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Theorem 7.13. If \( L/K \) is an algebraic field extension in characteristic \( p \) and if \( {K}_{s} \) is the subfield of all elements of \( L \) that are separable over \( K \), then \( L/{K}_{s} \) is a purely inseparable extension. | Proof. Let \( \alpha \) be an element of \( L \), and let \( f\left( X\right) \) be the minimal polynomial of \( \alpha \) over \( K \) . Then we can write \( f\left( X\right) = g\left( {X}^{{p}^{e}}\right) \), where \( {p}^{e} \) is the degree of inseparability of \( f \) . The polynomial \( g\left( X\right) \) is irr... | Yes |
Proposition 7.15. A field \( K \) is perfect if and only if every algebraic extension of \( K \) is separable. | Proof. We need to consider only the case that \( K \) has characteristic \( p \) . Suppose that \( x \mapsto {x}^{p} \) fails to be onto \( K \) . Choose \( \beta \) in \( K \) such that \( {X}^{p} - \beta \) has no root in \( K \) . Proposition 7.10 shows that \( {X}^{p} - \beta \) is irreducible over \( K \) . Since ... | Yes |
Lemma 7.16. Let \( R \) be a commutative ring with identity, let \( f\\left( X\\right) \) and \( g\\left( X\\right) \) be members of \( R\\left\\lbrack X\\right\\rbrack \) of respective degrees \( m \) and \( n \), and let \( a \) be the leading coefficient of \( g\\left( X\\right) \) . For the integer \( k = \\max \\l... | Proof. If \( m < n \), then \( k = 0 \), and the displayed formula holds with \( q\\left( X\\right) = 0 \) and \( r\\left( X\\right) = f\\left( X\\right) \) . For \( m \\geq n - 1 \), we proceed by induction on \( m \) . The base case of the induction is \( m = n - 1 \), which we have already handled. For the inductive... | Yes |
Lemma 7.17. Let \( L/K \) be a field extension, let \( {x}_{1},\ldots ,{x}_{n},{x}_{n + 1} \) be elements of \( L \), and suppose that \( {x}_{1},\ldots ,{x}_{n} \) are algebraically independent over \( K \) but that \( {x}_{1},\ldots ,{x}_{n},{x}_{n + 1} \) are not algebraically independent. Then the ideal \( I \) of ... | Proof. The algebraic dependence implies that \( I \) contains nonzero polynomials. Let \( g\left( {{X}_{1},\ldots ,{X}_{n},{X}_{n + 1}}\right) \) be one whose degree in \( {X}_{n + 1} \) is as small as possible, say \( l \) . Expand \( g \) as\n\n\[ g = {c}_{0}\left( {{X}_{1},\ldots ,{X}_{n}}\right) {X}_{n + 1}^{l} + {... | Yes |
Lemma 7.19. Suppose that \( L \) is a field extension of transcendence degree \( r \) over a field \( K \) and that \( L \) is not separably generated over \( K \) . If \( {x}_{1},\ldots ,{x}_{n} \) are elements of \( L \) such that \( L = K\left( {{x}_{1},\ldots ,{x}_{n}}\right) \), then for a suitable relabeling of t... | Proof. We fix \( K \) and \( r \), and we proceed by induction on \( n \) . The base case is that \( n = r + 1 \), and then there is nothing to prove. For the inductive step, suppose that the lemma has been proved for \( n - 1 \) when \( n > r + 1 \) . We prove the lemma for \( n \) . Since \( r < n \), we can renumber... | Yes |
Lemma 7.21. Let \( R \) be a commutative ring with identity, let \( {S}^{-1}R \) be the localization relative to a multiplicative system \( S \) in \( R \), let \( I \) be an ideal in \( R \), and let \( \bar{S} \) be the image of \( S \) in \( R/I \) . Then\n\n\[ \n{S}^{-1}R/{S}^{-1}I \cong {\bar{S}}^{-1}\left( {R/I}\... | Proof. Let \( q : R \rightarrow R/I \) and \( \bar{q} : {S}^{-1}R \rightarrow {S}^{-1}R/{S}^{-1}I \) be the quotient homomorphisms, and let \( \eta : R \rightarrow {S}^{-1}R \) and \( \bar{\eta } : R/I \rightarrow {\bar{S}}^{-1}\left( {R/I}\right) \) be the canonical homomorphisms of \( R \) and \( R/I \) into their lo... | Yes |
Theorem 7.23 (Zariski’s Theorem). With \( K \) algebraically closed, let \( I \) be a prime ideal in \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), let \( R = K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /I \), and let \( V\left( I\right) \) be the locus of common zeros of \( I \) in \( {K}^{n} \).... | REMARKS. We are going to prove for each point \( x \) of \( V\left( I\right) \) that\n\n\[ \n{\dim }_{K}\left( {{M}_{x}/{M}_{x}^{2}}\right) = {\dim }_{K}\left( {{\mathfrak{m}}_{x}/{\mathfrak{m}}_{x}^{2}}\right) \n\]\n\nand that\n\n\[ \n{\dim }_{K}\left( {{\mathfrak{m}}_{x}/{\mathfrak{m}}_{x}^{2}}\right) + \operatorname... | No |
Proposition 7.24. The following conditions on a field \( L \) with \( K \subseteq L \subseteq {K}_{\text{alg }} \) are equivalent:\n\n(a) \( L \) is a normal extension of \( K \) ,\n\n(b) \( \operatorname{Gal}\left( {{K}_{\mathrm{{alg}}}/K}\right) \) carries \( L \) to itself,\n\n(c) any \( K \) isomorphism of \( L \) ... | Proof. If (a) holds, let \( L \) be generated by \( K \) and elements \( {\alpha }_{ij} \) as in the paragraph before the proposition. If \( \varphi \) is in \( \operatorname{Gal}\left( {{K}_{\mathrm{{alg}}}/K}\right) \), then \( \varphi \left( {\alpha }_{ij}\right) \) is a root of \( {f}_{i}^{\varphi } = {f}_{i} \) be... | Yes |
Proposition 7.25. Every member of \( \operatorname{Gal}\left( {{K}_{\mathrm{{alg}}}/K}\right) \) carries \( {K}_{\text{sep }} \) into itself, any two members of \( \operatorname{Gal}\left( {{K}_{\mathrm{{alg}}}/K}\right) \) that agree on \( {K}_{\text{sep }} \) are equal on \( {K}_{\mathrm{{alg}}} \), and any field map... | Proof. The first statement has three conclusions to it. For the first conclusion, if \( \varphi \) is in \( \operatorname{Gal}\left( {{K}_{\mathrm{{alg}}}/K}\right) \) and if \( {x}_{0} \) is in \( {K}_{\text{sep }} \), let \( f \) be the minimal polynomial of \( {x}_{0} \) over \( K \) . By separability, \( f \) is a ... | Yes |
Corollary 7.26. Let \( L \) be a field with \( K \subseteq L \subseteq {K}_{\text{sep }} \), form \( \operatorname{Gal}\left( {L/K}\right) \), and let \( {L}^{\operatorname{Gal}\left( {L/K}\right) } \) be the fixed field\n\n\[ \n{L}^{\operatorname{Gal}\left( {L/K}\right) } = \{ x \in L \mid {\gamma x} = x\text{ for all... | Proof. Let \( L \) be normal over \( K \), let \( x \) be in \( {L}^{\operatorname{Gal}\left( {L/K}\right) } \), and let \( f \) be the minimal polynomial of \( x \) over \( K \) . Since \( L \) is normal, \( f \) splits in \( L \) . Since \( L \subseteq {K}_{\text{sep }} \), the roots of \( f \) in \( L \) all have mu... | Yes |
Proposition 7.28. In the category of compact topological groups, any two inverse limits for an inverse system \( \left( {I,\left\{ {G}_{i}\right\} ,\left\{ {f}_{ij}\right\} }\right) \) are canonically isomorphic. | Proof. This is a special case of the uniqueness in category theory of objects having a specific universal mapping property, as established in Basic Algebra. | No |
Proposition 7.30. With the above notation, the group \( \operatorname{Gal}\left( {L/K}\right) \) may be identified with the underlying abstract group of the inverse limit \( \mathop{\lim }\limits_{{L \leftarrow E}}\operatorname{Gal}\left( {E/K}\right) \) , taken over finite normal extensions \( E/K \) with \( E \subset... | Proof. Let \( G = \mathop{\lim }\limits_{{L \leftarrow E}}\operatorname{Gal}\left( {E/K}\right) \), put \( {G}_{E} = \operatorname{Gal}\left( {E/K}\right) \), and regard \( G \) as the standard inverse limit given as in Proposition 7.27:\n\n\[ G = \left\{ {{\left( {\gamma }_{E}\right) }_{E} \in \mathop{\prod }\limits_{... | Yes |
