C}^{\left( p\right) } \n\]\n\nBy Prop... | Yes |
Theorem 4.1. \( {Q}^{Q}{Q}_{\sigma } = {Q}_{\sigma }{Q}^{Q} \) . | Proof. It is clear, in fact, that \( {B}_{1} \) is obtained from \( B \) by first cutting down to the subobject \( {\gamma }^{-1}\left( {C}_{1}\right) \) and then factoring out \( \beta {\sigma }^{-1}\left( 0\right) \), whereas \( {\bar{B}}_{1} \) is obtained by the opposite process, that is, first factoring out \( \be... | Yes |
Consider the diagram\n\n\n\nwhere \( {\gamma \beta } \) factors through \( \varrho {\varrho }_{1} \) . Then, if \( \left( {{\beta }^{\prime },{\gamma }^{\prime }}\right) = {Q}^{\varrho }\left( {\beta ,\gamma }\right)... | Proof. Let \( {\gamma \beta } = \varrho {\varrho }_{1}\delta \) . Then \( \varrho {\gamma }^{\prime }{\beta }^{\prime } = \varrho {\varrho }_{1}\delta \) so that \( {\gamma }^{\prime }{\beta }^{\prime } = {\varrho }_{1}\delta \) . Then (4.10) follows either by observing that the juxtaposition of two pull-back squares i... | Yes |
Theorem 4.4. If the bottom row of (4.4) is exact, so are all rows of (4.4) and (4.7). | Proof. That the middle row of (4.4) is exact is plain, since\n\n\[ \n{\gamma }^{-1}\left( 0\right) \leqq {\gamma }^{-1}\left( {C}_{1}\right) \n\]\n\nThe rest of the statement of the theorem follows by duality. | No |
Theorem 5.1. There is a natural equivalence \( {R}_{2}{R}_{1}^{J} \cong {R}_{1}{R}_{2}^{I} \) . Setting either equal to \( R \), we have \( P \dashv R,{RP} = 1 \) and the counit \( \delta : {PR} \rightarrow 1 \) is given by \( \delta = {\delta }_{1}^{J} \circ {P}_{1}^{J}{\delta }_{2}{R}_{1}^{J} \) if \( R = {R}_{2}{R}_... | Proof. The first assertion is a special case of Theorem II.8.6. The rest follows readily from Proposition II.7.1 and we leave the details to the reader. \( ▱ \) | No |
Theorem 5.2. If each square in (5.6) is a pull-back, then (5.7) is a pullback. | Proof. We have to show that (5.7) is a pull-back diagram. Suppose then given \( \psi : X \rightarrow {A}_{0} \) and \( \chi : X \rightarrow {B}_{\infty } \), with \( {\varphi }_{0}\psi = \bar{\beta }\chi \) . We then obtain morphisms \( {\psi }_{0} = \psi : X \rightarrow {A}_{0} \) and \( {\chi }_{1} = {\delta }_{1}\ch... | No |
Theorem 5.3. In the notation of Theorem 4.6 we have\n\n\[ \n{E}_{\infty } = \mathop{\lim }\limits_{m}\mathop{\lim }\limits_{n}\left( {{E}_{m, n};{\varrho }_{m, n},{\sigma }_{m, n}}\right) = \mathop{\lim }\limits_{n}\mathop{\lim }\limits_{m}\left( {{E}_{m, n};{\varrho }_{n, n},{\sigma }_{m, n}}\right) .\n\] | Proof. Let us execute \( {Q}_{\eta }^{v} \) as \( {Q}_{\eta }{Q}^{v} \) . We thus obtain\n\n\n\n(5.14)\n\nHowever, by Theorem 5.2, \( {E}_{0,\infty } = \mathop{\lim }\limits_{ \leftarrow }{E}_{0, n} = \mathop{\bigcap... | Yes |
Theorem 6.2. In the Rees system (6.9) the spectral sequences of the couples \( \odot \) and \( \odot \) coincide. | Proof. The relations (6.10) assert that we have a morphism of EC,\n\n\[ \left( {\xi ,1}\right) : \text{②} \rightarrow \text{①.} \]\n\nApplying the spectral sequence functor, we get\n\n\[ {SS}\left( {\xi ,1}\right) : {SS} \oslash \rightarrow {SS} \odot . \]\n\nBut \( {SS}\left( {\xi ,1}\right) \) is then a morphism of s... | Yes |
Proposition 6.4. Given a Rees system (6.9) and its derived system (6.16), we have \( \bar{\gamma }\beta = {\overline{\varrho \gamma }}_{1}{\beta }_{1}\sigma \) . | Proof. This follows immediately from the definitions of \( {\beta }_{1},{\bar{\gamma }}_{1} \) given in Section 1. A proof valid in any abelian category is given in [10; Prop. 7.16]. | No |
Proposition 6.5. If (6.9) is special then (6.16) is special with the same \( \theta : \Gamma \cong \Gamma \) . | Proof. Suppose given \( \theta \) satisfying (6.11). Then\n\n\[ \theta {\varphi }_{1}\sigma = {\theta \varphi } = {\varphi \alpha } = {\varphi }_{1}{\sigma \alpha } = {\varphi }_{1}{\alpha }_{1}\sigma ,\]\n\nso that \( \theta {\varphi }_{1} = {\varphi }_{1}{\alpha }_{1} \) . Similarly \( {\bar{\varphi }}_{1}\theta = {\... | Yes |
Proposition 6.6. For the triple \( F\left( C\right) \), we have\n\n\[ \n{Gr} \circ H\left( C\right) = {i}_{A}H\left( A\right) /{i}_{B}H\left( B\right) = \ker {j}_{A}/\ker {j}_{B}.\n\] | Proof. Plainly, if \( G = G\left( C\right), A = A\left( C\right) \), then\n\n\[ \n{i}_{A}H\left( A\right) = {\bigoplus }_{p}\operatorname{im}H\left( {C}^{\left( p\right) }\right) ,\n\]\n\n\[ \n{i}_{B}H\left( B\right) = {\bigoplus }_{p}\operatorname{im}H\left( {C}^{\left( p - 1\right) }\right) .\n\]\n\nThus \( {i}_{A}H\... | Yes |
