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Corollary 1.7. Let \( \mathrm{T} : \mathcal{C} \rightarrow \mathcal{S} \) be a covariant functor from a category \( \mathcal{C} \) to the category S of sets. If \( \left( {\mathrm{A},\alpha }\right) \) and \( \left( {\mathrm{B},\beta }\right) \) are representations of \( \mathrm{T} \), then there is a unique equivalenc... | PROOF. Let \( u = {\alpha }_{A}\left( {1}_{A}\right) \) and \( v = {\beta }_{B}\left( {1}_{B}\right) \) . By Theorem 1.6 \( \left( {A, u}\right) \) and \( \left( {B, v}\right) \) are universal elements of \( T \), whence by Lemma I.7.10 there is a unique equivalence \( f : A \rightarrow B \) in \( \mathcal{C} \) such t... | Yes |
Corollary 1.8. (Yoneda) Let \( \\mathrm{T} : \\mathcal{C} \\rightarrow \\mathcal{S} \) be a covariant functor from a category \( \\mathcal{C} \) to the category \( \\mathcal{S} \) of sets and let \( \\mathrm{A} \) be an object of \( \\mathcal{C} \) . Then there is a one-to-one correspondence between the set \( \\mathrm... | SKETCH OF PROOF. Define a function \( \\psi = {\\psi }_{A} : \\operatorname{Nat}\\left( {{h}_{A}, T}\\right) \\rightarrow T\\left( A\\right) \) by\n\n\[ \n\\alpha \\mapsto {\\alpha }_{A}\\left( {1}_{A}\\right) \\in T\\left( A\\right) \n\] \n\nand a function \( \\phi : T\\left( A\\right) \\rightarrow \\operatorname{Nat}... | No |
Theorem 1.9. Let \( \mathcal{C} \) and \( \mathcal{D} \) be categories and \( \mathbf{T} \) a functor from the product category \( \operatorname{\mathcal{C} } \times \operatorname{\mathcal{D} } \) to the category \( \operatorname{\mathsf{S} } \) of sets, contravariant in the first variable and covariant in the second, ... | PROOF OF 1.9. The object function of the functor \( S \) is defined by \( S\left( C\right) = {A}_{C} \) for each object \( C \) of \( \mathcal{C} \) . The morphism function of \( S \) is defined as follows. For each object \( C \) of \( \& {\alpha }^{C}{}_{AC} : {\hom }_{\mathfrak{D}}\left( {{A}_{C},{A}_{C}}\right) \ri... | Yes |
Proposition 2.2. A covariant functor \( \mathrm{T} : \mathfrak{D} \rightarrow \mathcal{C} \) has a left adjoint if and only if for each object \( \mathrm{C} \) in \( \mathcal{C} \) the functor home \( \left( {\mathrm{C},\mathrm{T}\left( -\right) }\right) : \mathfrak{D} \rightarrow \mathcal{S} \) is representable. | PROOF. If \( S : \mathcal{C} \rightarrow \mathfrak{D} \) is a left adjoint of \( T \), then there is for each object \( C \) of \( \mathcal{C} \) and \( D \) of \( \mathfrak{D} \) a bijection\n\n\[{\alpha }_{C, D} : {\hom }_{\mathfrak{D}}\left( {S\left( C\right), D}\right) \rightarrow {\hom }_{\mathfrak{C}}\left( {C, T... | Yes |
Corollary 2.3. A covariant functor \( \mathrm{T} : \mathfrak{D} \rightarrow \mathcal{C} \) has a left adjoint if and only if there exists for each object \( \mathrm{C} \) of \( \mathcal{C} \) an object \( \mathrm{S}\left( \mathrm{C}\right) \) of \( \mathfrak{D} \) and a morphism \( {\mathrm{u}}_{\mathrm{C}} : \mathrm{C... | PROOF. Exercise; see Theorem 1.6. ∎ | No |
Corollary 2.4. Any two left adjoints of a covariant functor \( \mathrm{T} : \mathfrak{D} \rightarrow \mathcal{C} \) are naturally isomorphic. | PROOF. If \( {S}_{1} : \mathcal{C} \rightarrow \mathcal{D} \) and \( {S}_{2} : \mathcal{C} \rightarrow \mathcal{D} \) are left adjoints of \( T \), then there are natural isomorphisms\n\n\[ \alpha : {\hom }_{\mathfrak{D}}\left( {{S}_{1}\left( -\right) , - }\right) \rightarrow {\hom }_{\mathfrak{e}}\left( {-, T\left( -\... | Yes |
Proposition 3.2. Let \( \mathrm{f} : \mathrm{B} \rightarrow \mathrm{C} \) and \( \mathrm{g} : \mathrm{C} \rightarrow \mathrm{D} \) be morphisms of a category \( \mathrm{C} \) . (i) \( \mathrm{f} \) and \( \mathrm{g} \) monic \( \Rightarrow \mathrm{{gf}} \) monic; (ii) gf monic \( \Rightarrow \mathrm{f} \) monic; (iii) ... | PROOF. Exercise. | No |
Proposition 3.4. Let \( \mathcal{C} \) be a category which has a zero object 0 . Then for each pair \( \mathrm{C},\mathrm{D} \) of objects of \( \mathrm{C} \) there is a unique morphism \( {0}_{\mathrm{C},\mathrm{D}} : \mathrm{C} \rightarrow \mathrm{D} \) such that\n\n\[ \mathrm{f} \circ {0}_{\mathrm{C},\mathrm{D}} = {... | PROOF OF 3.4. (Uniqueness) If \( \left\{ {0}_{C, D}^{\prime }\right\} \) and \( \left\{ {0}_{C, D}\right\} \) are two families of morphisms with the stated properties, then for each pair \( C, D \)\n\n\[ {0}_{C, D} = {0}_{D, D}^{\prime }{0}_{C, D} = {0}_{C, D}^{\prime }. \]\n\n(Existence) For each object \( A \) of \( ... | Yes |
Proposition 3.6. Let \( \mathrm{f} : \mathrm{C} \rightarrow \mathrm{D} \) and \( \mathrm{g} : \mathrm{C} \rightarrow \mathrm{D} \) be morphisms of a category \( \mathrm{C} \) . (i) If \( \mathrm{i} : \mathrm{B} \rightarrow \mathrm{C} \) is a difference kernel of \( \left( {\mathrm{f},\mathrm{g}}\right) \), then \( \mat... | PROOF. (i) Let \( h, k : F \rightarrow B \) be morphisms such that \( {ih} = {ik} \) . Then \( f\left( {ih}\right) = \left( {fi}\right) h = \left( {gi}\right) h = g\left( {ih}\right) \) . Since \( i \) is a difference kernel of \( \left( {f, g}\right) \), there is a unique morphism \( t : F \rightarrow B \) such that \... | Yes |
Lemma 1.1. Let \( V \) be a non-zero finite-dimensional sub \( F \) -space of \( {F}^{S} \) . There is a basis \( \left( {{v}_{1},\ldots ,{v}_{n}}\right) \) of \( V \) and a corresponding subset \( \left( {{s}_{1},\ldots ,{s}_{n}}\right) \) of \( S \) such that \( {v}_{i}\left( {s}_{j}\right) = {\delta }_{ij} \) for al... | Proof. Suppose that we have already found elements \( {s}_{1},\ldots ,{s}_{k} \) of \( S \) and a basis \( \left( {{v}_{1, k},\ldots ,{v}_{n, k}}\right) \) of \( V \) such that the \( {v}_{i, k} \) ’s and the \( {s}_{j} \) ’s satisfy the requirements of the lemma for each \( i \) from \( \left( {1,\ldots, n}\right) \) ... | Yes |
Proposition 1.2. The canonical morphism \( \pi : {F}^{S} \otimes {F}^{T} \rightarrow {F}^{S \times T} \) is injective, and its image consists of all functions \( h \) with the property that the \( F \) -space spanned by the partial functions \( {h}_{t} \), where \( t \) ranges over \( T \) and \( {h}_{t}\left( s\right)... | Proof. Let \( \mathop{\sum }\limits_{{j = 1}}^{m}{f}_{j} \otimes {g}_{j} \) be an element of the kernel of \( \pi \), and let \( V \) be the sub \( F \) -space of \( {F}^{S} \) spanned by \( {f}_{1},\ldots ,{f}_{m} \) . If \( V = \left( 0\right) \) then our element is 0 . Otherwise choose \( \left( {{v}_{1},\ldots ,{v}... | Yes |
Proposition 1.3. The image of the morphism of F-algebras\n\n\\[ \n{\\pi }^{-1} \\circ {m}^{ * } : {\\mathcal{R}}_{F}\\left( G\\right) \\rightarrow {F}^{G} \\otimes {F}^{G}\n\\]\n\nactually lies in \\( {\\mathcal{R}}_{F}\\left( G\\right) \\otimes {\\mathcal{R}}_{F}\\left( G\\right) \\) . | Proof. Let \\( f \\) be an element of \\( {\\mathcal{R}}_{F}\\left( G\\right) \\) . Proceeding as in the proof of Proposition 1.2, we find elements \\( {s}_{1},\\ldots ,{s}_{n} \\) in \\( G \\) and elements \\( {v}_{1},\\ldots ,{v}_{n} \\) in \\( {F}^{G} \\) as in Lemma 1.1 such that we may write\n\n\\[ \n{\\pi }^{-1}\... | Yes |
The map \( \tau \mapsto {\tau }_{\mathrm{f}} \) is an injective morphism of \( F \) -algebras from \( {C}^{ \circ } \) to End \( {}_{F}\left( C\right) \), and the map \( \tau \mapsto {\tau }_{1} \) is an injective antimorphism of \( F \) -algebras from \( {C}^{ \circ } \) to \( {\operatorname{End}}_{F}\left( C\right) \... | Throughout the computations below, we identify \( F \otimes F \) with \( F \) , \( C \otimes F \) with \( C, F \otimes C \) with \( C \), etc. Let \( \sigma \) and \( \tau \) be elements of \( {C}^{ \circ } \) . We have\n\n\[ \delta \circ {\sigma }_{\mathrm{f}} = \delta \circ \left( {{i}_{C} \otimes \sigma }\right) \ci... | Yes |
