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Lemma 4.2.7. Given any finite set of points in \( {\mathbb{P}}^{n} \), there exists a hyperplane not containing any of them. | Proof. The vanishing of the linear form \( \mathop{\sum }\limits_{i}{a}_{i}{X}_{i} \) at a point forces \( \left( {{a}_{0},\ldots ,{a}_{n}}\right) \) to lie in an \( n \) -dimensional subspace of \( {k}^{n + 1} \) . So the lemma follows from (A.0.14). | No |
Theorem 4.3.2. Let \( \phi : {\mathbb{P}}_{K} \rightarrow V \subseteq {\mathbb{P}}^{n} \) be a birational map and let \( a \in V \) . Let \( {R}_{a} \) be the integral closure of \( {\ddot{\mathcal{O}}}_{a} \) in \( k\left( V\right) \) . Then every ideal I of \( {\mathcal{O}}_{a} \) contains a nonzero ideal of \( {R}_{... | Proof. By (4.2.6) and (1.1.8), \( {R}_{a} \) is the intersection of finitely many valuation rings \( {\mathcal{O}}_{{P}_{1}},\ldots ,{\mathcal{O}}_{{P}_{m}} \) of \( k\left( V\right) \), where \( \left\{ {{P}_{1},\ldots ,{P}_{m}}\right\} = {\phi }^{-1}\left( a\right) \) . Let \( \mathcal{V} \) denote the corresponding ... | Yes |
Corollary 4.3.3. Let \( \phi : {\mathbb{P}}_{K} \rightarrow V \subseteq {\mathbb{P}}^{n} \) be a projective map and let \( a \in V \) . In order that \( \phi \) be nonsingular at a it is necessary and sufficient that \( {\mathcal{O}}_{a} \) be contained in a unique valuation ring \( {\mathcal{O}}_{P} \) of \( K \) and ... | Proof. The necessity of the conditions being obvious, we argue that they are sufficient. Since \( {\mathcal{O}}_{P} \) is the unique valuation ring of \( K \) containing \( {\mathcal{O}}_{a} \), it is the integral closure of \( {\mathcal{O}}_{a} \) in \( K \) by (1.1.8). By (4.3.2) we have \( {P}^{m} \subseteq {\mathca... | Yes |
Theorem 4.3.4 (Hilbert Basis Theorem). If every ideal of the commutative ring \( R \) is finitely generated, then the same is true for \( R\left\lbrack X\right\rbrack \) . In particular, every ideal of \( k\left\lbrack {{X}_{0},\ldots ,{X}_{n}}\right\rbrack \) is finitely generated. | Proof. Let \( I \) be an ideal in \( R\left\lbrack X\right\rbrack \) . To show that \( I \) is finitely generated, we choose a sequence of polynomials \( {f}_{i} \in I \) of degree \( {d}_{i} \) and leading coefficient \( {a}_{i}\left( {i = 0,1,\ldots }\right) \) as follows. Choose \( {f}_{0} \neq 0 \) of minimal degre... | Yes |
A projective curve has only finitely many singularities. A birational map is nonsingular at almost all points. | The two statements are evidently equivalent. Let \( \phi = \left( {1,{\phi }_{1},\ldots ,{\phi }_{n}}\right) : {\mathbb{P}}_{K} \rightarrow \) \( V \) be a birational map. We consider first the special case \( n = 2 \) . By (4.1.10) we have \( V = V\left( f\right) \) for some irreducible polynomial \( f\left( {{X}_{0},... | Yes |
Lemma 4.3.8. Let \( \phi \) be an effective projective map with \( \phi \left( P\right) = a \) and assume that \( \phi \) is normalized at \( P \) . Then \( {\mathcal{O}}_{a} \) contains a local parameter at \( P \) if and only if \( \langle \phi \rangle \) contains a local parameter at \( P \) . | Proof. Make a linear change of basis so that \( a = \left( {1 : 0 : \cdots : 0}\right) \) . This amounts to choosing a basis \( \left( {1,{\phi }_{1},\ldots ,{\phi }_{n}}\right) \) for \( \langle \phi \rangle \) such that \( {\phi }_{i}\left( P\right) = 0 \) for \( i \geq 1 \) . Now we can write\n\n\[ \mathop{\sum }\li... | Yes |