Theorem 7.1. The sequences\n\n\[ \n\bar{U}\overset{{\xi }_{U}}{ \rightarrow }U\overset{{\varphi }_{U}}{ \rightarrow }\Gamma \n\]\n\n\[ \n\Gamma \xrightarrow[]{{\bar{\varphi }}_{\bar{I}}}\bar{I}\xrightarrow[]{{\xi }_{I}}I \n\]\n\nare exact. | Proof. Since \( {\xi }_{U},{\varphi }_{U} \) are induced by \( \xi ,\varphi \) by passing to quotient objects, the exactness of \( \bar{U}\xrightarrow[]{{\xi }_{U}}U\xrightarrow[]{{\varphi }_{U}}\Gamma \) follows immediately from that of \( \bar{D}\overset{\xi }{ \rightarrow }D\xrightarrow[]{\varphi }\Gamma \) . Simila... | Yes |
Proposition 7.2.\n\n\[ \n{\Gamma }^{ + } = {\varphi }_{U}U/{\varphi }_{U}{\alpha }^{\prime }U \]\n\n\[ \n{\Gamma }^{ - } = \ker {\bar{\alpha }}^{\prime \prime }{\bar{\varphi }}_{I}/\ker {\bar{\varphi }}_{I}. \]\n | Proof. Obviously \( {\varphi }_{U}U = {\varphi D},{\varphi }_{U}{\alpha }^{\prime }U = {\varphi \alpha D} \) . As to the second expression, we may appeal to duality (the two expressions actually are dual, although their expressions disguise the fact!), or invoke the diagram\n\n\n\n(7.5)\n\nwith exact rows and columns. | Proof. The exact sequence (5.18) yields the exact sequences involving \( {\beta }_{ * },{\gamma }_{ * };{\bar{\beta }}_{ * },{\bar{\gamma }}_{ * } \) . The diagram\n\n\n\nimmediately yields, by passing to cokernels, ... | Yes |
Corollary 7.4. The homomorphic relation \( \Theta \) from \( {E}_{\infty } \) to \( {\Gamma }^{ + } \) is an isomorphism if and only if\n\n\[ \operatorname{coker}{\bar{\alpha }}^{\prime } = 0,\;\ker {\alpha }^{\prime \prime } = 0. \] | Thus the conditions (7.7) are the necessary and sufficient conditions for the validity of (6.21); and hence of (6.17), \( {E}_{\infty } \cong {i}_{A}H\left( A\right) /{i}_{B}H\left( B\right) \), for a Rees system arising from a triple \( \left( {G, A,\theta }\right) \) . | Yes |
Theorem 7.5. In the spectral sequence arising from a filtered differential object \( C \), the homomorphic relation \( \Theta \) from \( {E}_{\infty } \) to \( {Gr} \circ H\left( C\right) \) is an isomorphism if and only if\n\n\[ \n{I}_{p} \cap {\alpha }^{-1}\left( 0\right) = 0,\;{\bar{\alpha }}^{\prime }{\bar{U}}_{p} ... | Let us finally observe how condition (3.9), that \( C \) be homologically finite, automatically - indeed, trivially - guarantees (7.8). For, in this case, we are dealing with a filtered graded differential object \( C \) and (3.9) (i) implies that \( {I}_{p, q} = 0 \) for all \( p, q \) so that \( {I}_{p} = 0 \) for al... | Yes |
Proposition 8.2. If \( \left( {X;{\eta }_{p}}\right) = \underline{\lim }\left( {{X}_{p},{\xi }_{p}}\right) \), then \( \left( {Y;{\eta }_{p}^{\prime }}\right) = \underline{\lim }\left( {{Y}_{p},{\xi }_{p}^{\prime }}\right) \) . | Proof. The right hand square of (8.6) is a pull-back since \( \ker {\xi }_{p}^{\prime } = \ker {\xi }_{p} \) . It thus follows from Theorem 5.2 that\n\n\[ \n{Y}_{-\infty }\xrightarrow[]{{\eta }_{p}^{\prime \prime }}{Y}_{p} \n\]\n\n\[ \n\downarrow \mu - \infty \; \downarrow \mu \n\]\n\n\[ \nX\xrightarrow[]{{\eta }_{p}}{... | Yes |
Theorem 8.4. Let \( \psi : X \rightarrow {X}^{\prime } \) be a morphism of filtered objects in the abelian category \( \mathfrak{A} \) . Thus\n\n\[ \n{X}^{p}\overset{{v}^{p}}{ \rightarrow }X\overset{{\eta }_{p}}{ \rightarrow }{X}_{p} \]\n\n\[ \n\downarrow {}^{\psi {}^{p}}\; \downarrow {}^{\psi }\; \downarrow {}^{\psi {... | Proof. We are given that \( \psi \) induces an isomorphism\n\n\[ \n{\psi }_{ * } : {X}^{p}/{X}^{p - 1}\overset{ \sim }{ \rightarrow }{X}^{\prime p}/{X}^{\prime p - 1}. \]\n\nIt then follows by induction on \( p - q \) that \( \psi \) induces an isomorphism \( {\psi }_{ * } : {X}_{q}^{p} \rightarrow {X}_{q}^{\prime p} \... | Yes |
Theorem 8.5. The completion is a complete filtration of \( Y = {\left( {X}^{\infty }\right) }_{-\infty } \) and \( \operatorname{Gr}Y = \operatorname{Gr}X \) . | Proof. By construction the filtration of \( Y \) is right complete. That it is left complete follows from the dual of Proposition 8.2.\n\nNow, given (8.4), we obtain \( \operatorname{Gr}X \) either by\n\n\[{\left( \operatorname{Gr}X\right) }_{p} = \operatorname{coker}{\xi }^{p}\]\n\nor by\n\n\[{\left( \operatorname{Gr}... | Yes |
For the (second) spectral sequence associated with the filtration (9.5) we have\n\n\[ \n{}_{2}{E}_{0}^{p, q} = {H}_{q - p}\left( {{B}_{*, p},{\partial }^{\prime }}\right) ,\;{}_{2}{E}_{1}^{p, q} = {H}_{p}\left( {{H}_{q - p}\left( {B,{\partial }^{\prime }}\right) ,{\partial }^{\prime \prime }}\right) .\n\]\n\n(9.7) | Proof. We prove (9.7) only, and so permit ourselves to write \( {F}^{p} \) for \( {}_{2}{F}^{p} \) .\n\nClearly, \( {F}^{p}{\left( \operatorname{Tot}\mathbf{B}\right) }_{q}/{F}^{p - 1}{\left( \operatorname{Tot}\mathbf{B}\right) }_{q} = {B}_{q - p, p} \) . Moreover the differential \( \partial = {\partial }^{\prime } + ... | Yes |