Each of \( {C}_{1}^{ \circ } \) and \( {C}_{1}^{ \circ } \) is the commuting algebra of the other, in \( {\operatorname{End}}_{F}\left( C\right) \) . An element e of \( {\operatorname{End}}_{F}\left( C\right) \) belongs to \( {C}_{\mathrm{I}}^{ \circ } \) if and only if \( \delta \circ e = \left( {{i}_{C} \otimes e}\ri... | We have already shown that the elements of \( {C}_{\mathrm{I}}^{ \circ } \) and \( {C}_{\mathrm{J}}^{ \circ } \) commute with each other. Now suppose that \( e \) is any element of \( {\operatorname{End}}_{F}\left( C\right) \) that commutes with every element of \( {C}_{1}^{ \circ } \) . This means that, for every elem... | Yes |
Proposition 4.1. Let \( G \) be an algebraic group, \( K \) a submonoid of \( G \) . Then the closure of \( K \) in \( G \) is an algebraic subgroup of \( G \) . | Proof. From the fact that \( K \) is a submonoid of \( G \), it follows immediately that \( {J}_{K} \) is a biideal of \( \mathcal{P}\left( G\right) \) . This implies that the annihilator, \( {K}^{\prime } \) say, of \( {J}_{K} \) in \( G \) is a submonoid of \( G \) . Now let \( x \) be an element of \( {K}^{\prime } ... | Yes |
Lemma 1.2. Let \( F \) be a field, \( S \) an irreducible affine algebraic \( F \) -set, \( B \) an integral domain F-algebra. Then \( \mathcal{P}\left( S\right) \otimes B \) is an integral domain. | Proof. Let \( u \) and \( v \) be elements of \( \mathcal{P}\left( S\right) \otimes B \) such that \( {uv} = 0 \) . Write\n\n\[ u = \mathop{\sum }\limits_{{i = 1}}^{n}{u}_{i} \otimes {b}_{i}\;\text{ and }\;v = \mathop{\sum }\limits_{{i = 1}}^{n}{v}_{i} \otimes {b}_{i}, \]\n\nwhere the \( {b}_{i} \) ’s are \( F \) -line... | Yes |
Proposition 1.3. If \( S \) and \( T \) are irreducible algebraic sets, so is \( S \times T \) . | Proof. We know from the beginning of this Section that \( \mathcal{P}\left( S\right) \) and \( \mathcal{P}\left( T\right) \) are integral domains. By Lemma 1.2, it follows that \( \mathcal{P}\left( S\right) \otimes \mathcal{P}\left( T\right) \) is an integral domain. Since this is \( \mathcal{P}\left( {S \times T}\righ... | Yes |
Theorem 1.4. Let \( G \) be an affine algebraic group. The irreducible components of \( G \) are mutually disjoint. The component \( {G}_{1} \) containing the neutral element of \( G \) is a closed normal subgroup of \( G \), and the irreducible components of \( G \) are the cosets of \( {G}_{1} \) in \( G \) . Moreove... | Proof. Suppose that \( U \) and \( V \) are irreducible components of \( G \) and that each contains the neutral element of \( G \) . The product set \( {UV} \) in \( G \) is the image of \( U \times V \) under the composition map of \( G \) . By Propositions 1.3 and 1.1, it is therefore irreducible. Since it contains ... | Yes |
Theorem 2.1. Let \( G \) be an affine algebraic \( F \) -group, \( H \) an algebraic subgroup of \( G \). There is a finite subset \( E \) of \( \mathcal{P}\left( G\right) \), and an element \( f \) of \( \mathcal{P}\left( G\right) \) whose restriction to \( H \) is a group homomorphism from \( H \) to \( {F}^{ * } \),... | Proof. Let \( I \) denote the annihilator of \( H \) in \( \mathcal{P}\left( G\right) \). There is a finite-dimensional left \( G \) -stable sub \( F \) -space \( V \) of \( \mathcal{P}\left( G\right) \) such that \( V \cap I \) generates \( I \) as an ideal. Let \( d \) denote the dimension of \( V \cap I \), and cons... | Yes |
Lemma 3.1. Let \( R \) be a subring of a field \( K \), and suppose that \( J \) is a proper ideal of \( R \). For every element \( u \) of \( K \), if \( R\left\lbrack u\right\rbrack J = R\left\lbrack u\right\rbrack \) then | Proof. Suppose this is false. Then there is an element \( u \) in \( K \) for which we have relations\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{m}{a}_{i}{u}^{i} = 1 = \mathop{\sum }\limits_{{j = 0}}^{n}{b}_{j}{u}^{-j} \]\n\nwhere the \( {a}_{i} \)’s and \( {b}_{j} \)’s are elements of \( J \). We assume that the relations... | Yes |
Theorem 3.3. Let \( R \) be a subring of a field \( K \), and let \( P \) be a finite subset of \( K \) . For every non-zero element \( u \) of \( R\left\lbrack P\right\rbrack \), there is a non-zero element \( {u}^{\prime } \) in \( R \) such that every homomorphism from \( R \) to an algebraically closed field \( F \... | Proof. Evidently, the statement of the theorem is adapted to an induction on the cardinality of \( P \) . Therefore, we suppose without loss of generality that \( P \) consists of a single element \( p \) . First, we deal with the case where \( p \) is not algebraic over the field of fractions of \( R \), which we deno... | Yes |
Lemma 3.4. Let \( B \) be a commutative ring. The intersection of the family of all prime ideals of \( B \) coincides with the set of all nilpotent elements. | Proof. Evidently, every nilpotent element of \( B \) belongs to every prime ideal. Conversely, suppose that \( b \) is an element of \( B \) that belongs to every prime ideal of \( B \) . Consider the polynomial ring \( B\left\lbrack x\right\rbrack \), where \( x \) is an auxiliary variable. The assumption on \( b \) c... | Yes |
Theorem 3.5. Let \( L \) be a field, \( B \) a finitely generated \( L \) -algebra having no nilpotent elements other than 0 . Let \( F \) be an algebraically closed field containing L. Then the L-algebra homomorphisms from B to \( F \) separate the elements of \( B \) . | Proof. By Lemma 3.4, the assumption on \( B \) means that the intersection of the family of all prime ideals of \( B \) is (0). Hence, if \( b \) is any non-zero element of \( B \), there is a prime ideal \( J \) in \( B \) not containing \( b \) . We identify \( L \) with its canonical image in the integral domain \( ... | Yes |
Proposition 3.6. Let \( F \subset L \subset K \) be a tower of fields. Suppose that \( K \) is finitely field-generated over \( F \) . Then the same is true for \( L \) . | Proof. There is a transcendence basis \( \left( {{s}_{1},\ldots ,{s}_{m},{t}_{1},\ldots ,{t}_{n}}\right) \) for \( K \) over \( F \) such that \( \left( {{s}_{1},\ldots ,{s}_{m}}\right) \) is a transcendence basis for \( L \) over \( F \) and \( \left( {{t}_{1},\ldots ,{t}_{n}}\right) \) is one for \( K \) over \( L \)... | Yes |
Proposition 3.7 (Artin-Tate). Let \( R, B, A \) be commutative rings, with \( R \subset B \subset A \). Suppose that \( R \) is Noetherian, that \( A \) is finitely generated as an \( R \) -algebra and also that \( A \) is finitely generated as a \( B \) -module. Then \( B \) is finitely generated as an R-algebra. | Proof. Exhibiting the assumptions on the generation of \( A \), we write\n\n\[ R\left\lbrack {{a}_{1},\ldots ,{a}_{n}}\right\rbrack = A = B{u}_{1} + \cdots + B{u}_{m} \]\n\nchoosing \( {u}_{1} = 1 \). Then we have\n\n\[ {a}_{i} = \mathop{\sum }\limits_{{j = 1}}^{m}{b}_{ij}{u}_{j}\;\text{ and }\;{u}_{i}{u}_{j} = \mathop... | Yes |