Corollary 4.3.9. Let \( V \subseteq {\mathbb{P}}^{n} \) be a projective curve. Then \( V \) is nonsingular at a if and only if some hyperplane has intersection multiplicity 1 at a. | Proof. Choose notation so that \( {X}_{0}\left( a\right) \neq 0 \) and let \( \phi : {\mathbb{P}}_{k\left( V\right) } \rightarrow V \) be the natural map. Suppose \( V \) is nonsingular at \( a \) and let \( \phi \left( P\right) = a \) . Since \( \phi \) is effective and \( a \) is not at infinity, there is a function ... | Yes |
Theorem 4.3.10. For any \( x \in K \), there exists \( y \in K \) such that \( \phi \mathrel{\text{:=}} \left( {1, x, y}\right) \) is birational and \( \langle \phi \rangle \) contains a local parameter at every point \( P \in {\mathbb{P}}_{K} \) . Thus, the only singularities of \( \phi \) are multiple points. | Proof. This is an easy consequence of the weak approximation theorem. Let \( S \) be the (finite) set of all points of \( K \) that are either ramified over \( k\left( x\right) \) or lie over \( \left( {x}^{-1}\right) \) . Let \( x - a \) be a point of \( k\left( x\right) \) that is unramified in \( K \) and let \( {P}... | Yes |
Theorem 4.3.11. Let \( \phi \mathrel{\text{:=}} \left( {1, x, y}\right) \) be a birational map to \( {\mathbb{P}}^{2} \) with no singularities at infinity. Then there exists \( z \in K \) such that \( \left( {1, x, y, z}\right) \) is nonsingular. | Proof. Let \( \left\{ {{P}_{1},\ldots ,{P}_{n}}\right\} \) be the set of singularities of \( \phi \) . Choose distinct elements \( {a}_{1},\ldots ,{a}_{n} \in k \), and let \( {P}_{\infty } \) be a nonsingular finite point of \( \phi \) . By (2.2.13) there exists \( z \in K \) such that \( {v}_{{P}_{i}}\left( {z - {a}_... | Yes |
Corollary 4.3.13. Suppose that \( K \) is a function field, \( \phi : {\mathbb{P}}_{K} \rightarrow V \subseteq {\mathbb{P}}^{n} \) is a projective map, and \( P \in {\mathbb{P}}_{K} \) . Then \( \phi \) is nonsingular at \( P \) if and only if for every point \( Q \in {\mathbb{P}}_{K} \) we have \( \operatorname{codim}... | Proof. Since the statement of the theorem depends only on the equivalence class of \( \langle \phi \rangle \), we may assume that \( \phi \) is effective and normalized at \( P \) . Suppose \( Q \neq P \) . Then \( \phi \left( P\right) \neq \phi \left( Q\right) \) if and only if there exists a hyperplane of \( {\mathbb... | Yes |
Lemma 4.3.14. Suppose that \( K \) has genus \( g > 0 \), and \( D \in \operatorname{Div}\left( K\right) \) such that either \( \deg D \geq {2g} \) or \( D \) is canonical. Then \( D = \left\lbrack {\phi }_{D}\right\rbrack \) . | Proof. The condition we need is that \( \dim L\left( D\right) > \dim L\left( {D - P}\right) \) for all \( P \) . This is immediate from Riemann-Roch when \( D - P \) is nonspecial, and in particular when \( \deg D \geq {2g} \) . If \( D \) is canonical, then Riemann-Roch yields\n\n\[ \dim L\left( {D - P}\right) = {2g} ... | Yes |