Corollary 7.31. With the notation of Proposition 7.30, give \( \operatorname{Gal}\left( {L/K}\right) \) the inverse-limit topology. If \( F \) is a finite normal extension of \( K \) contained in \( L \) , then \( \operatorname{Gal}\left( {L/F}\right) \) is a closed subgroup of \( \operatorname{Gal}\left( {L/K}\right) ... | Proof. We still need to prove that \( \operatorname{Gal}\left( {L/F}\right) \) has finite index in \( \operatorname{Gal}\left( {L/K}\right) \) . Proposition 7.24 shows that the restriction to \( F \) of any member of \( \operatorname{Gal}\left( {L/K}\right) \) is an automorphism of \( F \) . Since \( F \) is a finite e... | Yes |
Corollary 7.32. With the notation of Proposition 7.30, \( \operatorname{Gal}\left( {L/K}\right) \) has a system of open normal subgroups with intersection \( \{ 1\} \) . Hence the same thing is true of any closed subgroup of \( T \) of \( \operatorname{Gal}\left( {L/K}\right) \) . Moreover, if \( U \) is any open neigh... | Proof. The open normal subgroups in the first conclusion are the subgroups \( \operatorname{Gal}\left( {L/F}\right) \) as in Corollary 7.31. Since every member of \( L \) lies in some finite normal extension of \( K \) within \( L \), a member of \( \operatorname{Gal}\left( {L/K}\right) \) cannot lie in every \( \opera... | Yes |
Theorem 7.34. Let \( K \) be a perfect field, and \( L \) be an algebraically closed field containing \( K \) . Then the only members of \( L \) fixed by every element of \( \operatorname{Gal}\left( {L/K}\right) \) are the members of \( K \) . | Proof. Proposition 7.15 shows that \( {K}_{\text{sep }} = {K}_{\text{alg }} \), and Corollary 7.26 implies that the only members of \( {K}_{\text{alg }} \) fixed by \( \operatorname{Gal}\left( {{K}_{\text{alg }}/K}\right) \) are the members of \( K \) . Thus we are done unless \( L \) contains elements not in \( {K}_{\... | Yes |
Theorem 8.1. If \( A \) is a unique factorization domain and if \( f \) and \( g \) are nonzero members of \( A\left\lbrack X\right\rbrack \) of the form \( f\left( X\right) = \mathop{\sum }\limits_{{i = 0}}^{m}{f}_{i}{X}^{i} \) and \( g\left( X\right) = \mathop{\sum }\limits_{{j = 0}}^{n}{g}_{j}{X}^{j} \) with \( m > ... | Proof. Let us prove the equivalence of (a) and (b). Suppose that (a) holds. If \( u \) is a nonconstant polynomial in \( X \) that divides both \( f \) and \( g \), let us write \( f = {bu} \) and \( g = - {au} \) . Then \( {af} + {bg} = 0 \) . Also, \( \deg a + \deg u = \deg g \) ; since \( \deg u > 0,\deg a < \deg g ... | Yes |
Lemma 8.3. Any polynomial factor of a homogeneous polynomial over a field \( K \) is homogeneous. | Proof. Write \( F = {F}_{1}{F}_{2} \) nontrivially. Let \( {d}_{1} \) and \( {e}_{1} \) be the highest and lowest total degrees of terms in \( {F}_{1} \), and let \( {d}_{2} \) and \( {e}_{2} \) be the highest and lowest total degrees of terms in \( {F}_{2} \) . The product of the terms of total degree \( {d}_{1} \) in... | Yes |
Theorem 8.5 (Bezout’s Theorem). Let \( K \) be a field, let \( {K}_{\text{alg }} \) be an algebraic closure, and suppose that \( F \) in \( K{\left\lbrack X, Y, W\right\rbrack }_{m} \) and \( G \) in \( K{\left\lbrack X, Y, W\right\rbrack }_{n} \) are projective plane curves. Then their locus \( V\left( F\right) \cap V... | Proof. Without loss of generality, we may assume throughout that \( K \) is algebraically closed. Write \( F \) and \( G \) in the form\n\n\[ F\left( {X, Y, W}\right) = {f}_{0} + {f}_{1}W + \cdots + {f}_{m}{W}^{m}\;\text{ with }{f}_{j} \in K{\left\lbrack X, Y\right\rbrack }_{m - j}, \]\n\n\[ G\left( {X, Y, W}\right) = ... | Yes |
Lemma 8.6. In the above notation, \( f\left( {X, Y}\right) \) equals\n\n\[{\left( rX + sY + 1\right) }^{d - 1}{f}_{1}^{\prime }\left( {{\alpha X} + {\beta Y},{\gamma X} + {\delta Y}}\right)\]\n\n\[+ {\left( rX + sY + 1\right) }^{d - 2}{f}_{2}^{\prime }\left( {{\alpha X} + {\beta Y},{\gamma X} + {\delta Y}}\right)\]\n\n... | Proof. For the first conclusion, let us justify the following computation:\n\n\[f\left( {X, Y}\right) = \left( {F \circ {\Psi }^{-1}}\right) \left( {\Psi \circ {\Phi }^{-1}}\right) \left( {X, Y,1}\right)\]\n\n\[= \left( {F \circ {\Psi }^{-1}}\right) \left( {{\alpha X} + {\beta Y},{\gamma X} + {\delta Y},{rX} + {sY} + 1... | Yes |
Over the field \( K \) if a projective line \( L \) and a projective plane curve \( F \) meet at a point \( P \) in \( {\mathbb{P}}_{K}^{2} \), then \( I\left( {P, L \cap F}\right) = + \infty \) if and only if \( L \) divides \( F \) . | Proof. If \( L \) divides \( F \), then in the above notation the local expression \( l\left( {X, Y}\right) \) divides \( f\left( {X, Y}\right) \) . Since \( l\left( {\varphi \left( t\right) }\right) \) is the 0 polynomial, so is \( f\left( {\varphi \left( t\right) }\right) \) .\n\nConversely suppose that \( f\left( {\... | Yes |
Theorem 8.10. Let \( K \) be an algebraically closed field, let \( P \) be in \( {\mathbb{P}}_{K}^{2} \), and let \( F \) and \( G \) be projective plane curves over \( K \) . Then the intersection multiplicity \( I\left( {P, F \cap G}\right) \) has the following properties:\n\n(a) \( I\left( {P, F \cap G}\right) = I\l... | REMARKS. Properties (a) and (b) are evident. Properties (c) and (d) are conversational and will be proved in these remarks. Properties (e), (f), and (g) require proofs, and we give those proofs after computing an example. For (c), if \( P \) lies in \( {V}_{K}\left( F\right) \cap {V}_{K}\left( G\right) \), then the loc... | No |
Lemma 8.11. If \( R \) is a commutative Noetherian ring and \( I \) is an ideal in \( R \) , then \( {\left( \sqrt{I}\right) }^{m} \subseteq I \) for some integer \( m \geq 1 \) . | Proof. Since \( R \) is Noetherian, the ideal \( \sqrt{I} \) is finitely generated. Let \( \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \) be a set of generators for it. By definition of radical, choose integers \( {k}_{1},\ldots ,{k}_{n} \) such that \( {a}_{j}^{{k}_{j}} \) is in \( I \) for \( 1 \leq j \leq n \), and pu... | Yes |
Corollary 8.13. Let \( K \) be an algebraically closed field, and let \( I \) be an ideal in the polynomial ring \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) whose locus of common zeros in \( {K}^{n} \) is a finite set \( \left\{ {{P}_{1},\ldots ,{P}_{k}}\right\} \) . Then \( K\left\lbrack {{X}_{1},\ldots... | Proof. This is a corollary partly of the statement of Theorem 8.12 and partly of the proof. Let \( m \) be as in the proof. If \( {I}_{0} \) is the maximal ideal \( \left( {{X}_{1},\ldots ,{X}_{n}}\right) \) of \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), then \( {I}_{0}^{m} \) is the ideal generated by ... | Yes |
Lemma 8.14. For any field \( K \), let \( {\left\{ {L}_{i}\right\} }_{i \geq 1} \) be a system of nonzero homogeneous polynomials in \( K\left\lbrack {X, Y}\right\rbrack \) of the form \( {L}_{i} = {a}_{i}X + {b}_{i}Y \), let \( {\left\{ {M}_{j}\right\} }_{j \geq 1} \) be another such system with \( {M}_{j} = {c}_{j}X ... | Proof. The set \( \left\{ {{B}_{0},\ldots ,{B}_{n}}\right\} \) has \( n + 1 \) elements, and \( n + 1 \) is the dimension of \( K{\left\lbrack X, Y\right\rbrack }_{n} \) because \( \left\{ {{X}^{n},{X}^{n - 1}Y,\ldots ,{Y}^{n}}\right\} \) is a basis. Thus it is enough to show that \( \left\{ {{B}_{0},\ldots ,{B}_{n}}\r... | Yes |