Lemma 3.8. Let \( A \) be a commutative algebra over the field \( F \) that can be generated as such by a finite set of cardinality \( n \) . Then every chain of prime ideals of \( A \) has length at most \( n \) . If \( P \) and \( Q \) are prime ideals of \( A \) such that \( P \) is properly contained in \( Q \) the... | Proof. It is clear that the transcendence degree of \( \left\lbrack {A/P}\right\rbrack \) cannot exceed \( n \) . Hence, it suffices to prove the second assertion of the lemma.\n\nThere is a transcendence basis \( \left( {{y}_{1},\ldots ,{y}_{t}}\right) \) of \( \left\lbrack {A/Q}\right\rbrack \) relative to \( F \) co... | Yes |
Lemma 4.2. Let \( F \) be a field, \( G \) an irreducible affine algebraic \( F \) -group, \( B \) a sub Hopf algebra of \( \mathcal{P}\left( G\right) \) . Then \( \left\lbrack B\right\rbrack \cap \mathcal{P}\left( G\right) = B \) . | Proof. By Note I.2, \( B \) is the union of a family of sub Hopf algebras that are finitely generated as \( F \) -algebras. Therefore, we assume, without loss of generality, that \( B \) is finitely generated as an \( F \) -algebra.\n\nLet \( L \) be an algebraically closed field containing \( F \), and consider the ex... | Yes |
Theorem 4.3. Let \( G \) be an affine algebraic \( F \) -group, \( B \) a sub Hopf algebra of \( \mathcal{P}\left( G\right) \) . Then \( B \) is finitely generated as an \( F \) -algebra. If \( F \) is algebraically closed then the restriction map \( G \rightarrow \mathcal{G}\left( B\right) \) is surjective. | Proof. First, we deal with the case where \( G \) is irreducible. In that case, we see from Proposition 3.6 that \( \left\lbrack B\right\rbrack \) is finitely field-generated over \( F \) . Let \( \left( {{u}_{1}{v}_{1}^{-1},\ldots ,{u}_{n}{v}_{n}^{-1}}\right) \) be a finite system of field generators for \( \left\lbra... | Yes |
Theorem 4.4. Let \( F \) be an algebraically closed field, \( G \) an affine algebraic F-group, \( H \) a normal algebraic subgroup of \( G \). Then \( G/H \) has the structure of an affine algebraic F-group such that, via the transpose of the canonical morphism \( \pi : G \rightarrow G/H \), the Hopf algebra \( \mathc... | Proof. Clearly, \( \mathcal{P}{\left( G\right) }^{H} \) is stable under the left and right actions of \( G \) on \( \mathcal{P}\left( G\right) \), as well as under the antipode. Hence, \( \mathcal{P}{\left( G\right) }^{H} \) is a sub Hopf algebra of \( \mathcal{P}\left( G\right) \). It follows from Theorem 2.2 that the... | Yes |
Proposition 1.2. Let \( F \subset K \subset L \) be a tower of fields. If \( K \) is separable over \( F \) and \( L \) is separable over \( K \) then \( L \) is separable over \( F \) . If \( L \) is separable over \( F \), so is \( K \) . | Proof. The first part is clear from the definition. In order to prove the second part, let \( \tau \) be a derivation from \( F \) to a \( K \) -space \( S \) . We form the \( L \) - space \( L \otimes S \), and write it as a direct \( K \) -space sum \( S + T \) . Now we may view \( \tau \) as a derivation from \( F \... | Yes |
Let \( K \) be a field, \( F \) a subfield of \( K \), and \( u \) an element of \( K \) . Let \( \tau \) be a derivation from \( F \) to an \( F\left( u\right) \) -space \( S \) . If \( u \) is not algebraic over \( F \) then, for every element \( s \) of \( S \), there is one and only one extension of \( \tau \) to a... | First, consider the case where \( u \) is not algebraic over \( F \) . Clearly, there is one and only one derivation \( \sigma \) from \( F\left\lbrack u\right\rbrack \) to \( S \) sending \( u \) onto \( s \) and coinciding with \( \tau \) on \( F \) . In fact, \( \sigma \) is given by\n\n\[ \sigma \left( {\mathop{\su... | Yes |
Lemma 1.4. Let \( F \) be a field of non-zero characteristic \( p \), let \( S \) be an \( F \) -space, \( s \) an element of \( S \), and \( u \) an element of \( F \) that is not the \( p \) -th power of an element of \( F \) . There is a derivation \( \tau \) from \( F \) to \( S \) such that \( \tau \left( u\right)... | Proof. Let \( {F}^{\left\lbrack p\right\rbrack } \) denote the subfield of \( F \) consisting of the \( p \) -th powers of the elements of \( F \), let \( L \) be a subfield of \( F \) containing \( {F}^{\left\lbrack p\right\rbrack } \), and let \( v \) be an element of \( F \) not belonging to \( L \) . Let \( f\left(... | Yes |
Proposition 1.5. Let \( F \) be a field of non-zero characteristic \( p \), and let \( K \) be a field extension of \( F \) . Then \( K \) is separable over \( F \) if and only if, for every \( F \) -linearly independent subset \( U \) of \( K \), the set \( {U}^{\left\lbrack p\right\rbrack } \) of \( p \) -th powers o... | Proof. First, suppose that the condition is satisfied. Then the multiplication map from \( F{ \otimes }_{{F}^{\left\lbrack p\right\rbrack }}{K}^{\left\lbrack p\right\rbrack } \) to \( K \) is injective, so that the subfield \( F\left\lbrack {K}^{\left\lbrack p\right\rbrack }\right\rbrack \) of \( K \) is \( F \) -algeb... | Yes |
Theorem 2.3. Let \( K \) be a field, and let \( A \) be a group of field automorphisms of \( K \) . Then \( K \) is separable over its \( A \) -fixed part \( {K}^{A} \) . | Proof. By Theorem 2.2, it suffices to prove that \( K{ \otimes }_{{K}^{A}}L \) has no nilpotent element other than 0, for every field \( L \) containing \( {K}^{A} \) . Since there is nothing to prove if \( K \) is of characteristic 0, we assume that \( K \) is of non-zero characteristic \( p \) . Suppose that the resu... | Yes |
Theorem 3.1. Let \( G \) be an affine algebraic F-group. The \( \mathcal{P}\left( G\right) \) -module homomorphism\n\n\[ \mathcal{P}\left( G\right) \otimes \mathcal{L}\left( G\right) \rightarrow {\mathcal{D}}_{F}\left( {\mathcal{P}\left( G\right) }\right) \]\n\ndetermined by the condition that it send each element \( \... | Proof. Let \( \mathop{\sum }\limits_{{i = 1}}^{n}{a}_{i} \otimes {\tau }_{i} \) be an element of \( \mathcal{P}\left( G\right) \otimes \mathcal{L}\left( G\right) \) whose image in \( \mathcal{D}\left( {\mathcal{P}\left( G\right) }\right) \) is 0, where the \( {\tau }_{i} \) ’s are \( F \) -linearly independent elements... | Yes |
Theorem 3.2. Let \( G \) be an irreducible algebraic group. The dimension of \( \mathcal{L}\left( G\right) \) is equal to the degree of transcendence of \( \left\lbrack {P\left( G\right) }\right\rbrack \) over the base field. | Proof. By the isomorphism established just above, the dimension of \( \mathcal{L}\left( G\right) \) as a vector space over the base field \( F \) equals the dimension of \( {\mathcal{D}}_{F}\left( \left\lbrack {\mathcal{P}\left( G\right) }\right\rbrack \right) \) as a vector space over \( \left\lbrack {\mathcal{P}\left... | Yes |
Theorem 3.3. Let \( \rho : G \rightarrow H \) be a morphism of algebraic groups, where \( G \) and \( H \) are irreducible. Suppose that \( \rho \left( G\right) \) is dense in \( H \), and that \( \left\lbrack {\mathcal{P}\left( G\right) }\right\rbrack \) is separable over \( \left\lbrack {\mathcal{P}\left( H\right) \c... | Proof. Since \( \rho \left( G\right) \) is dense in \( H \), the transpose of \( \rho \) is injective from \( \mathcal{P}\left( H\right) \) to \( \mathcal{P}\left( G\right) \) . Accordingly, we identify the elements \( g \) of \( \mathcal{P}\left( H\right) \) with their images \( g \circ \rho \) in \( \mathcal{P}\left(... | Yes |
Lemma 4.1. Let \( G \) be an affine algebraic \( F \) -group, and suppose that\n\n\[ \rho : G \rightarrow {\operatorname{Aut}}_{F}\left( V\right) \]\n\nis the structure of a polynomial G-module. If \( {\rho }^{ * } \) denotes the corresponding comodule structure, then the following equations hold for every element \( \... | Proof. Consider an element \( \gamma /v \) of \( {\operatorname{End}}_{F}{\left( V\right) }^{ \circ } \), where \( \gamma \) is an element of \( {V}^{ \circ } \) and \( v \) is an element of \( V \) . The composite function \( \left( {\gamma /v}\right) \circ \rho \) is an element of \( \mathcal{P}\left( G\right) \), an... | Yes |