Proposition 9.2. If \( \mathbf{B} \) is positive, then both the first and the second spectral sequences (9.6), (9.7) converge finitely to the graded object associated with \( \left\{ {{H}_{n}\left( {\operatorname{Tot}\mathbf{B}}\right) }\right\} \), suitably (finitely) filtered. | Proof. By Theorem 3.5 we only have to verify that the filtrations (9.4), (9.5) are finite. But plainly, given (9.8), \[ {}_{1}{F}^{p}{\left( \operatorname{Tot}\mathbf{B}\right) }_{n} = 0\;\text{ if }\;p \leqq {n}_{0} - 1, \] \[ {}_{1}{F}^{p}{\left( \operatorname{Tot}\mathbf{B}\right) }_{n} = {\left( \operatorname{Tot}\... | Yes |
Theorem 9.3 (Grothendieck spectral sequence). Given \( F : \mathfrak{A} \rightarrow \mathfrak{B} \) , \( G : \mathfrak{B} \rightarrow \mathfrak{C} \), assume that if \( I \) is an injective object of \( \mathfrak{A} \), then \( F\left( I\right) \) is \( G \) -acyclic. Then there is a spectral sequence \( \left\{ {{E}_{... | Proof. Take an injective resolution of \( A \) in \( \mathfrak{A} \) , \[ \mathbf{I} : {I}_{0} \rightarrow {I}_{1} \rightarrow {I}_{2} \rightarrow \cdots . \] Apply \( F \) to obtain the cochain complex in \( \mathfrak{B} \) , \[ F{I}_{0} \rightarrow F{I}_{1} \rightarrow F{I}_{2} \rightarrow \cdots . \] Suppose we have... | Yes |
Lemma 9.4. We may resolve (9.18) as\n\n\n\n(9.19)\n\nwhere each column is an (augmented) injective resolution of the object appearing at its head, and\n\n\\[ {K}_{r, s} \rightarrowtail {J}_{r, s} \rightarrow {L}_{r +... | Proof. We already know (Lemma III. 5.4; see also the proof of Theorem IV.6.1) how to resolve \\( {Z}_{0} \rightarrowtail {F}_{0} \rightarrow {B}_{1} \\) . Given the resolution of \\( {B}_{1} \\), we choose an arbitrary resolution of \\( {Z}_{1}/{B}_{1} \\) and resolve \\( {B}_{1} \mapsto {Z}_{1} \rightarrow {Z}_{1}/{B}... | Yes |
Theorem 9.5 (Lyndon-Hochschild-Serre). Given the short exact sequence of groups\n\n\\[ \nN\\overset{i}{ \\rightarrow }K\\overset{p}{ \\rightarrow }Q \n\\]\n\nand a K-module \\( A \\), there is a natural action of \\( Q \\) on the cohomology groups \\( {H}^{m}\\left( {N, A}\\right) \\) . Moreover, there is a spectral se... | Proof. We must first verify the hypotheses of Theorem 9.3 for the functors \\( F \\) and \\( G \\) of (9.24). We have already remarked that \\( F \\) and \\( G \\) are additive, so it remains to show that, if \\( I \\) is an injective \\( K \\) -module, then \\( {I}^{N} \\) is \\( G \\) -acyclic; in fact, we show that ... | Yes |
Proposition 1.2. A closed class of epimorphisms contains every projection \( \pi : A \oplus B \rightarrow A \) . | Proof. Every object is projective rel \( \pi \) . | No |
Theorem 1.3. Let \( \mathbf{K} : \cdots \rightarrow {K}_{n} \rightarrow {K}_{n - 1} \rightarrow \cdots \rightarrow {K}_{0} \), and\n\n\[ L : \cdots \rightarrow {L}_{n} \rightarrow {L}_{n - 1} \rightarrow \cdots \rightarrow {L}_{0} \]\n\nbe two complexes in \( \mathfrak{A} \) . If \( \mathbf{K} \) is \( \mathcal{E} \) -... | 1.2. Prove Theorem 1.3. | No |
Theorem 2.1. Let \( 0 \rightarrow {A}^{\prime } \rightarrow A \rightarrow {A}^{\prime \prime } \rightarrow 0 \) be a short E-exact sequence in \( \mathfrak{A} \) . Then, for any additive functor \( T : \mathfrak{A} \rightarrow \mathfrak{B} \) there exist connecting homomorphisms \( {\omega }_{n} : {L}_{n}^{\varepsilon ... | Proof. As in the absolute case, the proof hinges on the following key lemma. | No |
Proposition 2.4. A right E-exact functor is additive. | Proof. Since zero objects of \( \mathfrak{A} \) are precisely those \( A \) such that \( A\overset{1}{ \rightarrow }A\overset{1}{ \rightarrow }A \rightarrow 0 \) is exact (and hence \( \mathcal{E} \) -exact), it follows that if \( T \) is right \( \mathcal{E} \) -exact then \( T\left( 0\right) = 0 \) . The proof is now... | No |
Proposition 2.6. For any additive functor \( T,{L}_{0}T \) is right \( \mathcal{E} \) -exact. | Proof. Apply Proposition 2.5 and Theorem 2.1. | No |
Theorem 2.7. For any additive functor \( T : \mathfrak{A} \rightarrow \mathfrak{B} \) there is a natural transformation \( \tau : {L}_{0}T \rightarrow T \) which is an equivalence if, and only if, \( T \) is right E-exact. | Proof. Let \( \cdots \rightarrow {P}_{1} \rightarrow {P}_{0} \) be an \( \mathcal{E} \) -projective resolution of \( A \) . Then\n\n\[ T{P}_{1} \rightarrow T{P}_{0} \rightarrow {L}_{0}T\left( A\right) \rightarrow 0 \]\n\n is exact, by definition; and\n\n\[ T{P}_{1} \rightarrow T{P}_{0} \rightarrow {TA} \]\n\n is differ... | Yes |
Proposition 3.1. If \( T \) is an \( \mathcal{E} \) -connected sequence of functors, the sequence\n\n\[ \cdots \rightarrow {T}_{j}\left( {A}^{\prime }\right) \rightarrow {T}_{j}\left( A\right) \rightarrow {T}_{j}\left( {A}^{\prime \prime }\right) \overset{{\omega }_{j}}{ \rightarrow }{T}_{j - 1}\left( {A}^{\prime }\rig... | Proof. The sequence is certainly differential at \( {T}_{i}\left( A\right) \) . That it is differential at \( {T}_{j}\left( {A}^{\prime \prime }\right) \) follows by naturality from the diagram\n\n\n\nthat it is diff... | Yes |