Lemma 8.17. If \( I \) is an ideal in \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) generated by monomials and if \( f\left( {{X}_{1},\ldots ,{X}_{n}}\right) \) is in \( I \), then each monomial appearing in the expansion of \( f \) with nonzero coefficient lies in \( I \) . Consequently \( I \) has a fini... | Proof. Let \( \left\{ {M}_{\alpha }\right\} \) be the set of monomials that generates \( I \) . If \( f \) is in \( I \), then we can write \( f = \mathop{\sum }\limits_{{j = 1}}^{k}{h}_{j}{M}_{{\alpha }_{j}} \) for polynomials \( {h}_{j} \) . Let \( {h}_{j} = \mathop{\sum }\limits_{{i = 1}}^{{l}_{j}}{c}_{ij}{M}_{ij} \... | Yes |
Proposition 8.19. If \( K \) is a field, if a monomial ordering is specified for \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), and if \( \left\{ {{g}_{1},\ldots ,{g}_{k}}\right\} \) is a Gröbner basis for a nonzero ideal \( I \) of \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), then \( \left\... | Proof. First we prove that if \( f \neq 0 \) is in \( I \), then there exist a \( {g}_{j} \), a monomial \( {M}_{0} \), and a nonzero scalar \( c \) such that \( \operatorname{LM}\left( {f - c{M}_{0}{g}_{j}}\right) < \operatorname{LM}\left( f\right) \) . To see this, we use the hypothesis that \( \left\{ {{g}_{1},\ldot... | Yes |
Proposition 8.20 (generalized division algorithm). Let \( \left( {{f}_{1},\ldots ,{f}_{s}}\right) \) be a fixed enumeration of a set of nonzero members of \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), and let \( f \) be an arbitrary nonzero member of \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack... | Proof. We shall do a kind of induction involving decompositions of \( f \) of the form\n\n\[ f = \left( {{a}_{1}{f}_{1} + \cdots + {a}_{s}{f}_{s}}\right) + p + r, \]\n\n\( \left( *\right) \)\n\nwhere \( {a}_{1},\ldots ,{a}_{s}, p, r \) are polynomials with the properties that\n\n(i) \( \operatorname{LM}\left( p\right) ... | Yes |
If \( \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \) is a Gröbner basis of a nonzero ideal \( I \) of \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) and if \( f \) is any nonzero member of \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), then there exist polynomials \( g \) and \( r \) such that \(... | For existence, let \( \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \) be a Gröbner basis of \( I \), and apply Proposition 8.20 to \( f \) and the ordered set \( \left( {{g}_{1},\ldots ,{g}_{s}}\right) \) . Then the existence follows immediately.\n\nFor uniqueness, suppose that \( f = {g}_{1} + {r}_{1} = {g}_{2} + {r}_{2}... | Yes |
Lemma 8.22. Let \( M \) and \( {M}_{1},\ldots ,{M}_{s} \) be monomials, let \( {f}_{1},\ldots ,{f}_{s} \) be nonzero polynomials, and suppose that \( {M}_{i}\operatorname{LM}\left( {f}_{i}\right) = M \) for all \( i \) . If \( {c}_{1},\ldots ,{c}_{s} \) are constants such that \( \operatorname{LM}\left( {\mathop{\sum }... | Proof. Let us write \( {L}_{ij} = \operatorname{LCM}\left( {\operatorname{LM}\left( {f}_{i}\right) ,\operatorname{LM}\left( {f}_{j}\right) }\right) \) for \( i \neq j \) . We may assume that all the \( {c}_{i} \) are nonzero, and we proceed by induction on \( s \) . There is nothing to prove for \( s = 1 \) . The key s... | Yes |
Theorem 8.23. Let \( \\left\\{ {{g}_{1},\\ldots ,{g}_{s}}\\right\\} \) be a set of generators of a nonzero ideal \( I \) of \( K\\left\\lbrack {{X}_{1},\\ldots ,{X}_{n}}\\right\\rbrack \), and assume that \( {g}_{i} \\neq 0 \) for all \( i \) . Then the following conditions on \( \\left\\{ {{g}_{1},\\ldots ,{g}_{s}}\\r... | Proof. We prove that (a) implies (b) and that (c) implies (a). Since (b) certainly implies (c), the proof will be complete.\n\nLet (a) hold, i.e., let \( \\left\\{ {{g}_{1},\\ldots ,{g}_{s}}\\right\\} \) be a Gröbner basis. If \( S\\left( {{g}_{j},{g}_{k}}\\right) \\neq 0 \), then \( S\\left( {{g}_{j},{g}_{k}}\\right) ... | Yes |
Corollary 8.24 (Buchberger’s algorithm). \( {}^{15} \) Each nonzero ideal in the polynomial ring \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) has a Gröbner basis. Such a basis can be obtained by the following procedure: Start from any set \( \left\{ {{f}_{1},\ldots ,{f}_{t}}\right\} \) of nonzero generato... | Proof. At the stage of the iteration that works with the set \( \left\{ {{f}_{1}^{\prime },\ldots ,{f}_{{t}^{\prime }}^{\prime }}\right\} \) of generators, any nonzero remainder \( r \) that arises has the property that no monomial occurring in \( r \) is divisible by any \( \operatorname{LM}\left( {f}_{j}^{\prime }\ri... | Yes |
Corollary 8.25 (solution of the ideal-membership problem). If \( I \) is a nonzero ideal in \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) and \( f \) is a polynomial, then a procedure for deciding whether \( f \) lies in \( I \) is as follows: introduce a monomial ordering, construct a Gröbner basis \( \le... | Proof. Corollary 8.24 produces the Gröbner basis, and Corollary 8.21 affirms that this procedure decides whether \( f \) lies in \( I \) . | Yes |
Corollary 8.26 (solution of the proper-ideal problem). If \( I \) is a nonzero ideal in \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), then a procedure for deciding whether \( I = K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) is to compute a Gröbner basis for \( I \) and to see whether one of its... | Proof. If \( I \) has a nonzero scalar as one of its generators, then 1 lies in \( I \) , and hence \( I \) certainly equals \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) . Conversely if \( I \) is given, then Corollary 8.24 produces a Gröbner basis \( \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \) . Since \... | Yes |
Lemma 8.27. If \( \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \) is a Gröbner basis for a nonzero ideal \( I \) in \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) and if \( \operatorname{LM}\left( {g}_{1}\right) \) lies in the ideal \( \left( {\operatorname{LT}\left( {g}_{2}\right) ,\ldots ,\operatorname{LT}\l... | Proof. By hypothesis, \( \left( {\operatorname{LT}\left( {g}_{2}\right) ,\ldots ,\operatorname{LT}\left( {g}_{s}\right) }\right) = \left( {\operatorname{LT}\left( {g}_{1}\right) ,\ldots ,\operatorname{LT}\left( {g}_{s}\right) }\right) = \operatorname{LT}\left( I\right) \) . Therefore \( \left\{ {{g}_{2},\ldots ,{g}_{s}... | Yes |
Theorem 8.28 (uniqueness of reduced Gröbner basis). If \( I \) is a nonzero ideal in \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), then \( I \) has a unique reduced Gröbner basis, and this can be obtained algorithmically starting from any minimal Gröbner basis. | Proof of uniqueness. Let \( \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \) be any Gröbner basis. Since \( \operatorname{LT}\left( I\right) = \) \( \left( {\operatorname{LT}\left( {g}_{1}\right) ,\ldots ,\operatorname{LT}\left( {g}_{s}\right) }\right) \), Lemma 8.17 shows that any \( \operatorname{LM}\left( f\right) \) fo... | Yes |
Corollary 8.29 (solution of the ideal-equality problem). Let \( I \) and \( J \) be two nonzero ideals in \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) specified in terms of finite sets of generators. Then \( I = J \) if and only if the reduced Gröbner bases of \( I \) and \( J \) relative to a single mono... | Proof. This result is immediate from Corollary 8.24 (constructive existence of Gröbner bases) and Theorem 8.28. | Yes |