Proposition 4.2. Let \( A \) and \( B \) be polynomial modules for an algebraic group \( G \) . Let \( \alpha ,\beta ,\alpha \otimes \beta \) denote the representations of \( G \) on \( A, B, A \otimes B \), respectively. For every element \( \tau \) of \( \mathcal{L}\left( G\right) \), one has \[ {\left( \alpha \otime... | Proof. By (1) of Lemma 4.1, we have \[ {\left( \alpha \otimes \beta \right) }^{\prime }\left( \tau \right) = \left( {{i}_{A} \otimes {i}_{B} \otimes \tau }\right) \circ {\left( \alpha \otimes \beta \right) }^{ * }. \] In Section I.2, we saw that \[ {\left( \alpha \otimes \beta \right) }^{ * } = \left( {{i}_{A} \otimes ... | Yes |
Proposition 4.3. Let \( V \) be a finite-dimensional polynomial module for an affine algebraic \( F \) -group \( G \), and let \( \rho \) denote the representation of \( G \) on \( V \) . For every element \( \tau \) of \( \mathcal{L}\left( G\right) \), the endomorphism \( {\rho }^{\prime }\left( \tau \right) \) of \( ... | Proof. Suppose that \( \alpha \) is an element of \( {\operatorname{End}}_{F}{\left( V\right) }^{ \circ } \) that annihilates every endomorphism of the form \( \rho \left( x\right) - {i}_{V} \) . This means that the representative function \( \alpha \circ \rho \) is a constant, whence \( \tau \left( {\alpha \circ \rho ... | Yes |
Lemma 1.2. Let \( K \) be a field of characteristic 0, and let \( K\left\lbrack \left\lbrack t\right\rbrack \right\rbrack \) be the \( K \) - algebra of integral power series in the variable \( t \) . Let \( \left( {{a}_{1},\ldots ,{a}_{q}}\right) \) be a subset of \( K \) that is linearly independent over the field of... | Proof. Every polynomial relation among the elements figuring in the lemma may be written in the form\n\n\[ \sum c\left( {{e}_{0},\ldots ,{e}_{q}}\right) {t}^{{e}_{0}}\exp \left( {\left( {{e}_{1}{a}_{1} + \cdots + {e}_{q}{a}_{q}}\right) t}\right) = 0, \]\n\nwhere the summation goes over a finite set of \( q \) -tuples \... | Yes |
Theorem 2.2. Let \( G \) be an algebraic group over a field of characteristic 0 . For every sub Lie algebra \( L \) of \( \mathcal{L}\left( G\right) \), let \( {G}_{L} \) denote the intersection of the family of all algebraic subgroups of \( G \) whose Lie algebras contain \( L \) . Then \( {G}_{L} \) is an irreducible... | Proof. Evidently, \( {G}_{L} \) is an algebraic subgroup of \( G \) . By the first part of Theorem 2.1, we have \( {G}_{\tau } \subset {G}_{L} \) for every element \( \tau \) of \( L \) . This implies that \( \mathcal{L}\left( {G}_{\tau }\right) \subset \mathcal{L}\left( {G}_{L}\right) \), and now the last part of Theo... | Yes |
Theorem 2.3. Let \( \rho : G \rightarrow H \) be a morphism of algebraic groups over a field of characteristic 0, and let \( K \) be the kernel of \( \rho \) . Then the kernel of the differential \( {\rho }^{ \cdot } \) coincides with \( \mathcal{L}\left( K\right) \) . | Proof. In any characteristic, it is clear from the definitions that \( \mathcal{L}\left( K\right) \) is contained in the kernel of \( {\rho }^{ \cdot } \) . Conversely, let \( \tau \) be an element of the kernel of \( \rho \) . Let \( f \) be an element of \( \mathcal{P}\left( H\right) \), and let \( x \) be an element... | Yes |
Theorem 4.1. Let \( \alpha \) denote the adjoint representation of the algebraic group \( G \) on its Lie algebra \( \mathcal{L}\left( G\right) \). Then, for every element \( x \) of \( G \) and every element \( \tau \) of \( \mathcal{L}\left( G\right) \), we have\n\n\[ \alpha \left( x\right) \left( \tau \right) = {x\t... | The extended differential of \( \alpha \) is given by\n\n\[ {\alpha }^{\prime }\left( \tau \right) = {D}_{\tau } \] | No |
Theorem 4.2. If \( G \) is an irreducible algebraic group over a field of characteristic 0, then the kernel of the adjoint representation of \( G \) coincides with the center of \( G \) . | Proof. Let \( x \) be an element of the kernel of the adjoint representation. It is evident from Theorem 4.1 that \( x \) commutes with every element of \( \mathcal{L}\left( G\right) \) . By Proposition I.2.1, this implies that \( {x}_{\mathrm{I}} \) commutes with \( {\tau }_{\mathrm{I}} \) for every element \( \tau \)... | Yes |
Theorem 4.3. Let \( F \) be a field, \( G \) an irreducible affine algebraic \( F \) -group. If \( G \) is abelian, so is \( \mathcal{L}\left( G\right) \) . Conversely, if \( \mathcal{L}\left( G\right) \) is abelian and \( F \) is of characteristic 0, then \( G \) is abelian. | Proof. Suppose that \( G \) is abelian. Then the adjoint representation of \( G \) is trivial, whence its extended differential is the 0-map. By Theorem 4.1, this implies that \( \mathcal{L}\left( G\right) \) is abelian.\n\nNow suppose that \( F \) is of characteristic 0, and that \( \mathcal{L}\left( G\right) \) is ab... | Yes |
Theorem 4.4. Let \( F \) be a field, \( G \) an irreducible algebraic \( F \) -group, \( K \) an irreducible algebraic subgroup of \( G \) . If \( K \) is normal in \( G \), then \( \mathcal{L}\left( K\right) \) is an ideal of \( \mathcal{L}\left( G\right) \) . Conversely, if \( \mathcal{L}\left( K\right) \) is an idea... | Proof. Suppose that \( K \) is normal in \( G \) . Let \( I \) be the annihilator of \( K \) in \( \mathcal{P}\left( G\right) \) , let \( x \) be an element of \( G \) and let \( \tau \) be an element of \( \mathcal{L}\left( K\right) \) . Then \( {x}^{-1} \cdot I \cdot x \subset I \) , i.e., \( {x}_{1}{x}_{1}^{-1} \) s... | Yes |
Proposition 5.1. In the notation of Theorem II.2.1, \( \mathcal{L}\left( H\right) \) consists precisely of those elements \( \tau \) of \( \mathcal{L}\left( G\right) \) which satisfy \( {\tau }_{\mathrm{I}}\left( e\right) \in \) Fe for every \( e \) in the finite set \( E \) of \( H \) -semi-invariants constructed in t... | Proof. It is clear from Proposition III.4.3 that every element of \( \mathcal{L}\left( H\right) \) satisfies the stated condition. Now suppose that \( \tau \) is an element of \( \mathcal{L}\left( G\right) \) satisfying this condition. Recall from the proof of Theorem II.2.1 that the elements of \( E \) are the functio... | Yes |
Proposition 5.2. In the notation of Theorem II.2.2, \( \mathcal{L}\left( H\right) \) consists precisely of those elements \( \tau \) of \( \mathcal{L}\left( G\right) \) which satisfy \( {\tau }_{1}\left( q\right) = 0 \) for every element \( q \) of the finite set \( Q \) of \( H \) -invariants constructed in the proof ... | Proof. One sees immediately from Proposition III.4.3 that every element of \( \mathcal{L}\left( H\right) \) satisfies the stated condition. In order to prove the converse, we must consider the differential of the representation \( \sigma \) of \( G \) on \( U \) used in the proof of Theorem II.2.2.\n\nRecall that we co... | Yes |
Theorem 5.4. Let \( G \) be an algebraic group over an algebraically closed field, and let \( H \) be a normal algebraic subgroup of \( G \) . Then\n\n\[ \dim \left( G\right) = \dim \left( {G/H}\right) + \dim \left( H\right) \] | Proof. We have \( {H}_{1} \subset {G}_{1} \), and \( {H}_{1} \) is of finite index in \( {G}_{1} \cap H \) . Hence, \( {H}_{1} = {\left( {G}_{1} \cap H\right) }_{1} \), so that \( \dim \left( H\right) = \dim \left( {{G}_{1} \cap H}\right) \) . The injection \( {G}_{1} \rightarrow G \) , followed by the canonical map \(... | Yes |