Theorem 3.2. If \( H : \mathfrak{A} \rightarrow \mathfrak{B} \) is a right \( \mathcal{E} \) -exact functor, where \( \mathcal{E} \) is a projective class of epimorphisms in \( \mathfrak{A} \), then the \( \mathcal{E} \) -connected sequence of functors \( \left\{ {{L}_{j}^{\mathcal{E}}H}\right\} \) is the left \( \math... | Proof. Since \( H \) is right \( \mathcal{E} \) -exact, \( H = {L}_{0}H \) . Thus we suppose given an \( \mathcal{E} \) -connected sequence of functors \( \mathbf{T} \) and a natural transformation \( \varphi : {T}_{0} \rightarrow {L}_{0}H \) and have to show that there exist unique natural transformations \( {\varphi ... | Yes |
Lemma 3.3. Let\n\n\n\nbe a morphism of \( \\mathcal{E} \) -projective presentations. Let \( {\\varphi }_{j + 1} : {T}_{j + 1}A \\rightarrow {L}_{j + 1}{HA} \) be defined by means of the top row, \( {\\varphi }_{j + 1... | Proof of Lemma. Embed (3.2) in the cube\n\n\n\nAll remaining faces of the cube commute and \( {\\omega }_{j + 1} : {L}_{j + 1}H{A}^{\\prime } \\rightarrow {L}_{j}H{K}^{\\prime } \) is monomorphic. Thus the face (3.2)... | Yes |
Corollary 3.4. If \( H : \mathfrak{A} \rightarrow \mathfrak{B} \) is an additive functor, then\n\n\[ \n{S}_{j}^{\varepsilon }\left( {{L}_{0}H}\right) = {L}_{j}^{\varepsilon }H,\;j \geqq 0.\n\] | Proof. It is sufficient to observe that \( {L}_{0}H \) is right \( \mathcal{E} \) -exact and \( {L}_{j}{L}_{0}H = {L}_{j}H \), since \( {L}_{0}{HP} = {HP} \) for any \( \mathcal{E} \) -projective \( P \) . | Yes |
Theorem 3.5. If \( H : \mathfrak{A} \rightarrow \mathfrak{B} \) is a left \( \mathcal{M} \) -exact functor, where \( \mathcal{M} \) is an injective class of monomorphisms in \( \mathfrak{A} \), then the \( \mathcal{M} \) -connected sequence of functors \( \left\{ {{R}_{j}^{\mathcal{M}}H}\right\} \) is the right \( \mat... | \[ {R}_{j}^{\mathcal{M}}H = {S}_{j}^{\mathcal{M}}H \] | Yes |
Proposition 5.2. If \( J : \mathfrak{U} \rightarrow \mathfrak{V} \) is a full embedding, then, for anv \( T : \mathfrak{U} \rightarrow \mathfrak{A},\widetilde{J}T \) does extend \( T \) in the sense that \( \left( {\widetilde{J}T}\right) J = T \) . | Proof. Let \( U \in \mathfrak{U} \) and consider the category \( {\mathfrak{J}}_{JU} \) . There is a subcategory of \( {\mathfrak{J}}_{JU} \) consisting of just the object \( \left( {U,1}\right) \) and its identity morphism. Now, given any object \( \left( {{U}_{1},{\psi }_{1}}\right) \) of \( {\mathfrak{J}}_{JU} \), t... | Yes |
Proposition 5.3. The Kan extension \( {\widetilde{K}}_{V} : \mathfrak{A} \rightarrow \left\lbrack {\mathfrak{V},\mathfrak{A}}\right\rbrack \) is given by\n\n\[ \left( {{\widetilde{K}}_{V}A}\right) {V}^{\prime } = \mathop{\coprod }\limits_{{v \in \mathfrak{B}\left( {V,{V}^{\prime }}\right) }}{A}_{v},\;{A}_{v} = A, \]\n\... | Proof. Of course it is possible to prove the implied adjointness relation directly. However, we shall apply the general construction of Theorem 5.1. So let \( J = {K}_{V} \), then \( {\mathfrak{J}}_{V} \) is the category with objects\n\n\[ \left( {\mathbf{1}, v}\right) = v : J\left( \mathbf{1}\right) = V \rightarrow {V... | Yes |
Lemma 5.5. Let \( {F}_{i} : {\mathfrak{C}}_{i} \rightarrow \mathfrak{D} \) be a left adjoint of \( {G}_{i} : \mathfrak{D} \rightarrow {\mathfrak{C}}_{i} \), and suppose that \( \mathfrak{D} \) has coproducts. Define \( G : \mathfrak{D} \rightarrow \mathop{\prod }\limits_{i}{\mathfrak{C}}_{i} \) by \( {GD} = \left\{ {{G... | \[ \text{Proof.}\mathfrak{D}\left( {F\left\{ {C}_{i}\right\}, D}\right) = \mathfrak{D}\left( {\mathop{\coprod }\limits_{i}{F}_{i}{C}_{i}, D}\right) = \mathop{\prod }\limits_{i}\mathfrak{D}\left( {{F}_{i}{C}_{i}, D}\right) \cong \mathop{\prod }\limits_{i}{\mathfrak{C}}_{i}\left( {{C}_{i},{G}_{i}D}\right) \] \[ = \left( ... | Yes |
Theorem 5.6. If \( \mathfrak{A} \) has enough projectives then the J-homology\n\n\[ \n{H}_{ * }\left( {J, - }\right) : \left\lbrack {\mathfrak{U},\mathfrak{A}}\right\rbrack \rightarrow \left\lbrack {\mathfrak{B},\mathfrak{A}}\right\rbrack \n\]\n\nexists. It may be computed as the left \( {\mathcal{E}}_{1}^{\prime } \) ... | (5.12)\n\nWe remark that Theorem 4.1 and Corollary 5.4 yield the form of the projectives. A functor \( S : \mathfrak{U} \rightarrow \mathfrak{A} \) is, by the last part of Theorem 4.1, \( {\mathcal{E}}_{1}^{\prime } \) -projective if and only if it is a direct summand of a functor \( \widetilde{I}\left\{ {P}_{U}\right\... | Yes |