Theorem 4.3.15. Suppose that \( K \) has genus \( g \) and \( D \in \operatorname{Div}\left( K\right) \) . If \( \deg D \geq {2g} + 1 \) , then \( {\phi }_{D} \) is an embedding. If \( D \) is canonical, then \( {\phi }_{D} \) is an embedding unless \( K \) contains a rational subfield \( k\left( x\right) \) with \( \l... | Proof. Put \( \phi \mathrel{\text{:=}} {\phi }_{D} \) . We have \( {L}_{\phi }\left( {P + Q}\right) = L\left( {D - P - Q}\right) \) for all \( P, Q \in {\mathbb{P}}_{K} \) by (4.3.14). If \( \deg \left( {D - P - Q}\right) \geq {2g} - 1 \), then \( D - P - Q \) is nonspecial. Thus codim \( {L}_{\phi }\left( {P + Q}\righ... | Yes |
Lemma 4.4.1. Denote by \( \tau \left( \phi \right) \) the projective map \( \left( {\tau \left( {\phi }_{0}\right) ,\ldots ,\tau \left( {\phi }_{n}\right) }\right) : {\mathbb{P}}_{K} \rightarrow {\mathbb{P}}^{n} \) . Then \( \deg \tau \left( \phi \right) = \deg \tau \deg \phi \) . | Proof. Because they lie in \( \tau \left( K\right) \), the coordinate functions \( \left( {\tau \left( {\phi }_{0}\right) ,\ldots ,\tau \left( {\phi }_{n}\right) }\right) \) define a map \( {\phi }^{\prime } : \tau \left( K\right) \rightarrow {\mathbb{P}}^{n} \) . Since \( \tau : K \rightarrow \tau \left( K\right) \) i... | Yes |
1. If \( {\phi }_{i}^{\prime } \mathrel{\text{:=}} \mathop{\sum }\limits_{j}{a}_{ij}{\phi }_{j}\left( {0 \leq i \leq n}\right) \), where \( {a}_{ij} \in k \) and \( A \mathrel{\text{:=}} \left( {a}_{ij}\right) \) is nonsingular, then \( {J}_{s}\left( {{\phi }^{\prime },\tau }\right) = {J}_{s}\left( {\phi ,\tau }\right)... | 1) If \( {H}^{\prime } \mathrel{\text{:=}} H\left( {{\phi }^{\prime }, s,\tau }\right) \), then \( {H}^{\prime } = {AH} \) because the Hasse derivatives and the map \( \tau \) are all \( k \) -linear. | No |
Lemma 4.4.6. Suppose that \( \phi = \left( {{\phi }_{0},\ldots ,{\phi }_{n}}\right) \) is a projective map, \( P \in {\mathbb{P}}_{K} \) , \( t \) is a local parameter at \( P \), and the \( {\phi }_{i} \) are defined at \( P \) and linearly independent over \( k \) . Then the matrix \( {h}_{ij}\left( P\right) \mathrel... | Proof. We use (2.5.13) to conclude:\n\n1. If all the \( \phi \) are defined at \( P \), so are all the Hasse derivatives, so the statement of the lemma makes sense.\n\n2. If \( {\alpha }_{0},\ldots ,{\alpha }_{n} \in k \), then \( \mathop{\sum }\limits_{{i = 0}}^{n}{\alpha }_{i}{h}_{ij}\left( P\right) \) is the coeffic... | Yes |
Corollary 4.4.7. With notation as in (4.4.4), if \( {w}_{s}\left( {\phi ,\tau }\right) = 0 \), then the \( {\phi }_{i} \) are linearly dependent over \( k \) . | Proof. Choose any \( P \in {\mathbb{P}}_{K} \) . Using (4.4.5) to replace \( \phi \) by an equivalent map if necessary, we may assume that \( s \) is a local parameter at \( P \) and that \( \phi \) is normalized at \( P \) . By definition, \( {w}_{s}\left( {\phi ,\tau }\right) = 0 \) precisely when the \( K \) -rank o... | Yes |
Corollary 4.4.8. Suppose \( {j}_{1}^{\prime } < \cdots < {j}_{n}^{\prime } \) is a strictly increasing sequence of nonnegative integers, \( s \in K \) is a separating variable, and \( \det {D}_{s}^{\left( {j}_{l}^{\prime }\right) }\left( {\phi }_{i}\right) \neq 0 \) . If \( J\left( {\phi ,\tau }\right) = \left( {{j}_{1... | Proof. This is now immediate from (4.4.3) because for every \( l \), the nonvanishing of the determinant guarantees that the \( K \) -rank of \( {H}^{\left( {j}_{1}^{\prime },\ldots ,{j}_{l}^{\prime }\right) } \) must be \( l + 1 \) . | Yes |