Proposition 1.1. Let \( R \) be a ring, \( M \) a semisimple \( R \) -module, \( C = {\operatorname{End}}_{R}\left( M\right) \) . Assume that \( M \) is finitely generated as a \( C \) -module. Then the image of \( R \) in \( \operatorname{End}\left( M\right) \) coincides with \( {\operatorname{End}}_{C}\left( M\right)... | Proof. Choose a finite system \( \left( {{m}_{1},\ldots ,{m}_{q}}\right) \) of \( C \) -module generators of \( M \) , and let \( S \) be the direct sum of \( q \) copies of the \( R \) -module \( M \), so that this system may be viewed as an element of the \( R \) -module \( S \) . For \( i = 1,\ldots, q \), let\n\n\[... | Yes |
Proposition 1.2. Let \( V \) be a vector space over a field \( F \), and let \( S \) be a sub \( F \) - algebra of \( {\operatorname{End}}_{F}\left( V\right) \) . Let \( K \) be an extension field of \( F \) . If \( V \otimes K \) is semisimple with respect to \( S \otimes K \), then \( V \) is semisimple with respect ... | Proof. Suppose that \( V \otimes K \) is semisimple, and consider a sub \( S \) -module \( U \) of \( V \) . The assumption implies that there is an \( S \) -module projection \( \mu \) from \( V \otimes K \) to \( U \otimes K \) . Choose an \( F \) -space complement \( C \) of \( F \) in \( K \) . For \( v \) in \( V ... | Yes |
Lemma 1.3. Let \( F \) be a perfect field, \( x \) a variable over \( F \). Let \( f \) be an element of \( F\left\lbrack x\right\rbrack \smallsetminus F \), and let \( g \) be a product of mutually inequivalent prime elements of \( F\left\lbrack x\right\rbrack \), including the prime factors of \( f \). There is an \(... | Proof. Since \( F \) is perfect, \( g \) has no multiple roots in any extension field of \( F \), so that \( g \) is relatively prime to its formal derivative \( {g}^{\prime } \). Thus, there are elements \( u \) and \( v \) in \( F\left\lbrack x\right\rbrack \) such that\n\n\[ u{g}^{\prime } + {vg} = 1\text{.}\]\n\nLe... | Yes |
Theorem 1.4. Let \( F \) be a perfect field, \( V \) a finite-dimensional \( F \) -space, e an \( F \) -linear endomorphism of \( V \) . There are \( F \) -linear endomorphisms \( {e}^{\left( n\right) } \) and \( {e}^{\left( s\right) } \) of \( V \) satisfying the following conditions: \( {e}^{\left( n\right) } \) is n... | Proof. Let \( \rho \) be the \( F \) -algebra homomorphism from \( F\left\lbrack x\right\rbrack \) to \( {\operatorname{End}}_{F}\left( V\right) \) sending \( x \) onto \( e \) . The kernel of \( \rho \) is a principal ideal \( F\left\lbrack x\right\rbrack f \) . If \( f \) is a multiple of \( x \), we define \( g \) a... | Yes |
Theorem 2.1. Let \( F \) be a field, and let \( \rho \) be a representation of a group \( G \) by linear automorphisms of a finite-dimensional F-space \( V \) . If \( \rho \left( x\right) - {i}_{V} \) is nilpotent for every element \( x \) of \( G \), then \( \rho \) is a unipotent representation. | Proof. If \( K \) is an algebraically closed field containing \( F \), we can extend the given structure in the canonical fashion so as to obtain a representation of \( G \) by \( K \) -linear automorphisms of \( V \otimes K \) that still has the property assumed for \( \rho \) . Therefore, we suppose without loss of g... | Yes |
Proposition 2.2. Let \( G \) be a group of linear automorphisms of a finite-dimensional vector space \( V \). Let \( S \) and \( T \) be subgroups of \( G \), with \( T \) normal in \( G \). If \( V \) is unipotent as an S-module and as a T-module, then \( V \) is unipotent as an ST-module. | Proof. For convenience of notation, let us work with the ordinary group algebra \( \mathcal{L}\left\lbrack G\right\rbrack \) of \( G \) over the ring \( \mathcal{L} \) of integers, and let us view \( V \) as a \( \mathcal{L}\left\lbrack G\right\rbrack \) - module in the evident fashion. If \( s \in S \) and \( t \in T ... | Yes |
Theorem 4.1. Let \( F \) be an algebraically closed field of characteristic 0 . Let \( G \) be an affine algebraic \( F \) -group, and let \( B \) be a sub Hopf algebra of \( \mathcal{P}\left( G\right) \) . If the restriction morphism \( \rho : G \rightarrow \mathcal{G}\left( B\right) \) is injective then \( B \) coinc... | Proof. Under the present assumptions, \( \mathcal{G}\left( B\right) \) is an affine algebraic \( F \) -group with \( \mathcal{P}\left( {\mathcal{G}\left( B\right) }\right) = B \), and \( \rho \) is a bijective morphism of affine algebraic \( F \) - groups. Clearly, this implies that \( \rho \) maps the set of irreducib... | Yes |
Theorem 5.2. If \( F \) is an algebraically closed field then every abelian irreducible linearly reductive affine algebraic F-group is an F-toroid. | Proof. Let \( G \) be such a group, and let \( V \) be a finite-dimensional \( G \) -stable sub \( F \) -space of \( \mathcal{P}\left( G\right) \) that generates \( \mathcal{P}\left( G\right) \) as an \( F \) -algebra. The assumptions on \( G \) imply that \( V \) is the direct \( G \) -module sum of a finite family of... | Yes |
Corollary 1.2. If \( G \) is as in Theorem 1.1 then \( G/{G}_{u} \) is abelian, and every unipotent element of \( G \) belongs to \( {G}_{u} \) . | Proof. By Theorem 1.1, the commutator subgroup of \( G \) acts trivially on every simple polynomial \( G \) -module. Therefore, it must be contained in \( {G}_{u} \) , so that \( G/{G}_{u} \) is abelian.\n\nNow let \( x \) be a unipotent element of \( G \), and let \( H \) be the subgroup of \( G \) that is generated b... | Yes |
Proposition 2.2. Let \( F \) be an algebraically closed field, \( G \) an abelian linearly reductive affine algebraic F-group, \( M \) a unipotent abelian affine algebraic F-group having a \( G \) -module structure such that the defining map from \( G \times M \) to \( M \) is a polynomial map. Let \( f \) be a polynom... | Proof. By Theorem V.5.2, the irreducible component \( {G}_{1} \) of the neutral element in \( G \) is an \( F \) -toroid. Since \( F \) is algebraically closed, \( {G}_{1} \) is therefore a divisible group. Since \( G \) is abelian, it follows from this that \( G \) is the direct product of \( {G}_{1} \) and a finite g... | Yes |
Theorem 4.1. Let \( F \) be an algebraically closed field, \( G \) an irreducible solvable affine algebraic F-group. There is a chain of irreducible normal algebraic subgroups \( {U}_{i} \) of \( G \), starting with \( {U}_{0} = \left( 1\right) \) and ending with \( {U}_{k} = {G}_{u} \), such that each \( {U}_{i}/{U}_{... | Proof. Let \( V \) be a finite-dimensional sub \( G \) -module of \( \mathcal{P}\left( G\right) \) that generates \( \mathcal{P}\left( G\right) \) as an \( F \) -algebra, and let \( \rho \) denote the representation of \( G \) on \( V \) . Then \( \rho \) is an isomorphism of affine algebraic \( F \) -groups from \( G ... | Yes |
Proposition 4.2. Let \( F \) be an algebraically closed field, \( M \) a finite-dimensional \( F \) -space, \( G \) a group of linear automorphisms of \( M \) . Suppose that \( M \) is simple as a G-module, and that there is a positive integer \( e \) such that \( {x}^{e} = {i}_{M} \) for every element \( x \) of \( G ... | Proof. Applying Proposition V.1.1 to the subalgebra of \( {\operatorname{End}}_{F}\left( M\right) \) that is generated by \( G \), we see that \( G \) contains an \( F \) -basis \( \left( {{x}_{1},\ldots ,{x}_{q}}\right) \) of \( {\operatorname{End}}_{F}\left( M\right) \) . Let \( T \) denote the trace function on \( {... | Yes |