Theorem 8.30 (Elimination Theorem). Let \( K \) be any field, let \( I \) be a nonzero ideal in \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), let \( 0 \leq k \leq n \), and fix a monomial ordering of \( k \) -elimination type. If \( \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \) is a Gröbner basis of \( I \... | Proof. Relabeling the members of \( \left\{ {{g}_{1},\ldots ,{g}_{s}}\right\} \), we may assume that the \( {g}_{j} \) ’s lying in \( J \) are \( {g}_{1},\ldots ,{g}_{t} \) . The first step is to show that \( J = \left( {{g}_{1},\ldots ,{g}_{t}}\right) \) . If \( f \in J \) is given, we apply the generalized division a... | Yes |
Theorem 8.31 (Extension Theorem). Let \( K \) be an algebraically closed field, let \( I = \left( {{f}_{1},\ldots ,{f}_{s}}\right) \) be an ideal in \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \), and let \( J \) be the first elimination ideal of \( I \) in \( K\left\lbrack {{X}_{2},\ldots ,{X}_{n}}\right\r... | Proof of THEOREM 8.31. Let us abbreviate \( \bar{X} = \left( {{X}_{2},\ldots ,{X}_{n}}\right) \) and \( \bar{c} = \) \( \left( {{c}_{2},\ldots ,{c}_{n}}\right) \) ; we shall write \[ \left( {{X}_{1},\bar{X}}\right) = \left( {{X}_{1},\ldots ,{X}_{n}}\right) \;\text{ and }\;\left( {{X}_{1},\bar{c}}\right) = \left( {{X}_{... | Yes |
Let \( \mathbb{F} \) be a function field in one variable over \( \mathbb{k} \), and let \( {\mathbb{k}}^{\prime } \) be the subfield of all elements in \( \mathbb{F} \) algebraic over \( \mathbb{k} \) . If \( x \) is in \( {\mathbb{F}}^{ \times } \), then every discrete valuation of \( \mathbb{F} \) defined over \( \ma... | Proof. If \( x \in \mathbb{F} \) is transcendental over \( \mathbb{k} \), then the observation at the end of Section 1 produces discrete valuations of \( \mathbb{F} \) defined over \( \mathbb{k} \) that take nonzero values on \( x \) . Conversely if \( x \in {\mathbb{F}}^{ \times } \) is algebraic over \( \mathbb{k} \)... | Yes |
Proposition 9.2. Let \( v \) be any discrete valuation of \( \mathbb{F} \) defined over \( \mathbb{k} \), let \( {R}_{v} \) be the valuation ring, and let \( {P}_{v} \) be the valuation ideal. Then \( {R}_{v} \) and \( {P}_{v} \) are \( \mathbb{k} \) vector spaces, and \( {\dim }_{\mathbb{k}}{R}_{v}/{P}_{v} \) is finit... | Proof. The fact that \( {R}_{v} \) and \( {P}_{v} \) are \( \mathbb{k} \) vector spaces is immediate from Proposition 9.1. Since \( v \) is not identically zero, there exists some \( x \in \mathbb{F} \) with \( v\left( x\right) \neq 0 \), and \( x \) is transcendental by Proposition 9.1. Possibly replacing \( x \) by \... | Yes |
Theorem 9.3. The degree of every principal divisor is 0 . In more detail, if \( \left( x\right) \) is a principal divisor with \( x \) not in \( \mathbb{k} \), then \( \deg {\left( x\right) }_{0} = \deg {\left( x\right) }_{\infty } = {\dim }_{\mathbb{k}\left( x\right) }\mathbb{F} \), and hence \( \deg \left( x\right) =... | Proof. If \( x \) is in \( {\mathbb{k}}^{ \times } \), then Proposition 9.1 shows that \( v\left( x\right) = 0 \) for every \( v \in {\mathbb{V}}_{\mathbb{F}} \), and hence \( \deg \left( x\right) = 0 \) . Thus we may assume that \( x \) is transcendental over \( \mathbb{k} \) . Applying the observation at the end of S... | Yes |
Corollary 9.4. \( L\left( 0\right) = \mathbb{k} \), and \( L\left( A\right) = 0 \) if \( A \) is a nonzero divisor with \( A \leq 0 \) . | Proof. If \( A \leq 0 \) is nontrivial and if \( x \in {\mathbb{F}}^{ \times } \) were to have \( \left( x\right) \geq - A \), then we would have \( \deg \left( x\right) \geq - \deg A > 0 \), in contradiction to the conclusion \( \deg \left( x\right) = 0 \) of Theorem 9.3. Thus \( L\left( A\right) = 0 \) . Next, we hav... | Yes |
Lemma 9.5. If \( A \) is a divisor and \( {v}_{0} \) is in \( {\mathbb{V}}_{\mathbb{F}} \), then\n\n\[ \n{\dim }_{\mathbb{k}}L\left( {A + {v}_{0}}\right) /L\left( A\right) \leq {f}_{{v}_{0}} = \deg {v}_{0}.\n\] | Proof. Put \( f = {f}_{{v}_{0}} \), let \( {R}_{{v}_{0}} \) be the valuation ring of \( {v}_{0} \), and let \( {P}_{{v}_{0}} \) be the valuation ideal of \( {v}_{0} \) . Since \( {v}_{0} \) carries \( {\mathbb{F}}^{ \times } \) onto \( \mathbb{Z} \), we can choose an element \( y \in {\mathbb{F}}^{ \times } \) with \( ... | Yes |
Theorem 9.6. If \( A \) and \( B \) are divisors such that \( A \leq B \), then \( L\left( B\right) /L\left( A\right) \) is finite-dimensional over \( \mathbb{k} \) with\n\n\[{\dim }_{\mathbb{k}}L\left( B\right) /L\left( A\right) \leq \deg B - \deg A.\]\n\nMoreover, \( L\left( A\right) \) and \( L\left( B\right) \) are... | Proof. The first conclusion is immediate from Lemma 9.5 by induction on \( \mathop{\sum }\limits_{v}\left( \right. {\operatorname{ord}}_{v}B - {\operatorname{ord}}_{v}A \) ). Fixing a reference point \( {v}_{0} \) in \( {\mathbb{V}}_{\mathbb{F}} \) and taking \( A = \)\n\n\( \mathop{\sum }\limits_{{{\operatorname{ord}}... | Yes |
Lemma 9.8. If \( A \) is any divisor and \( x \) is any member of \( {\mathbb{F}}^{ \times } \), then \( L\left( {\left( x\right) + A}\right) \cong \) \( L\left( A\right) \) canonically. Therefore \( \ell \left( {\left( x\right) + A}\right) = \ell \left( A\right) \) . In addition, \( \deg \left( {\left( x\right) + A}\r... | Proof. Define a \( \mathbb{k} \) linear mapping \( \varphi : L\left( A\right) \rightarrow \mathbb{F} \) by \( \varphi \left( y\right) = {x}^{-1}y \) . This is certainly one-one, and its image is contained in \( L\left( {\left( x\right) + A}\right) \) because any nonzero \( z \) in \( L\left( A\right) \) has \( \left( z... | Yes |
Theorem 9.9 (Riemann’s inequality). For each \( x \) in \( \mathbb{F} \) that is not in \( \mathbb{k} \), let \( {g}_{x} \) be the integer such that \( 1 - {g}_{x} \) is the largest possible \( {C}_{x} \) with\n\n\[ \ell \left( {p{\left( x\right) }_{\infty }}\right) - \deg \left( {p{\left( x\right) }_{\infty }}\right) ... | Proof. We begin by proving (c) with \( g \) replaced by \( {g}_{x} \) . Let \( {C}_{x} \) be any integer with the property that \( \ell \left( {p{\left( x\right) }_{\infty }}\right) - \deg \left( {p{\left( x\right) }_{\infty }}\right) \geq {C}_{x} \) for all \( p \) . If a divisor \( A \) is given, we can write \( A = ... | Yes |
Proposition 9.10. If \( A \) is any divisor such that \( L\left( A\right) \neq 0 \), then\n\n\[ \ell \left( A\right) - \deg A \leq 1 \]\n\nHence any divisor \( A \) with \( \deg A \leq - 1 \) has \( \ell \left( A\right) = 0 \) . | Proof. Let \( y \) be a member of \( {\mathbb{F}}^{ \times } \) that lies in \( L\left( A\right) \) . Then every \( v \in {\mathbb{V}}_{\mathbb{F}} \) has \( v\left( y\right) \geq - {\operatorname{ord}}_{v}A \) and hence \( 0 \geq - {\operatorname{ord}}_{v}A - v\left( y\right) = - {\operatorname{ord}}_{v}\left( {A + \l... | Yes |
Lemma 9.12. If \( A \) and \( B \) are divisors with \( A \leq B \), then there is an exact sequence in the category of \( \mathbb{k} \) vector spaces given by\n\n\[ 0 \rightarrow L\left( B\right) /L\left( A\right) \overset{\psi }{ \rightarrow }\mathcal{L}\left( B\right) /\mathcal{L}\left( A\right) \]\n\n\[ \overset{\v... | Proof. The map \( \psi \) is induced by the map \( \Delta : L\left( B\right) \rightarrow \mathcal{L}\left( B\right) \) followed by passage to the quotient. It descends to \( L\left( B\right) /L\left( A\right) \) because \( \Delta \left( {L\left( A\right) }\right) \subseteq \mathcal{L}\left( A\right) \) , and it is one-... | Yes |