Proposition 5.8. Let \( \mathfrak{A} \) have enough projectives and exact coproducts. If \( R \in \left\lbrack {\mathfrak{U},\mathfrak{A}}\right\rbrack \) is an \( {\mathcal{E}}_{0}^{\prime } \) -projective functor, then \( \left( {{L}_{n}^{{\mathcal{E}}_{1}^{\prime }}\widetilde{J}}\right) R = 0 \) for \( n \geqq 1 \) ... | Proof. Clearly every functor \( {\mathfrak{U}}_{d} \rightarrow \mathfrak{A} \) is \( {\mathcal{E}}_{0} \) -projective. Thus, since \( \left( {{L}_{n}^{{\mathcal{E}}_{1}}\widetilde{J}}\right) \) is additive and \( \mathfrak{Y} \) has exact coproducts, it is enough, by Theorem 4.1 and Corollary 5.4, to prove the assertio... | Yes |
Theorem 6.1. Let \( J : \mathfrak{U} \rightarrow \mathfrak{V}, I : \mathfrak{V} \rightarrow \mathfrak{W} \) be two functors between small categories, and let \( \mathfrak{A} \) be an abelian category with colimits and enough projectives. Then there is a spectral sequence\n\n\[ \n{E}_{1}^{p, q} = {H}_{p}\left( {I,{H}_{q... | Proof. We only have to show that projectives in \( \left\lbrack {\mathfrak{U},\mathfrak{A}}\right\rbrack \) are transformed by \( \widetilde{J} \) into \( \widetilde{I} \) -acyclic objects in \( \left\lbrack {\mathfrak{V},\mathfrak{A}}\right\rbrack \) . Since \( \widetilde{J} \) is additive it is enough to check this c... | No |
Theorem 1.1. If \( \pi \) has an element of finite order, then there can be no finite-dimensional model for \( K\left( {\pi ,1}\right) \) . | Proof. First consider \( K\left( {{C}_{m},1}\right) \), where \( {C}_{m} \) is a cyclic group of order \( m \geq 2 \) . Since \( \mathbf{C}\left( {\widetilde{K}\left( {{C}_{m},1}\right) }\right) \) is a free \( {C}_{m} \) -resolution of \( \mathbb{Z} \), and since \( {C}_{m} \) has nonzero integral homology groups in a... | Yes |
Proposition 2.1. Let \( N \) be a nilpotent group. Then there is a P-local group \( {N}_{P} \) and a map \( l : N \rightarrow {N}_{P} \) which P-localizes; and nil \( {N}_{P} \leq \) nil \( N \) . Moreover, a homomorphism \( \widetilde{l} : N \rightarrow M \) between nilpotent groups P-localizes if and only if the indu... | The group \( {N}_{P} \), as well as the map \( l \), may be constructed, as we have said, by induction on the nilpotency class of the group \( N \) . To start the induction, one has to show that, for an abelian group \( A \), the map \( l : A \rightarrow {\mathbb{Z}}_{P} \otimes A \) induces localizing maps \( {l}_{n} ... | Yes |
Theorem 4.3 (Alperin, Evens; Carlson). The complexity \( {c}_{G}\left( M\right) \) of a module \( M \) is the maximum of the complexities \( {c}_{E}\left( {M}_{E}\right) \) of the restrictions \( {M}_{E} \) of \( M \) to the elementary abelian p-subgroups \( E \) of \( G \) . | It follows immediately from the definition of the complexity that \( {c}_{G}\left( M\right) = 0 \) if and only if the module \( M \) is \( {kG} \) -projective. The above theorem thus implies as corollary the following surprising result, which had actually been obtained earlier: | No |
Corollary 4.4 (Chouinard). The module \( M \) is \( {kG} \) -projective if and only if, for every elementary abelian p-subgroup of \( G \), the restriction \( {M}_{E} \) is \( {kE} \) -projective. | It is obvious now that the class of indecomposable modules can be subdivided according to complexity. In this direction much work has already been done, and it remains an area of intensive research. | No |
Theorem 1.1. If \( \left( {{\omega }_{1},{\omega }_{2}}\right) \) is a fundamental pair of periods, then the triangle with vertices \( 0,{\omega }_{1},{\omega }_{2} \) contains no further periods in its interior or on its boundary. Conversely, any pair of periods with this property is fundamental. | Proof. Consider the parallelogram with vertices \( 0,{\omega }_{1},{\omega }_{1} + {\omega }_{2} \), and \( {\omega }_{2} \) , shown in Figure 1.2a. The points inside or on the boundary of this parallelogram have the form\n\n\[ z = \alpha {\omega }_{1} + \beta {\omega }_{2} \]\n\nwhere \( 0 \leq \alpha \leq 1 \) and \(... | Yes |
Theorem 1.3. A nonconstant elliptic function has a fundamental pair of periods. | Proof. If \( f \) is elliptic the set of points where \( f \) is analytic is an open connected set. Also, \( f \) has two periods with nonreal ratio. Among all the nonzero periods of \( f \) there is at least one whose distance from the origin is minimal (otherwise \( f \) would have arbitrarily small nonzero periods a... | Yes |
Theorem 1.4. If an elliptic function f has no poles in some period parallelogram, then \( f \) is constant. | Proof. If \( f \) has no poles in a period parallelogram, then \( f \) is continuous and hence bounded on the closure of the parallelogram. By periodicity, \( f \) is bounded in the whole plane. Hence, by Liouville’s theorem, \( f \) is constant. | Yes |
Theorem 1.5. If an elliptic function f has no zeros in some period parallelogram, then \( f \) is constant. | Proof. Apply Theorem 1.4 to the reciprocal \( 1/f \) . | Yes |