Lemma 4.4.9. If \( J\left( \phi \right) = \left( {{j}_{1},\ldots ,{j}_{n}}\right) ,\tau \neq 1 \), and \( \phi \) is birational, then there exists an integer \( m \geq 1 \) such that | \[ J\left( {\phi ,\tau }\right) = \left( {0,{j}_{1},\ldots ,{j}_{m - 1},{j}_{m + 1},\ldots ,{j}_{n}}\right) . \] | No |
Corollary 4.4.10. The divisor \( {W}_{s}\left( {\phi ,\tau }\right) \) is independent of \( s \) and depends only on \( \tau \) and the equivalence class of \( \phi \) . | Proof. Let \( {a}_{ij} \in k, y \in K \), and let \( t \) be any separating variable. Put \( {\phi }_{i}^{\prime } \mathrel{\text{:=}} \mathop{\sum }\limits_{j}{a}_{ij}{\phi }_{j} \) and let \( {\phi }^{\prime } \mathrel{\text{:=}} \left( {{\phi }_{0}^{\prime },\ldots ,{\phi }_{n}^{\prime }}\right) \) . From (4.4.5) we... | Yes |
Theorem 4.4.11. The divisor \( W\left( {\phi ,\tau }\right) \) is nonnegative and\n\n\[ \deg W\left( {\phi ,\tau }\right) = \left( {\deg \tau + n}\right) \deg \phi + \left( {2{g}_{K} - 2}\right) j\left( {\phi ,\tau }\right) . \] | To illustrate the foregoing with an example, choose any \( x \in K \smallsetminus k{K}^{p} \) and let \( \phi \mathrel{\text{:=}} \left( {1, x}\right) \) . Taking any separating variable \( t \), the matrix of Hasse derivatives with respect to \( t \) is\n\n\[ \left\lbrack \begin{matrix} 1 & 0 & \ldots \\ x & {dx}/{dt}... | No |
Lemma 4.4.12. With the above notation, we have \( {j}_{l} \leq {j}_{l}^{\prime } \) for all \( l \) . Moreover, the following conditions are equivalent:\n\n1. \( {j}_{l}^{\prime } = {j}_{l} \) for all \( l \) .\n\n2. \( {w}_{t}\left( \phi \right) \left( P\right) \neq 0 \) .\n\n3. \( {v}_{P}\left( {W\left( \phi \right) ... | The orders of \( \phi \) at \( P \) have an important geometric intrepretation. Namely, if we row-reduce the matrix \( H\left( P\right) \) of Hasse derivatives at \( P \) and use \( k \) -linearity, we obtain a basis \( \left( {{\phi }_{0}^{\prime },\ldots ,{\phi }_{n}^{\prime }}\right) \) for \( \langle \phi \rangle \... | No |
Corollary 4.4.14. If \( \phi \) is nonsingular at \( P \), then the osculating filtration at \( P \) is\n\n\[ \langle \phi \rangle = {L}_{\phi }\left( 0\right) \supsetneq {L}_{\phi }\left( P\right) \supsetneq {L}_{\phi }\left( {2P}\right) \supsetneq \cdots . \]\n\nIn particular, \( {j}_{1}^{\prime } = 1 \) . | Proof. This follows from (4.3.13), since codim \( {L}_{\phi }\left( {2P}\right) = 2 \) . | No |
Lemma 4.4.16. If \( n \) is a gap number, then at least half of the positive integers less than \( n \) are also gap numbers. | Proof. Clearly, if \( j \) and \( k \) are pole numbers, then taking the product of the corresponding two functions shows that \( j + k \) is also a pole number. It follows that if \( n \) is a gap number and \( m < n \) is a pole number, then \( n - m \) must be a gap number. Thus, there are at least as many gap numbe... | Yes |
Corollary 4.4.17. Let \( {j}_{1}^{\prime },\ldots ,{j}_{g - 1}^{\prime } \) be the orders of the canonical map at some point \( P \in {\mathbb{P}}_{K} \) . Then \( {j}_{l}^{\prime } \leq {2l} \) for all \( l \) . | Proof. If \( {n}_{1},{n}_{2},\ldots ,{n}_{l} \) are the first \( l \) positive gap numbers, then we have \( l \geq \) \( \left( {{n}_{l} - 1}\right) /2 \) by (4.4.16). Note that \( {n}_{1} = 1 \) ; otherwise \( K \) is rational, and there is no canonical map. Thus, using (4.4.15), we see that \( {j}_{l - 1}^{\prime } =... | Yes |