Theorem 9.13. There exists a divisor \( C \) such that \( {\mathcal{A}}_{\mathbb{F}} = \mathcal{L}\left( C\right) + \Delta \left( \mathbb{F}\right) \) . For each divisor \( A \) , \[ \delta \left( A\right) = {\dim }_{\mathbb{k}}\left( {{\mathcal{A}}_{\mathbb{F}}/\left( {\mathcal{L}\left( A\right) + \Delta \left( \mathb... | Proof. Riemann’s inequality produces a divisor \( C \), specifically any sufficiently large positive power of a divisor \( {\left( x\right) }_{\infty } \), such that \( \delta \left( C\right) = 0 \) . If we can show that \( {\mathcal{A}}_{\mathbb{F}} = \mathcal{L}\left( C\right) + \Delta \left( \mathbb{F}\right) \), th... | Yes |
Theorem 9.16 (Riemann-Roch Theorem). Let \( \\mathbb{F} \) be a function field in one variable over a field \( \\mathbb{k} \), and suppose that every member of \( \\mathbb{F} \) not in \( \\mathbb{k} \) is transcendental over \( \\mathbb{k} \). If \( A \) is any divisor of \( \\mathbb{F} \) and \( C \) is any canonical... | Proof. Lemma 9.15 shows that there exists a nonzero differential \( {\\omega }_{0} \). Let \( {C}_{0} = \\operatorname{Div}\\left( {\\omega }_{0}\\right) \). Lemma 9.15 shows that \( C = {C}_{0} + \\left( {y}_{0}\\right) \) for some \( {y}_{0} \\in {\\mathbb{F}}^{ \\times } \). Then \( \\omega = {y}_{0}{\\omega }_{0} \... | Yes |
Corollary 9.17. If \( C \) is any canonical divisor, then \( \ell \left( C\right) = g \) . | Proof. Put \( A = 0 \) in Theorem 9.16, and use the fact given in Corollary 9.4 that \( \ell \left( 0\right) = 1 \) . | Yes |
Corollary 9.18. If \( C \) is any canonical divisor, then \( \deg C = {2g} - 2 \) . | Proof. Put \( A = C \) in Theorem 9.16, and apply Corollary 9.17 and Corollary 9.4. | No |
Corollary 9.19. Any divisor \( A \) with \( \deg A > {2g} - 2 \) has \( \delta \left( A\right) = 0 \), i.e., \( \ell \left( A\right) = \deg A + \left( {1 - g}\right) . | Proof. If \( \deg A > {2g} - 2 \), then it follows from Corollary 9.18 that \( \deg \left( {C - A}\right) < 0 \) . By Proposition 9.10, \( \ell \left( {C - A}\right) = 0 \) . Then the corollary is immediate from Theorem 9.16. | Yes |
Corollary 9.20. If \( A \) is a divisor with \( \deg A = {2g} - 2 \), then either \( A \) is a canonical divisor and \( \ell \left( A\right) = g \), or \( A \) is not a canonical divisor and \( \ell \left( A\right) = g - 1 \) . | Proof. If \( A \) is a canonical divisor, then \( \ell \left( A\right) = g \) by Corollary 9.17. Otherwise, the divisor \( C - A \), which has degree 0 by Corollary 9.18, is not a principal divisor. Any nonzero \( y \) in \( L\left( {C - A}\right) \) then would have \( \left( y\right) \geq - \left( {C - A}\right) \) an... | Yes |
Corollary 9.21. If \( {v}_{0} \) is in \( {\mathbb{V}}_{\mathbb{F}} \) and \( n > \max \left( {{2g} - 1,0}\right) \), then there exists a nonscalar \( x \) in \( {\mathbb{F}}^{ \times } \) with \( {\left( x\right) }_{\infty } \leq n{v}_{0} \) . | Proof. Let \( A = n{v}_{0} \), and let \( {f}_{{v}_{0}} \) be the residue class degree of \( {v}_{0} \) . Then \( \deg A = n{f}_{{v}_{0}} \geq n > \max \left( {{2g} - 1,0}\right) \), and Corollary 9.19 gives\n\n\[ \ell \left( A\right) = \deg A + \left( {1 - g}\right) = n{f}_{{v}_{0}} + \left( {1 - g}\right) \]\n\n\[ > ... | Yes |
Corollary 9.22. If \( \mathbb{k} \) is algebraically closed, if \( {v}_{0} \) is in \( {\mathbb{V}}_{\mathbb{F}} \), and if \( g = 1 \), then every \( x \) in \( \mathbb{F} \) with \( {\left( x\right) }_{\infty } \leq {v}_{0} \) is a scalar multiple of the identity. | Proof. Put \( A = {v}_{0} \) . We seek \( x \in \mathbb{F} \) with \( {v}_{0}\left( x\right) \geq - 1 = - {\operatorname{ord}}_{{v}_{0}}A \) and with \( v\left( x\right) \geq 0 = - {\operatorname{ord}}_{v}A \) for all other \( v \) . Thus we seek \( x \) in \( L\left( A\right) \) . This \( A \) has \( \deg A = 1 = g = ... | Yes |
Corollary 9.23. If \( \mathbb{k} \) is algebraically closed, if \( {v}_{0} \) is in \( {\mathbb{V}}_{\mathbb{F}} \), and if \( g > 1 \), then every \( x \) in \( \mathbb{F} \) with \( {\left( x\right) }_{\infty } \leq {v}_{0} \) is a scalar multiple of the identity. | Proof. We argue by contradiction. Suppose that \( x \) is a nonscalar element in \( L\left( {v}_{0}\right) \) . Take \( r = {2g} - 1 \), and let \( {c}_{1},\ldots ,{c}_{r} \) be distinct members of \( \mathbb{k} \) . For each \( j \) with \( 1 \leq j \leq r, x - {c}_{j} \) is in \( L\left( {v}_{0}\right) \) . Since \( ... | Yes |
Proposition 10.1. Affine algebraic sets in \( {\mathbb{A}}^{n} \) have the following properties:\n\n(a) \( V\left( \varnothing \right) = V\left( 0\right) = {\mathbb{A}}^{n} \) and \( V\left( A\right) = \varnothing \) ,\n\n(b) \( V\left( {\mathop{\bigcup }\limits_{\alpha }{S}_{\alpha }}\right) = \mathop{\bigcap }\limits... | Proof. Property (a) is immediate. For (b), we have\n\n\[ V\left( {\mathop{\bigcup }\limits_{\alpha }{S}_{\alpha }}\right) = \mathop{\bigcap }\limits_{{f \in \mathop{\bigcup }\limits_{\alpha }{S}_{\alpha }}}V\left( f\right) = \mathop{\bigcap }\limits_{\alpha }\mathop{\bigcap }\limits_{{f \in {S}_{\alpha }}}V\left( f\rig... | Yes |
Proposition 10.2. For fixed \( n \), the function \( I\left( \cdot \right) \) has the following properties:\n\n(a) \( I\left( \varnothing \right) = A \) and \( I\left( A\right) = 0 \) ,\n\n(b) \( I\left( {{E}_{1} \cup {E}_{2}}\right) = I\left( {E}_{1}\right) \cap I\left( {E}_{2}\right) \) if \( {E}_{1} \) and \( {E}_{2... | Proof. Property (a) is immediate. For (b), we have\n\n\[ I\left( {{E}_{1} \cup {E}_{2}}\right) = \mathop{\bigcap }\limits_{{P \in {E}_{1} \cup {E}_{2}}}I\left( {\{ P\} }\right) = \left( {\mathop{\bigcap }\limits_{{P \in {E}_{1}}}I\left( {\{ P\} }\right) }\right) \cap \left( {\mathop{\bigcap }\limits_{{P \in {E}_{2}}}I\... | Yes |
Corollary 10.4. The affine varieties in \( {\mathbb{A}}^{n} \) are characterized as those nonempty Zariski closed sets that cannot be written as the union of two proper closed subsets. | Proof. Let \( V\left( \mathfrak{p}\right) \) be an affine variety with \( \mathfrak{p} \) prime, and suppose that \( V\left( \mathfrak{p}\right) = \) \( {E}_{1} \cup {E}_{2} \) with \( {E}_{1} \) and \( {E}_{2} \) both closed and properly contained in \( V\left( \mathfrak{p}\right) \) . Application of \( I\left( \cdot ... | Yes |
Proposition 10.5. If \( X \) is a Noetherian topological space, then any closed subset is the finite union of irreducible closed subsets. This decomposition of a closed set as such a union may be chosen in such a way that none of the closed sets in the union contains another set in the union, and in this case the decom... | Proof. For existence of some decomposition of each closed set as a finite union of irreducible closed subsets, we argue by contradiction. Assuming that there exists some closed subset \( E \) of \( X \) that is not the finite union of irreducible closed subsets, we may assume by the Noetherian condition on \( X \) that... | Yes |
Corollary 10.6. Every affine algebraic set in \( {\mathbb{A}}^{n} \) can be expressed uniquely as the finite (possibly empty) union of affine varieties in such a way that none of the varieties contains another of the varieties. | Proof. We saw before Proposition 10.5 that \( {\mathbb{A}}^{n} \) is a Noetherian topological space, and Corollary 10.4 shows that the irreducible subsets are the affine varieties. The closed sets are the affine algebraic sets by definition, and hence the result is a special case of Proposition 10.5. | No |
Lemma 10.8. Every minimal nonzero prime ideal in \( A \) is principal. | Proof. Let \( \mathfrak{p} \) be a minimal nonzero prime ideal, let \( f \neq 0 \) be a nonzero member, and write \( f \) as the product of irreducible elements. Since \( \mathfrak{p} \) is prime, one of the irreducible elements, say \( g \), lies in \( \mathfrak{p} \). Since \( A \) is a unique factorization domain, \... | Yes |