Theorem 1.6. The contour integral of an elliptic function taken along the boundary of any cell is zero. | Proof. The integrals along parallel edges cancel because of periodicity. | No |
Theorem 1.7. The sum of the residues of an elliptic function at its poles in any period parallelogram is zero. | Proof. Apply Cauchy's residue theorem to a cell and use Theorem 1.6. | No |
Theorem 1.8. The number of zeros of an elliptic function in any period parallelogram is equal to the number of poles, each counted with multiplicity. | Proof. The integral\n\n\[ \n\frac{1}{2\pi i}{\int }_{C}\frac{{f}^{\prime }\left( z\right) }{f\left( z\right) }{dz} \]\n\ntaken around the boundary \( C \) of a cell, counts the difference between the number of zeros and the number of poles inside the cell. But \( {f}^{\prime }/f \) is elliptic with the same periods as ... | Yes |
Lemma 1. If \( \alpha \) is real the infinite series\n\n\[ \mathop{\sum }\limits_{\substack{{\omega \in \Omega } \\ {\omega \neq 0} }}\frac{1}{{\omega }^{\alpha }} \]\n\nconverges absolutely if, and only if, \( \alpha > 2 \) . | Proof. Refer to Figure 1.3 and let \( r \) and \( R \) denote, respectively, the minimum and maximum distances from 0 to the parallelogram shown. If \( \omega \) is any of the 8 nonzero periods shown in this diagram we have\n\n\[ r \leq \left| \omega \right| \leq R\;\text{ (for }8\text{ periods }\omega \text{ ). } \]\n... | Yes |
Lemma 2. If \( \\alpha > 2 \) and \( R > 0 \) the series\n\n\[ \n\\mathop{\\sum }\\limits_{{\\left| \\omega \\right| > R}}\\frac{1}{{\\left( z - \\omega \\right) }^{\\alpha }}\n\]\n\nconverges absolutely and uniformly in the disk \( \\left| z\\right| \\leq R \) . | Proof. We will show that there is a constant \( M \) (depending on \( R \) and \( \\alpha \) ) such that, if \( \\alpha \\geq 1 \), we have\n\n(1)\n\n\[ \n\\frac{1}{{\\left| z - \\omega \\right| }^{\\alpha }} \\leq \\frac{M}{{\\left| \\omega \\right| }^{\\alpha }}\n\]\n\nfor all \( \\omega \) with \( \\left| \\omega \\... | Yes |
Theorem 1.9. Let \( f \) be defined by the series\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{\omega \in \Omega }}\frac{1}{{\left( z - \omega \right) }^{3}}.\]\n\nThen \( f \) is an elliptic function with periods \( {\omega }_{1},{\omega }_{2} \) and with a pole of order 3 at each period \( \omega \) in \( \Omega ... | Proof. By Lemma 2 the series obtained by summing over \( \left| \omega \right| > R \) converges uniformly in the disk \( \left| z\right| \leq R \) . Therefore it represents an analytic function in this disk. The remaining terms, which are finite in number, are also analytic in this disk except for a 3rd order pole at e... | Yes |
Theorem 1.10. The function \( \wp \) so defined has periods \( {\omega }_{1} \) and \( {\omega }_{2} \). It is analytic except for a double pole at each period \( \omega \) in \( \Omega \). Moreover \( \wp \left( z\right) \) is an even function of \( z \). | Proof. Each term in the series has modulus\n\n\[\n\left| {\frac{1}{{\left( z - \omega \right) }^{2}} - \frac{1}{{\omega }^{2}}}\right| = \left| \frac{{\omega }^{2} - {\left( z - \omega \right) }^{2}}{{\omega }^{2}{\left( z - \omega \right) }^{2}}\right| = \left| \frac{z\left( {{2\omega } - z}\right) }{{\omega }^{2}{\le... | Yes |
Theorem 1.11. Let \( r = \min \{ \left| \omega \right| : \omega \neq 0\} \) . Then for \( 0 < \left| z\right| < r \) we have\n\n\[ \wp \left( z\right) = \frac{1}{{z}^{2}} + \mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {{2n} + 1}\right) {G}_{{2n} + 2}{z}^{2n} \]\n\nwhere\n\n\[ {G}_{n} = \mathop{\sum }\limits_{{\omega... | Proof. If \( 0 < \left| z\right| < r \) then \( \left| {z/\omega }\right| < 1 \) and we have\n\n\[ \frac{1}{{\left( z - \omega \right) }^{2}} = \frac{1}{{\omega }^{2}{\left( 1 - \frac{z}{\omega }\right) }^{2}} = \frac{1}{{\omega }^{2}}\left( {1 + \mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {n + 1}\right) {\left( \f... | Yes |
Theorem 1.12. The function \( \wp \) satisfies the nonlinear differential equation\n\n\[ \n{\left\lbrack {\wp }^{\prime }\left( z\right) \right\rbrack }^{2} = 4{\wp }^{3}\left( z\right) - {60}{G}_{4}\wp \left( z\right) - {140}{G}_{6}.\n\] | Proof. We obtain this by forming a linear combination of powers of \( \wp \) and \( {\wp }^{\prime } \) which eliminates the pole at \( z = 0 \) . This gives an elliptic function which has\n\nno poles and must therefore be constant. Near \( z = 0 \) we have\n\n\[ \n{\wp }^{\prime }\left( z\right) = - \frac{2}{{z}^{3}} ... | Yes |