Proposition 4.3. Let \( F \) be an algebraically closed field, \( G \) an irreducible affine algebraic \( F \) -group such that there is a positive integer \( e \) with \( {x}^{e} = {1}_{G} \) for every element \( x \) of \( G \) . Then \( G \) is unipotent. | Proof. Let \( V \) be a finite-dimensional sub \( G \) -module of \( \mathcal{P}\left( G\right) \) that generates \( \mathcal{P}\left( G\right) \) as an \( F \) -algebra. Let\n\n\[ \left( 0\right) = {V}_{0} \subset \cdots \subset {V}_{n} = V \]\n\nbe a composition series of \( V \) as a \( G \) -module. Let \( H \) be ... | Yes |
Lemma 5.1. Let \( A \) and \( B \) be Hopf algebras over a field. Then\n\n\[ {P}_{A \otimes B} = {P}_{A} + {P}_{B} \] | Proof. Let \( u = \sum a \otimes b \) be an element of \( {P}_{A \otimes B} \) . Then we have\n\n\[ \sum \delta \left( a\right) \otimes \delta \left( b\right) = {s}_{23}\left( {\delta \left( u\right) }\right) = \sum a \otimes {1}_{A} \otimes b \otimes {1}_{B} + \sum {1}_{A} \otimes a \otimes {1}_{B} \otimes b \]\n\nwhe... | Yes |
Lemma 5.2. Let \( F \) be a field, and let \( H \) be the polynomial Hopf algebra \( F\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \), where the \( {x}_{i} \)’s are algebraically independent primitive elements. Then \( {P}_{H} \) consists of the \( F \) -linear combinations of the powers \( {x}_{i}^{{p}^{t}} \),... | Proof. Lemma 5.1 reduces the problem to the case \( n = 1 \) . In this case, let us write \( x \) for \( {x}_{1} \), and let us consider a primitive element \( u \) . Using that \( \varepsilon \left( u\right) = 0 \), we write\n\n\[ u = \mathop{\sum }\limits_{{i = 1}}^{m}{c}_{i}{x}^{i} \]\n\nwith each \( {c}_{i} \) in \... | Yes |
Lemma 1.1. Let \( F \) be a field of characteristic 0, and let \( V \) be a finite-dimensional \( F \) -space. Let \( {x}_{1},\ldots ,{x}_{n} \) and \( {y}_{1},\ldots ,{y}_{n} \) be elements of \( {\operatorname{End}}_{F}\left( V\right) \), and let\n\n\[ e = \mathop{\sum }\limits_{{i = 1}}^{n}\left( {{x}_{i}{y}_{i} - {... | Proof. For every positive exponent \( s \), we have\n\n\[ {e}^{s} = \mathop{\sum }\limits_{{i = 1}}^{n}\left( {{x}_{i}\left( {{e}^{s - 1}{y}_{i}}\right) - \left( {{e}^{s - 1}{y}_{i}}\right) {x}_{i}}\right) \]\n\nwhere \( {e}^{0} \) stands for \( {i}_{V} \) . It is clear from this that the trace of \( {e}^{s} \) is equa... | Yes |
Theorem 1.2. Let \( L \) be a Lie algebra of linear endomorphisms of a finite-dimensional \( F \) -space \( V \), where \( F \) is a field of characteristic 0 . Suppose that \( V \) is semisimple as an L-module. If \( Z \) is the center of \( L \), then \( L/Z \) is a semisimple Lie algebra, \( \left\lbrack {L, L}\righ... | Proof. Let \( A \) denote the (associative) sub \( F \) -algebra of \( {\operatorname{End}}_{F}\left( V\right) \) that is generated by \( L \) . First, we show that \( A \) has no non-zero nilpotent left ideal.\n\nSuppose this is false. Then there is a non-zero left ideal \( B \) of \( A \) such that \( {BB} = \left( 0... | Yes |
Theorem 1.3. Let \( L \) be a solvable Lie algebra over a field of characteristic 0 . Then every finite-dimensional semisimple L-module is annihilated by \( \left\lbrack {L, L}\right\rbrack \) . | Proof. Let \( V \) be such an \( L \) -module. Let \( {L}^{\prime } \) denote the image of \( L \) in \( \operatorname{End}\left( V\right) \) , and let \( Z \) denote the center of \( {L}^{\prime } \) . By Theorem 1.2, \( {L}^{\prime }/Z \) has no non-zero abelian ideal. Since \( {L}^{\prime }/Z \) is solvable, this im... | Yes |
Lemma 1.4. Let \( L \) be a Lie algebra, \( V \) an \( L \)-module, \( S \) a subspace of \( L \) that is nilpotent on \( V \). Suppose that \( x \) is an element of \( L \) that is nilpotent on \( V \) and such that \( \left\lbrack {x, S}\right\rbrack \subset S \). Then \( x + S \) is nilpotent on \( V \). | Proof. Without loss of generality, we assume that \( L \) is given as a Lie algebra of linear endomorphisms of \( V \). Then the assumption is that there are positive integers \( p \) and \( q \) such that \( {S}^{p} = \left( 0\right) \) and \( {x}^{q} = 0 \). We show that \( {\left( x + S\right) }^{pq} = \left( 0\righ... | Yes |
Theorem 1.5. Let \( L \) be a Lie algebra, and let \( V \) be a finite-dimensional \( L \)-module. If every element of \( L \) is nilpotent on \( V \) then \( L \) is nilpotent on \( V \). | Proof. As above, we assume without loss of generality that \( L \) is given as a Lie algebra of linear endomorphisms of \( V \). Let \( x \) and \( y \) be elements of \( L \), and consider \( {D}_{x}^{m}\left( y\right) \) for positive integers \( m \). This is a sum of products \( \pm {x}^{p}y{x}^{q} \), where \( p + ... | "No" |
Theorem 2.1. Let \( L \) be a finite-dimensional Lie algebra over a field of characteristic 0 . If \( L \) is semisimple and \( \rho \) is an injective finite-dimensional representation of \( L \), then the trace form \( {\tau }_{\rho } \) is non-degenerate. If the trace form of the adjoint representation of \( L \) is... | Proof. Let \( H \) denote the set of all elements \( x \) of \( L \) such that \( {\tau }_{\rho }\left( {x, y}\right) = 0 \) for every \( y \) in \( L \) . The formal property of \( {\tau }_{\rho } \) noted above shows that \( H \) is an ideal of \( L \) . Theorem 1.6, applied to \( \rho \left( H\right) \), shows that ... | Yes |
Proposition 2.2. Let \( L \) be a finite-dimensional semisimple Lie algebra over a field of characteristic 0 , and let I be an ideal of L. There is one and only one ideal \( {I}^{\prime } \) of \( L \) such that \( L = I + {I}^{\prime } \) and \( I \cap {I}^{\prime } = \left( 0\right) \) . Also, \( L = \left\lbrack {L,... | Proof. Since \( L \) is semisimple, its center is (0), so that the adjoint representation of \( L \) is injective. By Theorem 2.1, the trace form of the adjoint representation of \( L \) is therefore non-degenerate. Let \( {I}^{\prime } \) be the set of all elements \( x \) of \( L \) such that \( T\left( {{D}_{x}{D}_{... | Yes |
Lemma 2.3. Let \( L \) be a finite-dimensional semisimple Lie algebra over a field of characteristic 0, and let \( V \) be a finite-dimension L-module. Then \[ V = L \cdot V + {V}^{L} \] | Proof. Let \( \rho \) denote the representation of \( L \) on \( V \), let \( I \) be the kernel of \( \rho \) , and let \( {I}^{\prime } \) be the ideal complementary to \( I \) figuring in Proposition 2.2. Clearly, \( {I}^{\prime } \) is a semisimple Lie algebra, \( L \cdot V = {I}^{\prime } \cdot V \) and \( {V}^{L}... | Yes |
Theorem 2.4. Let \( L \) be a finite-dimensional semisimple Lie algebra over a field of characteristic 0 . Then every finite-dimensional L-module is semisimple. | Proof. Let \( V \) be a finite-dimensional \( L \) -module, and let \( U \) be a sub \( L \) -module of \( V \) . We show that \( U \) has an \( L \) -module complement in \( V \) . Let \( H \) be the space of all linear maps \( f : V/U \rightarrow V \) . We make \( H \) into an \( L \) -module by\n\n\[ \left( {x \cdot... | Yes |
Corollary 2.5. Let \( L \) be a finite-dimensional semisimple Lie algebra over the field \( F \) of characteristic 0, and let \( V \) be a finite-dimensional L-module. Then every cocycle for \( L \) in \( V \) is a coboundary. | Proof. Let \( f \) be a cocycle for \( L \) in \( V \) . We define an \( L \) -module structure on the direct sum \( V + F \) by setting\n\n\[ x \cdot \left( {v, a}\right) = \left( {{af}\left( x\right) + x \cdot v,0}\right) \]\n\nfor every \( x \) in \( L \), every \( v \) in \( V \) and every \( a \) in \( F \) . In f... | Yes |