Proposition 10.9. Suppose that \( \mathfrak{p} \) is a prime ideal of \( A \) and \( V\left( \mathfrak{p}\right) \) is the corresponding affine variety. If \( \dim V\left( \mathfrak{p}\right) = n - 1 \), then \( \mathfrak{p} \) is principal, and hence \( V\left( \mathfrak{p}\right) \) is an irreducible hypersurface. | Proof. For any \( n \geq 1,\dim V\left( \mathfrak{p}\right) = n - 1 < n = \dim V\left( 0\right) \) implies \( \mathfrak{p} \neq 0 \) . Since \( \dim V\left( \mathfrak{p}\right) = n - 1 \), there exists a chain\n\n\[ 0 = {\mathfrak{q}}_{0} \subsetneqq {\mathfrak{q}}_{1} \subsetneqq \cdots \subsetneqq {\mathfrak{q}}_{n -... | Yes |
Lemma 10.10. If \( Y \) is a quasi-affine variety in \( {\mathbb{A}}^{n} \) and if \( E \) is a nonempty relatively closed subset of \( Y \), then \( E \) is irreducible \( {}^{8} \) for \( Y \) if and only if \( \bar{E} \) is irreducible for \( {\mathbb{A}}^{n} \). | Proof. First we check that \( E \) reducible implies \( \bar{E} \) reducible. If \( E \) is reducible, say is a union \( E = {E}_{1} \cup {E}_{2} \) with \( {E}_{1} \) and \( {E}_{2} \) relatively closed proper subsets of \( E \), then \( \bar{E} = \overline{{E}_{1}} \cup \overline{{E}_{2}} \) . Each of \( \overline{{E... | Yes |
Proposition 10.11. If \( Y \) is a quasi-affine variety in \( {\mathbb{A}}^{n} \), then \( \dim Y = \dim \bar{Y} \) . Here \( \dim \bar{Y} \) refers to the dimension of the affine variety \( \bar{Y} \) in any of the senses of Theorem 10.7. | Proof. Let \( {E}_{0} \subseteq {E}_{1} \subseteq \cdots \subseteq {E}_{d} \) be a strictly increasing sequence of relatively closed irreducible subsets of \( Y \) . Then \( {\bar{E}}_{0} \subseteq {\bar{E}}_{1} \subseteq \cdots \subseteq {\bar{E}}_{d} \) is an increasing sequence of closed subsets of \( {\mathbb{A}}^{... | Yes |
Corollary 10.13. The projective varieties in \( {\mathbb{P}}^{n} \) are characterized as those nonempty Zariski closed sets that cannot be written as the union of two proper closed subsets. | Proof. If \( V\left( \mathfrak{p}\right) \) is a projective variety, then the union of \( \{ 0\} \) and the subset of \( {\mathbb{k}}^{n + 1} \) whose equivalence classes are in \( V\left( \mathfrak{p}\right) \) is an affine variety in \( {\mathbb{A}}^{n + 1} \) . It is irreducible in \( {\mathbb{A}}^{n + 1} \) , and t... | Yes |
Lemma 10.16. If \( \mathfrak{a} \) is a homogeneous ideal in \( \widetilde{A} \) and \( \mathfrak{b} = {\beta }_{0}^{t}\left( \mathfrak{a}\right) \) is its image under \( {\beta }_{0}^{t} \), then \( {\beta }_{0}^{t} \) carries the set of homogeneous elements of \( \mathfrak{a} \) onto \( \mathfrak{b} \) . | Proof. Every member of \( \mathfrak{b} \) is the sum of the images under \( {\beta }_{0}^{t} \) of finitely many homogeneous members of \( \mathfrak{a} \) . If \( {F}_{1},\ldots ,{F}_{k} \) are these homogeneous members, then it is enough to produce \( {G}_{1},\ldots ,{G}_{k} \) in a all homogeneous of the same degree ... | Yes |
Lemma 10.17. Let \( \mathfrak{a} \) be a homogeneous ideal of \( \widetilde{A} \), and let \( \mathfrak{b} \) be the ideal of \( A \) given by \( \mathfrak{b} = {\beta }_{0}^{t}\left( \mathfrak{a}\right) \) . Then \( {\beta }_{0}\left( {V\left( \mathfrak{b}\right) }\right) = V\left( \mathfrak{a}\right) \cap {\beta }_{0... | Proof. If \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) is in \( V\left( \mathfrak{b}\right) \) and if \( F \) is a homogeneous member of \( \mathfrak{a} \) , then \( f = {\beta }_{0}^{t}\left( F\right) \) is in \( \mathfrak{b} \) with \( 0 = f\left( {{x}_{1},\ldots ,{x}_{n}}\right) = F\left( {{\beta }_{0}\left( {{x}_{... | Yes |
Proposition 10.18. Under the inclusion \( {\beta }_{0} : {\mathbb{A}}^{n} \rightarrow {\mathbb{P}}^{n} \), the Zariski topology of affine \( n \) -space \( {\mathbb{A}}^{n} \) coincides with the relative topology from \( {\mathbb{P}}^{n} \). | Proof. If we start from an affine algebraic set \( V\left( \mathfrak{b}\right) \) in \( {\mathbb{A}}^{n} \), then Lemma 10.17 shows that \( {\beta }_{0}\left( {V\left( \mathfrak{b}\right) }\right) = V\left( \mathfrak{a}\right) \cap {\beta }_{0}\left( {\mathbb{A}}^{n}\right) \) for the homogeneous ideal \( \mathfrak{a} ... | Yes |
Corollary 10.19. If \( V \) is a quasi-affine variety in \( {\mathbb{A}}^{n} \), then \( {\beta }_{0}\left( V\right) \) is a quasipro-jective variety in \( {\mathbb{P}}^{n} \) . Moreover, the geometric dimension of \( V \) as a quasi-affine variety equals the geometric dimension of \( {\beta }_{0}\left( V\right) \) as ... | Proof. Because of the homeomorphism given by Proposition 10.18, Lemma 10.10 as restated in the lemma’s remarks applies with \( Y = {\beta }_{0}\left( {\mathbb{A}}^{n}\right), X = {\mathbb{P}}^{n} \), and \( E \) equal to the closure of \( V \) in \( {\mathbb{A}}^{n} \) . The conclusion is that the closure of \( E \) in... | Yes |
Corollary 10.21. The inclusion \( {\beta }_{0} : {\mathbb{A}}^{n} \rightarrow {\mathbb{P}}^{n} \) sets up a one-one correspondence between the prime ideals in \( A \) and those prime homogeneous ideals in \( \widetilde{A} \) that do not contain \( {X}_{0} \) . | Proof. If \( \mathfrak{a} \) is a prime homogeneous ideal in \( \widetilde{A} \) and \( {X}_{0}F \) is in \( \mathfrak{a} \), then either \( {X}_{0} \) or \( F \) is in \( \mathfrak{a} \). If we can always exclude \( {X}_{0} \) from being in \( \mathfrak{a} \), then \( F \) is in \( \mathfrak{a} \), and the condition i... | No |
Corollary 10.22. Let \( \mathfrak{a} \) be a prime homogeneous ideal of \( \widetilde{A} \) not containing \( {X}_{0} \), and let \( \mathfrak{b} = {\beta }_{0}^{t}\left( \mathfrak{a}\right) \) be the corresponding prime ideal of \( A \) . Then the Zariski closure in \( {\mathbb{P}}^{n} \) of \( {\beta }_{0}\left( {V\l... | Proof. Corollary 10.19 shows that \( \overline{{\beta }_{0}\left( {V\left( \mathfrak{b}\right) }\right) } = V\left( {\mathfrak{a}}^{\prime }\right) \) for some prime homogeneous ideal of \( \widetilde{A} \) . Since \( {\beta }_{0}\left( {V\left( \mathfrak{b}\right) }\right) \subseteq V\left( \mathfrak{a}\right) \) by L... | Yes |
Proposition 10.24. If \( f \) is a rational function on the affine variety \( V = V\left( \mathfrak{p}\right) \) , then the pole set of \( f \) is the affine algebraic set \( V\left( \mathfrak{a}\right) \subseteq V\left( \mathfrak{p}\right) \) corresponding to the ideal \( \mathfrak{a} \supseteq \mathfrak{p} \) of all ... | Proof. The set \( \mathfrak{a} \) in the statement is an ideal in \( A \) that contains \( \mathfrak{p} \) . Hence \( V\left( \mathfrak{a}\right) \subseteq V\left( \mathfrak{p}\right) \) . If \( P \) is in \( V\left( \mathfrak{p}\right) \) and \( f \) is defined at \( P \), then there are members \( \bar{a} \) and \( \... | Yes |
Proposition 10.26. If a rational function \( f \) on the affine variety \( V \) is regular on the nonempty open set \( U \) of \( V \), then it is continuous from \( U \) into \( {\mathbb{A}}^{1} \) with the Zariski topology (in which the proper closed sets are the finite sets). | Proof. It is to be proved that \( {f}^{-1} \) of any finite subset of \( {\mathbb{A}}^{1} \) is relatively closed in \( U \) . Since the finite union of closed sets is closed, it is enough to consider \( {f}^{-1}\left( {\{ c\} }\right) \) for an element \( c \) of \( \mathbb{k} \) . This is the intersection with \( U \... | Yes |