Theorem 1.13. Each Eisenstein series \( {G}_{n} \) is expressible as a polynomial in \( {g}_{2} \) and \( {g}_{3} \) with positive rational coefficients. In fact, if \( b\left( n\right) = \left( {{2n} + 1}\right) {G}_{{2n} + 2} \) we have the recursion relations\n\n\[ \nb\left( 1\right) = {g}_{2}/{20},\;b\left( 2\right... | Proof. Differentiation of the differential equation for \( \wp \) gives another differential equation of second order satisfied by \( \wp \) ,\n\n(5)\n\n\[ \n{\wp }^{\prime \prime }\left( z\right) = 6{\wp }^{2}\left( z\right) - \frac{1}{2}{g}_{2} \n\]\n\nNow we write \( \wp \left( z\right) = {z}^{-2} + \mathop{\sum }\l... | Yes |
Theorem 1.14. We have\n\n\\[ 4{\\wp }^{3}\\left( z\\right) - {g}_{2}\\wp \\left( z\\right) - {g}_{3} = 4\\left( {\\wp \\left( z\\right) - {e}_{1}}\\right) \\left( {\\wp \\left( z\\right) - {e}_{2}}\\right) \\left( {\\wp \\left( z\\right) - {e}_{3}}\\right) . \\]\n\nMoreover, the roots \\( {e}_{1},{e}_{2},{e}_{3} \\) ar... | Proof. Since \\( \\wp \\) is even, the derivative \\( {\\wp }^{\\prime } \\) is odd. But it is easy to show that the half-periods of an odd elliptic function are either zeros or poles. In fact, by periodicity we have \\( {\\wp }^{\\prime }\\left( {-\\frac{1}{2}\\omega }\\right) = {\\wp }^{\\prime }\\left( {\\omega - \\... | Yes |
Theorem 1.15. The functions \( {g}_{2}\left( \tau \right) ,{g}_{3}\left( \tau \right) ,\Delta \left( \tau \right) \), and \( J\left( \tau \right) \) are analytic in \( H \) . | Proof. Since \( \Delta \left( \tau \right) \neq 0 \) in \( H \) it suffices to prove that \( {g}_{2} \) and \( {g}_{3} \) are analytic in \( H \) . Both \( {g}_{2} \) and \( {g}_{3} \) are given by double series of the form\n\n\[ \mathop{\sum }\limits_{\substack{{m, n = - \infty } \\ {\left( {m, n}\right) \neq \left( {... | Yes |
Theorem 1.16. If \( \tau \in H \) and \( a, b, c, d \) are integers with \( {ad} - {bc} = 1 \), then \( \left( {{a\tau } + b}\right) /\left( {{c\tau } + d}\right) \in H \) and | \[ J\left( \frac{{a\tau } + b}{{c\tau } + d}\right) = J\left( \tau \right) \] | No |
Theorem 1.17. If \( \tau \in H, J\left( \tau \right) \) can be represented by an absolutely convergent Fourier series | Proof. Introduce the change of variable\n\n\[ x = {e}^{2\pi i\tau } \]\n\nThen the upper half-plane \( H \) maps into the punctured unit disk\n\n\[ D = \{ x : 0 < \left| x\right| < 1\} .\n\n(See Figure 1.5.) Each \( \tau \) in \( H \) maps onto a unique point \( x \) in \( D \), but each \( x \) in \( D \) is the image... | Yes |
Lemma 3. If \( \tau \in H \) and \( n > 0 \) we have the Fourier expansions\n\n\[ \mathop{\sum }\limits_{{m = - \infty }}^{{+\infty }}\frac{1}{{\left( m + n\tau \right) }^{4}} = \frac{8{\pi }^{4}}{3}\mathop{\sum }\limits_{{r = 1}}^{\infty }{r}^{3}{e}^{2\pi irn\tau } \]\n\nand\n\n\[ \mathop{\sum }\limits_{{m = - \infty ... | Proof. Start with the partial fraction decomposition of the cotangent:\n\n\[ \pi \cot {\pi \tau } = \frac{1}{\tau } + \mathop{\sum }\limits_{\substack{{m = - \infty } \\ {m \neq 0} }}^{{+\infty }}\left( {\frac{1}{\tau + m} - \frac{1}{m}}\right) . \]\n\nLet \( x = {e}^{2\pi i\tau } \) . If \( \tau \in H \) then \( \left... | Yes |
Theorem 1.18. If \( \tau \in H \) we have the Fourier expansions\n\n\[ \n{g}_{2}\left( \tau \right) = \frac{4{\pi }^{4}}{3}\left\{ {1 + {240}\mathop{\sum }\limits_{{k = 1}}^{\infty }{\sigma }_{3}\left( k\right) {e}^{2\pi ik\tau }}\right\} \n\]\n\nand\n\n\[ \n{g}_{3}\left( \tau \right) = \frac{8{\pi }^{6}}{27}\left\{ {1... | Proof. We write\n\n\[ \n{g}_{2}\left( \tau \right) = {60}\mathop{\sum }\limits_{\substack{{m, n = - \infty } \\ {\left( {m, n}\right) \neq \left( {0,0}\right) } }}^{{+\infty }}\frac{1}{{\left( m + n\tau \right) }^{4}} \n\]\n\n\[ \n= {60}\left\{ {\mathop{\sum }\limits_{\substack{{m = - \infty } \\ {m \neq 0\left( {n = 0... | Yes |
Theorem 1.19. If \( \tau \in H \) we have the Fourier expansion\n\n\[ \Delta \left( \tau \right) = {\left( 2\pi \right) }^{12}\mathop{\sum }\limits_{{n = 1}}^{\infty }\tau \left( n\right) {e}^{2\pi in\tau } \]\n\nwhere the coefficients \( \tau \left( n\right) \) are integers, with \( \tau \left( 1\right) = 1 \) and \( ... | Proof. Let\n\n\[ x = {e}^{2\pi i\tau },\;A = \mathop{\sum }\limits_{{n = 1}}^{\infty }{\sigma }_{3}\left( n\right) {x}^{n},\;B = \mathop{\sum }\limits_{{n = 1}}^{\infty }{\sigma }_{5}\left( n\right) {x}^{n}. \]\n\nThen\n\n\[ \Delta \left( \tau \right) = {g}_{2}{}^{3}\left( \tau \right) - {27}{g}_{3}{}^{2}\left( \tau \r... | Yes |
Theorem 1.20. If \( \tau \in H \) we have the Fourier expansion\n\n\[ \n{12}^{3}J\left( \tau \right) = {e}^{-{2\pi i\tau }} + {744} + \mathop{\sum }\limits_{{n = 1}}^{\infty }c\left( n\right) {e}^{2\pi in\tau },\n\]\n\nwhere the \( c\left( n\right) \) are integers. | Proof. We agree to write \( I \) for any power series in \( x \) with integer coefficients. Then if \( x = {e}^{2\pi it} \) we have\n\n\[ \n{g}_{2}{}^{3}\left( \tau \right) = \frac{64}{27}{\pi }^{12}{\left( 1 + {240}x + I\right) }^{3} = \frac{64}{27}{\pi }^{12}\left( {1 + {720x} + I}\right) ,\n\]\n\n\[ \n\Delta \left( ... | Yes |