Proposition 2.6. Let \( L \) be a finite-dimensional semisimple Lie algebra over a field of characteristic 0 . Then every derivation of \( L \) is of the form \( {D}_{x} \), with \( x \) in \( L \) . | Proof. This is seen immediately from Corollary 2.5 by observing that a derivation is a cocycle for \( L \) in \( L \), with respect to the adjoint representation. | No |
Theorem 3.2. Let \( L \) be a finite-dimensional Lie algebra over a field of characteristic 0 . Then \( \left\lbrack {L, L}\right\rbrack \cap {L}_{r} = \left\lbrack {L,{L}_{r}}\right\rbrack \) . If \( V \) is a finite-dimensional \( L \) -module then \( \left\lbrack {L,{L}_{r}}\right\rbrack \) is nilpotent on \( V \) . | Proof. If we apply Theorem 3.1 to the canonical homomorphism \( L \rightarrow L/{L}_{r} \) , we see that there is a semisimple sub Lie algebra \( S \) of \( L \) that is mapped\nisomorphically onto \( L/{L}_{r} \) by the canonical map. Clearly, \( L = {L}_{r} + S \) and \( {L}_{r} \cap S = \left( 0\right) \) . Since \(... | Yes |
Theorem 3.3. Let \( L \) be a finite-dimensional Lie algebra over a field of characteristic 0, and let \( V \) be a finite-dimensional L-module. There is an ideal \( P \) of \( L \) that is nilpotent on \( V \) and contains every ideal of \( L \) that is nilpotent on \( V \) . Every element of \( {L}_{r} \) that is nil... | Proof. Let \( \left( 0\right) = {V}_{0} \subset \cdots \subset {V}_{n} = V \) be a composition series for the \( L \) - module \( V \) . Let \( {P}_{i} \) be the kernel of the induced representation of \( L \) on \( {V}_{i}/{V}_{i - 1} \) , and put \( P = \mathop{\bigcap }\limits_{{i = 1}}^{n}{P}_{i} \) . Clearly, \( P... | Yes |
Theorem 4.1. Let \( L \) be a finite-dimensional Lie algebra over a field of characteristic 0, and let \( S \) be a semisimple sub Lie algebra of \( L \) such that \( L = {L}_{r} + S \) (the existence of \( S \) is guaranteed by Theorem 3.1). Let \( T \) be any semisimple sub Lie algebra of \( L \) . There is an elemen... | Proof. Suppose that \( T \subset {L}_{r}^{\left\lbrack i\right\rbrack } + S \), for some index \( i \) . Then, since \( T = \left\lbrack {T, T}\right\rbrack \) , it follows that \( T \subset {L}_{r}^{\left\lbrack i + 1\right\rbrack } + S \) . Thus, we have \( T \subset {L}_{r}^{\left\lbrack \infty \right\rbrack } + S \... | Yes |
Theorem 1.1. Let \( F \) be a field of characteristic 0, let \( G \) be an affine algebraic \( F \) -group and \( T \) a unipotent algebraic subgroup of \( G \) . Then \( T \) is irreducible, and the map sending each element \( \tau \) of \( \mathcal{L}\left( T\right) \) onto the element \( \exp \left( \tau \right) \) ... | Proof. Consider the representation of \( T \) on \( \mathcal{P}{\left( T\right) }^{{T}_{1}} \) . This factors through the finite group \( T/{T}_{1} \) . Since \( F \) is of characteristic 0, it follows that the representation of \( T \) on \( \mathcal{P}{\left( T\right) }^{{T}_{1}} \) is semisimple. On the other hand, ... | Yes |
Theorem 1.2. Let \( F \) and \( G \) be as in Theorem 1.1, and let \( U \) be a unipotent affine algebraic F-group. Let \( \rho \) be a morphism of affine algebraic F-groups from \( U \) to \( G \) . Then \( \rho \left( U\right) \) is a unipotent algebraic subgroup of \( G \) . | Proof. It follows directly from the definitions that \( {\rho }^{ \cdot }\left( {\mathcal{L}\left( U\right) }\right) \) is a sub Lie algebra of \( \mathcal{L}\left( G\right) \) that is nilpotent on \( \mathcal{P}\left( G\right) \) . By Theorem 1.1, \( U = \exp \left( {\mathcal{L}\left( U\right) }\right) \) . From what ... | Yes |
Proposition 3.1. Let \( G \) be an affine algebraic group over a field of characteristic 0, and let \( L \) be a sub Lie algebra of \( \mathcal{L}\left( G\right) \) . Then \( \left\lbrack {L, L}\right\rbrack = \left\lbrack {{L}^{ + },{L}^{ + }}\right\rbrack \) . | Proof. Consider the adjoint representation of \( G \) on \( \mathcal{L}\left( G\right) \) . In the above remark, let \( W = \mathcal{L}\left( G\right), V = L \) and \( U = \left\lbrack {L, L}\right\rbrack \) . The remark shows that \( \left\lbrack {{L}^{ + }, L}\right\rbrack \subset \left\lbrack {L, L}\right\rbrack \) ... | Yes |
Lemma 4.1. Let \( G \) be a linearly reductive algebraic group, and let \( V \) be a finite-dimensional polynomial G-module. Then every polynomial cocycle for \( G \) in \( V \) is a coboundary. | Proof. We proceed in exact analogy with the proof of Corollary VII.2.5. Let \( f \) be a polynomial cocycle for \( G \) in \( V \), and let \( F \) denote the base field. We define an action of \( G \) on the direct sum \( V + F \) by\n\n\[ x \cdot \left( {v, a}\right) = \left( {{af}\left( x\right) + x \cdot v, a}\righ... | Yes |
Proposition 4.2. Let \( G \) be an algebraic group over the field \( F \) of characteristic 0, and suppose that there is a linearly reductive subgroup \( P \) of \( G \) such that \( {G}_{u}P = G \) . Let \( Q \) be any linearly reductive subgroup of \( G \) . Then there is an element \( t \) in \( {G}_{u} \) such that... | Proof. By Theorem V.4.2, \( P \) is an algebraic subgroup of \( G \), and \( G \) is the semidirect product \( {G}_{u} \bowtie P \) . If \( {G}_{u} \) is trivial, there is nothing to prove. If \( {G}_{u} \) is non-trivial, then the center of \( {G}_{u} \) is non-trivial, because \( {G}_{u} \) is nilpotent. Let \( C \) ... | Yes |
Theorem 4.4. Let \( \rho : G \rightarrow H \) be a morphism of algebraic groups over a field of characteristic 0 . Suppose that \( \rho \left( G\right) \) is dense in \( H \) . Then \( \rho \left( {G}_{u}\right) = {H}_{u} \) . | Proof. Clearly, \( \rho \left( {G}_{u}\right) \) is a normal unipotent subgroup of \( H \), so that\n\n\[ \rho \left( {G}_{u}\right) \subset {H}_{u} \]\n\nLet \( L \) denote the inverse image of \( {H}_{u} \) in \( G \) . Then \( L \) is a normal algebraic subgroup of \( G \) containing \( {G}_{u} \), whence \( {L}_{u}... | Yes |
Proposition 1.2. Let \( S \) be an irreducible affine variety over an algebraically closed field \( F \) . The rational functions that are regular on all of \( S \) are precisely the elements of \( \mathcal{P}\left( S\right) \), i.e., \( {\mathcal{F}}_{S}\left( S\right) = \mathcal{P}\left( S\right) \) . | Proof. Let \( f \) be a rational function that is regular on all of \( S \), and let \( {J}_{f} \) be the ideal of all elements \( v \) of \( \mathcal{P}\left( S\right) \) such that \( {vf} \) belongs to \( \mathcal{P}\left( S\right) \) . The assumption on \( f \) means that \( {J}_{f} \) has no zero in \( S \) . Since... | Yes |
Proposition 1.3. Suppose that \( \sigma \) is a map from a prevariety \( S \) to a prevariety \( T \) satisfying the following condition. There are open sets \( {U}_{1},\ldots ,{U}_{n} \) of \( S \) and \( {V}_{1},\ldots ,{V}_{n} \) of \( T \) such that \( S \) is the union of the \( {U}_{i} \) ’s and \( T \) is the un... | Proof. Each \( {U}_{i} \) is the union of a finite family of affine patches. Not requiring that the \( {V}_{i} \) ’s be mutually distinct, we enlarge the index set so as to achieve that each new \( {U}_{i} \) is an affine patch.\n\nLet \( {\sigma }_{i} \) denote the restriction of \( \sigma \) to \( {U}_{i} \), regardi... | Yes |