Lemma 10.27. If \( V \) is an affine variety, then any two members of the affine coordinate ring \( A\left( V\right) \) that are equal on a nonempty open subset of \( V \) are the same. | Proof. Subtracting, we may suppose that \( \bar{a} \in A\left( V\right) \) is 0 on the nonempty open subset \( U \) of \( V \) . By Proposition 10.26, \( \bar{a} \) is continuous from \( V \) into \( {\mathbb{A}}^{1} \) . The complement of \( {\bar{a}}^{-1}\left( {\{ 0\} }\right) \) has to be open in \( V \) and disjoi... | Yes |
Proposition 10.28. Let \( U \) be a nonempty open subset of the affine variety \( V \) in \( {\mathbb{A}}^{n} \) . Suppose that \( {f}_{0} : U \rightarrow \mathbb{k} \) is a function with the following property: for each \( P \) in \( U \), there exist an open subset \( W \) of \( U \) containing \( P \) and polynomial... | Proof of UNIQUENESS. If there are two such members of \( \mathbb{k}\left( V\right) \), then subtracting them gives a member \( g \) of \( \mathbb{k}\left( V\right) \) that is 0 on \( U \) . By definition of \( \mathbb{k}\left( V\right) \) , \( g = \bar{a}/\bar{b} \) with \( \bar{a} \) and \( \bar{b} \) in \( A\left( V\... | Yes |
Let \( U \) be a nonempty open subset of the affine variety \( V \) in \( {\mathbb{A}}^{n} \), and let \( P \) be in \( U \) . To each germ \( \left\{ \left( {{U}_{0},{f}_{0}}\right) \right\} \) of regular functions at \( P \) corresponds one and only one member \( f \) of \( \mathbb{k}\left( V\right) \) that is associ... | Proof. If \( \left( {{U}_{0},{f}_{0}}\right) \) and \( \left( {{U}_{0}^{\prime },{f}_{0}^{\prime }}\right) \) are two pairs in a germ at \( P \), then the definition of germ gives a pair \( \left( {W,{g}_{0}}\right) \) such that \( W \) is a neighborhood of \( P \) contained in \( {U}_{0} \cap {U}_{0}^{\prime } \) and ... | Yes |
Proposition 10.33. Let a prime ideal \( \mathfrak{a} \) of \( \widetilde{A} \) not containing \( {X}_{0} \) correspond to the prime ideal \( \mathfrak{b} \) of \( A \) under the formula \( \mathfrak{b} = {\beta }_{0}^{t}\left( \mathfrak{a}\right) \) as in Theorem 10.20, and let \( U = V\left( \mathfrak{b}\right) \) and... | Proof. Since \( {\beta }_{0}^{t} \) carries \( \widetilde{A} \) onto \( A \) and carries \( \mathfrak{a} \) into \( \mathfrak{b},{\beta }_{0}^{t} \) descends to a homomorphism \( \psi \) of \( \widetilde{A}/\mathfrak{a} = \widetilde{A}\left( V\right) \) onto \( A/\mathfrak{b} = A\left( U\right) \) . If \( \bar{F} \) an... | Yes |
Corollary 10.34. Let \( V \) be a projective variety, and let \( U \) be a nonempty open subset of \( V \) . Then each member of \( \mathcal{O}\left( U\right) \subseteq \mathbb{k}\left( V\right) \) is determined as an element in \( \mathbb{k}\left( V\right) \) by its restriction to \( U \) . | Proof. Subtracting two such members, we may assume that their difference \( h \) is 0 on \( U \) . We are to prove that \( h = 0 \) in \( \mathbb{k}\left( V\right) \) . For some \( j \) with \( 0 \leq j \leq n \) , \( {\beta }_{j}\left( {\mathbb{A}}^{n}\right) \cap V \) is nonempty, and we may assume that this is the c... | Yes |
Proposition 10.35. Let \( U \) be a nonempty open subset of the projective variety \( V \) in \( {\mathbb{P}}^{n} \). Suppose that \( {h}_{0} : U \rightarrow \mathbb{k} \) is a function with the following property: for each \( P \) in \( U \), there exist an open subset \( W \) of \( U \) containing \( P \) and homogen... | Proof. For each \( j \) with \( 0 \leq j \leq n \) such that \( {V}_{j} = {\beta }_{j}\left( {\mathbb{A}}^{n}\right) \cap V \) is nonempty, \( {\beta }_{j}^{-1}\left( {V}_{j}\right) \) is an affine variety, and \( {U}_{j} = U \cap {V}_{j} \) is a nonempty open subset such that \( {h}_{j,0} = {\left. {h}_{0}\right| }_{{... | Yes |
Proposition 10.36. If \( P \) and \( Q \) are distinct points of a variety \( V \), then there exists a rational function \( h \in \mathbb{k}\left( V\right) \) such that \( h \) is defined at both \( P \) and \( Q \), has \( h\left( P\right) = 0 \), and has \( h\left( Q\right) \neq 0 \) . | Proof. Without loss of generality, we may assume that \( V \) is projective. Say that \( V \subseteq {\mathbb{P}}^{n} \) . Let \( \mathfrak{p} \) be the prime homogeneous ideal in \( \widetilde{A} = \mathbb{k}\left\lbrack {{X}_{0},\ldots ,{X}_{n}}\right\rbrack \) such that \( \widetilde{A}\left( V\right) = \widetilde{A... | Yes |
Proposition 10.37. Let \( {\beta }_{0} : {\mathbb{A}}^{n} \rightarrow {\mathbb{P}}^{n} \) be the usual inclusion. If \( U \) is a quasi-affine variety in \( {\mathbb{A}}^{n} \), then \( {\beta }_{0} \) is an isomorphism of the quasi-affine variety \( U \) onto the quasiprojective variety \( {\beta }_{0}\left( U\right) ... | Proof. Proposition 10.18 shows that \( {\beta }_{0} \) is a homeomorphism of \( U \) onto its image. The last conclusion of Proposition 10.33 implies that the regular functions for \( U \) match those for \( {\beta }_{0}\left( U\right) \) under \( {\beta }_{0} \), and the result follows. | Yes |
Lemma 10.39. If \( U \) is a variety and \( V \) is an affine variety in \( {\mathbb{A}}^{n} \), then a function \( \psi : U \rightarrow V \) is a morphism if and only if \( \overline{{X}_{i}} \circ \psi \) is a regular function on \( U \) for the image \( \overline{{X}_{i}} \) in \( A\left( V\right) \) of each coordin... | Proof. If \( \psi \) is a morphism, then the definition of morphism forces \( \overline{{X}_{i}} \circ \psi \) to be a regular function.\n\nConversely suppose \( \psi \) has the property that each \( \overline{{X}_{i}} \circ \psi \) is a regular function. Then \( f \circ \psi \) is a regular function on \( U \) for eac... | Yes |
Corollary 10.40. If \( U \) and \( V \) are affine varieties, then the morphisms \( \varphi : U \rightarrow V \) are in one-one correspondence with the \( \mathbb{k} \) algebra homomorphisms \( \widetilde{\varphi } : A\left( V\right) \rightarrow A\left( U\right) \) via the formula\n\n\[ \widetilde{\varphi }\left( f\rig... | Proof. This is immediate from Theorem 10.38, since Corollary 10.25 shows that \( \mathcal{O}\left( U\right) = A\left( U\right) \) . | Yes |
Proposition 10.41. If \( U \) and \( V \) are varieties and if \( \varphi : U \rightarrow V \) and \( \psi : U \rightarrow V \) are morphisms such that \( {\left. \varphi \right| }_{E} = {\left. \psi \right| }_{E} \) for some nonempty open set \( E \) in \( U \), then \( \varphi = \psi \) . | Proof. Let \( h \) be a rational function on \( V \), and let \( {E}^{\prime } \) be the nonempty open subset of \( V \) on which \( h \) is regular. Since \( \varphi \) and \( \psi \) are morphisms, \( h \circ \varphi \) and \( h \circ \psi \) are regular on the respective nonempty open subsets \( {\varphi }^{-1}\left... | Yes |
Proposition 10.42. Suppose that \( U \) and \( V \) are varieties and that \( \varphi : U \rightarrow V \) is a morphism. If \( P \) is in \( U \), then \( \varphi \) induces a \( \mathbb{k} \) algebra homomorphism \( {\varphi }_{P}^{ * } : {\mathcal{O}}_{\varphi \left( P\right) }\left( V\right) \rightarrow {\mathcal{O... | Proof. Propositions 10.33 and 10.37 together imply that we may assume \( U \) and \( V \) to be quasi-affine. Let \( f \) in \( \mathbb{k}\left( V\right) \) be defined at \( \varphi \left( P\right) \) . Proposition 10.24 shows that the set \( E \) on which \( f \) is regular is open in \( V \) . Since \( \varphi \) is ... | Yes |