Lemma 1. Given \( {\omega }_{1}{}^{\prime },{\omega }_{2}{}^{\prime } \) with \( {\omega }_{2}{}^{\prime }/{\omega }_{1}{}^{\prime } \) not real, let\n\n\[ \n\Omega = \left\{ {m{\omega }_{1}{}^{\prime } + n{\omega }_{2}{}^{\prime } : m, n\text{ integers }}\right\} .\n\]\n\nThen there exists a fundamental pair \( \left(... | Proof. We arrange the elements of \( \Omega \) in a sequence according to increasing distances from the origin, say\n\n\[ \n\Omega = \left\{ {0,{w}_{1},{w}_{2},\ldots }\right\} \n\]\n\nwhere\n\n\[ \n0 < \left| {w}_{1}\right| \leq \left| {w}_{2}\right| \leq \cdots \text{ and }\arg {w}_{n} < \arg {w}_{n + 1}\text{ if }\l... | Yes |
Theorem 2.2. If \( {\tau }^{\prime } \in H \), there exists a complex number \( \tau \) in \( H \) equivalent to \( {\tau }^{\prime } \) under \( \Gamma \) such that\n\n\[ \left| \tau \right| \geq 1,\;\left| {\tau + 1}\right| \geq \left| \tau \right| \;\text{ and }\;\left| {\tau - 1}\right| \geq \left| \tau \right| . \... | Proof. Let \( {\omega }_{1}{}^{\prime } = 1,{\omega }_{2}{}^{\prime } = {\tau }^{\prime } \) and apply Lemma 1 to the set of periods \( \Omega = \left\{ {m + n{\tau }^{\prime } : m, n}\right. \) integers \( \} \) . Then there exists a fundamental pair \( {\omega }_{1},{\omega }_{2} \) with \( \left| {\omega }_{2}\right... | Yes |
Theorem 2.3. The open set\n\n\[ \n{R}_{\Gamma } = \{ \tau \in H : \left| \tau \right| > 1,\left| {\tau + \bar{\tau }}\right| < 1\}\n\]\n\nis a fundamental region for \( \Gamma \) . Moreover, if \( A \in \Gamma \) and if \( {A\tau } = \tau \) for some \( \tau \) in \( {R}_{\Gamma } \), then \( A = I \) . In other words,... | Proof. Theorem 2.2 shows that if \( {\tau }^{\prime } \in H \) there is a point \( \tau \) in the closure of \( {R}_{\Gamma } \) equivalent to \( {\tau }^{\prime } \) under \( \Gamma \) . To prove that no two distinct points of \( {R}_{\Gamma } \) are equivalent under \( \Gamma \), let \( {\tau }^{\prime } = {A\tau } \... | Yes |
Theorem 2.4. If f is modular and not identically zero, then in the closure of the fundamental region \( {R}_{\Gamma } \), the number of zeros of \( f \) is equal to the number of poles. | Proof. Assume first that \( f \) has no zeros or poles on the finite part of the boundary of \( {R}_{\Gamma } \) . Cut \( {R}_{\Gamma } \) by a horizontal line, \( \operatorname{Im}\left( \tau \right) = M \), where \( M > 0 \) is taken so large that all the zeros or poles of \( f \) are inside the truncated region whic... | Yes |
Theorem 2.5. If f is modular and not constant, then for every complex \( c \) the function \( f - c \) has the same number of zeros as poles in the closure of \( {R}_{\Gamma } \) . In other words, \( f \) takes on every value equally often in the closure of \( {R}_{\Gamma } \) . | Proof. Apply the previous theorem to \( f - c \) . | No |
Theorem 2.6. Iff is modular and bounded in \( H \) then \( f \) is constant. | Proof. Since \( f \) is bounded it omits a value so \( f \) is constant. | No |
Theorem 2.7. The function \( J \) takes every value exactly once in the closure of \( {R}_{\Gamma } \) . In particular, at the vertices we have\n\n\[ J\left( \rho \right) = 0,\;J\left( i\right) = 1,\;J\left( {i\infty }\right) = \infty . \]\n\nThere is a first order pole at \( i\infty \), a triple zero at \( \rho \), an... | Proof. First we verify that \( {g}_{2}\left( \rho \right) = 0 \) and \( {g}_{3}\left( i\right) = 0 \) . Since \( {\rho }^{3} = 1 \) and \( {\rho }^{2} + \rho + 1 = 0 \) we have\n\n\[ \frac{1}{60}{g}_{2}\left( \rho \right) = \mathop{\sum }\limits_{{m, n}}\frac{1}{{\left( m + n\rho \right) }^{4}} = \mathop{\sum }\limits_... | Yes |
Theorem 2.8. Every rational function of \( J \) is a modular function. Conversely, every modular function can be expressed as a rational function of \( J \) . | Proof. The first part is clear. To prove the second, suppose \( f \) has zeros at \( {z}_{1},\ldots ,{z}_{n} \) and poles at \( {p}_{1},\ldots ,{p}_{n} \) with the usual conventions about multiplicities. Let\n\n\[ g\left( \tau \right) = \mathop{\prod }\limits_{{k = 1}}^{n}\frac{J\left( \tau \right) - J\left( {z}_{k}\ri... | Yes |
Theorem 2.10. Every nonconstant entire function attains every complex value with at most one exception. | Proof. We assume \( f \) is an entire function which omits two values, say \( a \) and \( b, a \neq b \), and show that \( f \) is constant. Let\n\n\[ g\left( z\right) = \frac{f\left( z\right) - a}{b - a}. \]\n\nThen \( g \) is entire and omits the values 0 and 1 .\n\nThe upper half-plane \( H \) is covered by the imag... | Yes |
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