Proposition 2.1. A prevariety \( R \) is a variety if and only if it satisfies the following condition. For every pair \( \left( {\rho ,\sigma }\right) \) of morphisms from a prevariety \( S \) to \( R \), the set of points \( s \) in \( S \) such that \( \rho \left( s\right) = \sigma \left( s\right) \) is closed in \(... | Proof. Suppose the condition is satisfied. Choose \( \rho \) and \( \sigma \) to be the projections from \( R \times R \) to the first and second factor. Then the set described in the condition is the diagonal, whence \( R \) is a variety.\n\nConversely, suppose that \( R \) is a variety, and let \( \rho \) and \( \sig... | Yes |
Proposition 2.3. Suppose that \( S \) is a prevariety for which there exists an injective morphism into a variety. Then \( S \) is a variety. | Proof. Let \( \tau : S \rightarrow T \) be an injective morphism, where \( T \) is a variety. Let \( \alpha \) and \( \beta \) be morphisms from a prevariety \( V \) to \( S \) . Using the same notation as in the last proof, we have from Proposition 2.1 that \( {E}_{\tau \circ \alpha ,\tau \circ \beta } \) is closed. S... | No |
Proposition 2.4. Let \( \alpha \) be a morphism from a variety \( S \) to a variety \( T \) . Then the graph of \( \alpha \) is closed in \( S \times T \) . | Proof. Define the map \( \delta \) from \( S \times T \) to \( T \times T \) by\n\n\[ \delta \left( {s, t}\right) = \left( {\alpha \left( s\right), t}\right) \]\n\nThen the graph of \( \alpha \) is the inverse image, with respect to \( \delta \), of the diagonal in \( T \times T \) . Clearly, \( \delta \) is a morphism... | Yes |
Lemma 3.1. Suppose that \( V \) is a prevariety such that, for every pair \( \left( {x, y}\right) \) of points of \( V \), there is an affine patch of \( V \) containing \( x \) and \( y \) . Then \( V \) is a variety. | Proof. Let \( \rho \) and \( \sigma \) be morphisms from a prevariety \( W \) to \( V \), and let \( {E}_{\rho ,\sigma } \) be the set of all points \( w \) in \( W \) such that \( \rho \left( w\right) = \sigma \left( w\right) \) . By Proposition 2.1, it suffices to show that \( {E}_{\rho ,\sigma } \) is closed in \( W... | Yes |
Proposition 5.2. Let \( X \) and \( Y \) be varieties.\n\n(1) If \( X \) is complete and \( Y \) is a closed subvariety of \( X \) then \( Y \) is complete. | Proof. Evidently, (1) follows immediately from the definition. | No |
Theorem 1.2. Let \( F \) be a field, and let \( R \) be a finitely generated commutative \( F \)-algebra. There is a subset \( \left( {{z}_{1},\ldots ,{z}_{s}}\right) \) of \( R \) that is algebraically free over \( F \) and such that \( R \) is integral over \( F\left\lbrack {{z}_{1},\ldots ,{z}_{s}}\right\rbrack \). | Proof. Write \( R = F\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack /I \), where the \( {\mathrm{x}}_{i} \)’s are independent variables over \( F \). Let \( \left( {{b}_{1},\ldots ,{b}_{n}}\right) \) and \( r \) be as obtained from Proposition 1.1, with \( F\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) in... | Yes |
Proposition 1.3. Let \( X \) be an irreducible variety, and let \( Y \) be a closed irreducible subvariety of \( X \), other than \( X \) . Then \( \dim \left( Y\right) < \dim \left( X\right) \) . | Proof. There is an affine patch \( U \) of \( X \) such that \( Y \cap \mathrm{U} \neq \varnothing \) . Now \( Y \cap U \) is closed in \( U \), and \( Y \cap U \neq U \), because otherwise \( Y \) contains the closure of \( U \), which is \( X \) . Also, \( Y \cap U \) is open in \( Y \), and hence is irreducible. We ... | Yes |
Corollary 1.4. Let \( X \) be an irreducible affine variety, and let \( Y \) be a closed irreducible subset of \( X \) such that \( \dim \left( Y\right) = \dim \left( X\right) - 1 \) . Then, for every non-zero element fof \( \mathcal{P}\left( X\right) \) such that \( f\left( Y\right) = \left( 0\right), Y \) is an irred... | Proof. Evidently, the irreducible set \( Y \) is contained in some irreducible component, \( Z \) say, of the set of zeros of \( f \) . By Proposition 1.3, we have \( \dim \left( Z\right) < \dim \left( X\right) \), and \( \dim \left( Y\right) \leq \dim \left( Z\right) \) . Since \( \dim \left( X\right) - \dim \left( Y\... | Yes |
Corollary 1.6. Suppose \( X \) is an irreducible affine variety and \( Y \) is a closed irreducible subset of \( X \), with \( \dim \left( Y\right) = \dim \left( X\right) - r \), where \( r > 0 \) . There are closed irreducible subsets \( {Y}_{i} \) of \( X \) such that \( Y = {Y}_{r} \subset \cdots \subset {Y}_{1} \) ... | Proof. If \( r = 1 \) there is nothing to prove. Suppose \( r > 1 \) and the corollary established in the lower cases. Since \( Y \neq X \) there is a non-zero element \( f \) in \( \mathcal{P}\left( X\right) \) that vanishes on \( Y \) . Now \( Y \) is contained in some irreducible component \( {Y}_{1} \) of the set o... | No |
Corollary 1.7. Let \( X \) be an irreducible affine variety, and let \( {f}_{1},\ldots ,{f}_{r} \) be elements of \( \mathcal{P}\left( X\right) \). Suppose \( Y \) is an irreducible component of the set \( \mathcal{V}\left( {{f}_{1},\ldots ,{f}_{r}}\right) \) of common zeros in \( X \) of the \( {f}_{i} \)’s. Then \( \... | Proof. This follows by induction on \( r \), using Proposition 1.5 for the inductive step. In fact, \( Y \) is an irreducible closed subset of some irreducible component, \( Z \) say, of \( \mathcal{V}\left( {{f}_{1},\ldots ,{f}_{r - 1}}\right) \). Since \( Y \) is a maximal irreducible subset of \( \mathcal{V}\left( {... | Yes |
Lemma 1.8. Let \( T \) be a commutative ring, \( {P}_{1},\ldots ,{P}_{k} \) prime ideals of \( T \), and \( K \) a subset of \( T \) that is closed under addition and multiplication. Then, if the union of the family of \( {P}_{i} \) ’s contains \( K \), one of the \( {P}_{i} \) ’s contains \( K \) . | Proof. Making an induction on \( k \), we may suppose that \( k > 1 \) and that \( K \) is not contained in the union of any proper subset of the set of \( {P}_{i} \) ’s, and show that then \( K \) is not contained in \( \mathop{\bigcup }\limits_{{i = 1}}^{k}{P}_{i} \) . Choose \( {a}_{j} \) from \( K \smallsetminus \m... | Yes |
Theorem 1.9. Let \( X \) be an irreducible affine variety. Suppose that \( {Y}_{1},\ldots ,{Y}_{r} \) are irreducible closed subsets of \( X \) such that \( {Y}_{r} \subset \cdots \subset {Y}_{1} \) and\n\n\[ \dim \left( {Y}_{i}\right) = \dim \left( X\right) - i \]\n\nfor each i. Then there are elements \( {f}_{1},\ldo... | Proof. By Corollary 1.4, there is an element \( {f}_{1} \) in \( \mathcal{P}\left( X\right) \) such that \( {Y}_{1} \) is an irreducible component of \( \mathcal{V}\left( {f}_{1}\right) \) . By Proposition 1.5, every irreducible component of \( \mathcal{V}\left( {f}_{1}\right) \) has dimension \( \dim \left( X\right) -... | Yes |
Theorem 2.1. Suppose \( \sigma : X \rightarrow Y \) is a dominant morphism between irreducible varieties. Let \( W \) be a closed irreducible subset of \( Y \), and let \( Z \) be an irreducible component of \( {\sigma }^{-1}\left( W\right) \) such that \( \sigma \left( Z\right) \) is dense in \( W \). Then\n\n\[ \dim ... | Proof. There is an affine patch \( U \) of \( Y \) such that \( U \cap W \neq \varnothing \). Then \( U \cap W \) is a closed irreducible subset of the irreducible affine variety \( U \), and \( \dim \left( {U \cap W}\right) = \dim \left( W\right) \). Now \( \sigma \) induces a morphism of varieties from \( {\sigma }^{... | Yes |
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