Let \( V = V\left( f\right) \) be the hypersurface \( {}^{13} \) in \( {\mathbb{A}}^{n} \) defined by a nonconstant polynomial \( f \) in \( \mathbb{k}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) . Then the open set \( {\mathbb{A}}^{n} - V \) is isomorphic to an affine variety, specifically to the hypersurfac... | Let \( W = V\left( {{X}_{n + 1}f - 1}\right) \) . Let \( \varphi : W \rightarrow {\mathbb{A}}^{n} \) be the map defined by \( \varphi \left( {{x}_{1},\ldots ,{x}_{n + 1}}\right) = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) for \( \left( {{x}_{1},\ldots ,{x}_{n + 1}}\right) \) in \( W \) . Then \( {X}_{j} \circ \varphi ... | Yes |
Lemma 10.44. If \( V \) is a variety, then there is a base for the Zariski topology on \( V \) consisting of open sets that are isomorphic to affine varieties. | Proof. Let \( P \) be in \( V \), and let \( U \) be an open subset of \( V \) containing \( P \) . We are to produce an open subset \( W \) of \( U \) containing \( P \) that is isomorphic to an affine variety. Since any nonempty open set of a quasiprojective variety is a quasiprojective variety, \( U \) is a variety.... | Yes |
Theorem 10.45. Let \( U \) and \( V \) be varieties, and let \( \varphi \mapsto \widetilde{\varphi } \) be the function carrying dominant rational maps \( \varphi : U \rightarrow V \) to field mappings \( \widetilde{\varphi } : \mathbb{k}\left( V\right) \rightarrow \mathbb{k}\left( U\right) \) respecting the operations... | Proof. We begin by inverting \( \varphi \mapsto \widetilde{\varphi } \) . Lemma 10.44 shows that any variety is covered by open subvarieties isomorphic to affine varieties, and the function fields of the variety and the subvarieties may all be identified with one another. Thus there is no loss in generality in assuming... | Yes |
Corollary 10.46. If \( U \) and \( V \) are varieties, then the following conditions are equivalent:\n\n(a) \( U \) and \( V \) are birationally equivalent,\n\n(b) \( \mathbb{k}\left( U\right) \) and \( \mathbb{k}\left( V\right) \) are isomorphic as \( \mathbb{k} \) algebras,\n\n(c) there are nonempty open subsets \( E... | Proof. The equivalence of (a) and (b) follows from Theorem 10.45 and the fact that composition of dominant rational maps corresponds to composition of homomorphisms of \( \mathbb{k} \) algebras in the reverse order.\n\nLet us check that (c) implies (a). If (c) holds, let \( \varphi : E \rightarrow F \) and \( \psi : F ... | Yes |
Theorem 10.47 (Zariski's Theorem, rephrased). In the above notation, \[ {\dim }_{\mathbb{k}}\left( {{M}_{P}/{M}_{P}^{2}}\right) = {\dim }_{\mathbb{k}}\left( {{\mathfrak{m}}_{P}/{\mathfrak{m}}_{P}^{2}}\right) \geq \dim V\left( \mathfrak{p}\right) ,\] and \( P \) is nonsingular if and only if equality holds. The set of n... | Toward the proof of this theorem, we showed in Section VII. 5 for all \( P \in V\left( \mathfrak{p}\right) \) that (a) \[ {\dim }_{\mathbb{k}}\left( {{M}_{P}/{M}_{P}^{2}}\right) = {\dim }_{\mathbb{k}}\left( {{\mathfrak{m}}_{P}/{\mathfrak{m}}_{P}^{2}}\right) \] (b) \[ {\dim }_{\mathbb{k}}\left( {{\mathfrak{m}}_{P}/{\mat... | No |
Proposition 10.48. Any \( m \) -dimensional variety is birationally equivalent to an irreducible affine hypersurface \( H \) in \( {\mathbb{A}}^{m + 1} \) . | Proof. Let \( V \) be the variety in question. By definition of \( \dim V \), the function field \( \mathbb{k}\left( V\right) \) is a finitely generated extension field of \( \mathbb{k} \) of transcendence degree \( m \) over \( \mathbb{k} \) . Since algebraically closed fields are perfect, Theorem 7.20 shows that \( \... | No |
Lemma 10.49. Every point \( P \) in \( V\left( \mathfrak{p}\right) \) has \( 0 \leq {\dim }_{\mathbb{k}}\left( {{M}_{P}/{M}_{P}^{2}}\right) \leq n \), and the set of points \( P \) in \( V\left( \mathfrak{p}\right) \) with \( {\dim }_{\mathbb{k}}\left( {{M}_{P}/{M}_{P}^{2}}\right) \geq r \) is a Zariski closed subset f... | Proof. The entries of the matrix \( \left\lbrack \frac{\partial {f}_{i}}{\partial {X}_{j}}\right\rbrack \) are polynomials, and the set of points \( P \) of \( V\left( \mathfrak{p}\right) \) for which the matrix \( \left\lbrack {\frac{\partial {f}_{i}}{\partial {X}_{i}}\left( P\right) }\right\rbrack \) has rank \( \leq... | Yes |
Proposition 10.50. Let \( R \) be a Noetherian local ring that is an integral domain with the property that the only nonzero prime ideal is the maximal ideal. Let \( M \) be the unique maximal ideal of \( R \), let \( K \) be the field of fractions of \( R \), and let \( F = R/M \) be the quotient field. Under the assu... | Proof. If (a) holds, then \( R \) satisfies the three conditions (Noetherian, integrally closed, every nonzero prime ideal maximal) in the definition of Dedekind domain. Thus (a) implies (b). A Dedekind domain with only finitely many maximal ideals is a principal ideal domain by Corollary 8.62 of Basic Algebra, and thu... | Yes |
Corollary 10.51. Let \( C \) be an irreducible quasiprojective curve over \( \mathbb{k} \), and let \( \mathbb{k}\left( C\right) \) be its function field. If \( P \) is a point of \( C \), then the following conditions are equivalent:\n\n(a) \( P \) is a nonsingular point,\n\n(b) \( {\mathcal{O}}_{P}\left( C\right) \) ... | Proof. Let \( {M}_{P} \) be the unique maximal ideal of \( {\mathcal{O}}_{P}\left( C\right) \) . Zariski’s Theorem (Theorem 10.47) shows that (a) holds if and only if \( {\dim }_{\mathbb{k}}{M}_{P}/{M}_{P}^{2} = 1 \) . The corollary therefore follows from the equivalence of \( \left( \mathrm{f}\right) \) , \( \left( \m... | Yes |
Corollary 10.52. If \( C \) is an irreducible affine curve over \( \mathbb{k} \) with affine coordinate ring \( A\left( C\right) \), then the following conditions on \( C \) are equivalent:\n\n(a) \( A\left( C\right) \) is integrally closed,\n\n(b) \( {\mathcal{O}}_{P}\left( C\right) \) is integrally closed for each po... | Proof. If \( A\left( C\right) \) is integrally closed, then Corollary \( {8.48}\mathrm{c} \) of Basic Algebra shows that each localization \( {\mathcal{O}}_{P}\left( C\right) \) is integrally closed. Conversely if each \( {\mathcal{O}}_{P}\left( C\right) \) is integrally closed and if a member \( f \) of the function f... | Yes |
Theorem 10.53. Let \( C \) be an irreducible projective curve with function field \( \mathbb{k}\left( C\right) \) equal to \( \mathbb{K} \), and let \( v \) be a discrete valuation of \( \mathbb{K} \) defined over \( \mathbb{k} \). If \( {R}_{v} \) is the valuation ring of \( v \) and \( {\mathfrak{p}}_{v} \) is the va... | Proof of UNIQUENESS. Assume the contrary. If \( P \) and \( Q \) are distinct points with \( {M}_{P} \subseteq {\mathfrak{p}}_{v} \) and \( {M}_{Q} \subseteq {\mathfrak{p}}_{v} \), then Proposition 10.36 constructs a function \( h \) in \( \mathbb{k}\left( C\right) \) with \( h \) defined at \( P \) and \( Q, h\left( P... | Yes |
Corollary 10.55. If two nonsingular irreducible projective curves are birationally equivalent, then they are isomorphic as varieties. | Proof. This follows by applying Corollary 10.54 twice. | No |
Corollary 10.56. If \( C \) is a nonsingular irreducible projective curve with function field \( \mathbb{K} = \mathbb{k}\left( C\right) \), then the points of \( C \) are in one-one correspondence with the discrete valuations of \( \mathbb{K} \) defined over \( \mathbb{k} \) . | Proof. This is the correspondence given in one direction by Corollary 10.51 and in the reverse direction by Theorem 10.53. | No |
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