Corollary 4.4.18 (Clifford). For \( n \leq {2g} - 2 \) and \( P \in {\mathbb{P}}_{K} \) we have \( \dim L\left( {nP}\right) \leq \) \( 1 + n/2 \) . | Proof. By Riemann-Roch, \( \dim L\left( {\left( {{2g} - 2}\right) P}\right) \leq g \) . We may therefore assume, by descending induction on \( n \), that the formula holds for \( n + 1 \leq {2g} - 2 \) . If \( \dim L\left( {\left( {n + 1}\right) P}\right) = \dim L\left( {nP}\right) + 1 \), the result follows. Otherwise... | No |
Theorem 4.4.20. Let \( K/k \) be a function field and let \( \tau : K \rightarrow K \) be a \( k \)-embedding. Let \( P \in {\mathbb{P}}_{K} \) be a fixed point of \( \tau \), let \( \phi : {\mathbb{P}}_{K} \rightarrow {\mathbb{P}}^{n} \) be effective and normalized at \( P \), and let \( t \in K \) be a local paramete... | Proof. By (4.4.13) there is a basis \( {\phi }_{l}^{\prime } = \mathop{\sum }\limits_{j}{a}_{lj}{\phi }_{j} \) for \( \langle \phi \rangle \) such that \( {\phi }_{0}^{\prime } = 1 \) and \[ {\phi }_{l}^{\prime } = {t}^{{j}_{l}} + {t}^{{j}_{l} + 1}{v}_{l}\;\left( {0 < l \leq n}\right) \] where \( {v}_{l} \in {\mathcal{... | Yes |
Lemma 4.4.22. Assume that either \( \operatorname{char}\left( k\right) = 0 \) or \( \operatorname{char}\left( k\right) > \deg \phi \), and let \( J \mathrel{\text{:=}} \) \( \left( {{j}_{1},\ldots ,{j}_{n}}\right) \) . If \( I \mathrel{\text{:=}} \left( {1,2,\ldots, n}\right) \), then\n\n\[ \det \left( \begin{array}{l}... | Proof. Expanding the binomial coefficient, we have\n\n\[ \left( \begin{array}{l} {j}_{l} \\ m \end{array}\right) = \frac{{j}_{l}\left( {{j}_{l} - 1}\right) \cdots \left( {{j}_{l} - m + 1}\right) }{m!} = \frac{{j}_{l}^{m} + \mathop{\sum }\limits_{{i = 1}}^{{m - 1}}{v}_{lmi}{j}_{l}^{m - i}}{m!}, \]\n\nwhere the \( {v}_{l... | Yes |
Lemma 4.4.23. With the above notation, let \( P \) be a strong fixed point of \( \tau \), let \( J\left( {\phi ,\tau }\right) = : {j}_{1},{j}_{2},\ldots ,{j}_{n} \), and let \( {J}^{\prime } \mathrel{\text{:=}} {j}_{1}^{\prime },\ldots ,{j}_{n}^{\prime } \) be the orders of \( \phi \) at \( P \) . Assume that \( \tau \... | Proof. For any integer sequence \( I = {i}_{1},\ldots ,{i}_{n} \), define \( {I}^{ - } \mathrel{\text{:=}} {i}_{2},\ldots ,{i}_{n} \) and \( {I}^{ + } \mathrel{\text{:=}} \) \( 0,{i}_{1},\ldots ,{i}_{n} \) . Put \( \widetilde{J} \mathrel{\text{:=}} {j}_{2}^{\prime } - {j}_{1}^{\prime },\ldots ,{j}_{n}^{\prime } - {j}_{... | Yes |
Theorem 4.4.24 (Stöhr-Voloch). Let \( K/k \) be a function field of genus \( g \), let \( \tau \) : \( K \rightarrow K \) be a \( k \) -embedding, and let \( \phi : {\mathbb{P}}_{K} \rightarrow {\mathbb{P}}^{n} \) be a projective map. Assume that \( \tau \left( \phi \right) \) is not a \( K \) -multiple of \( \phi \) .... | Proof. Continuing the notation of the previous lemma, we have shown that \( {j}_{l} \leq \) \( {j}_{l}^{\prime } - {j}_{1}^{\prime } \) for \( 1 \leq l \leq n \) . We conclude from (4.4.20) that\n\n\[ \n{\nu }_{P}\left( {W\left( {\phi ,\tau }\right) }\right) \geq n{j}_{1}^{\prime } \geq n\n\]\n\nbecause \( {j}_{1}^{\pr... | Yes |
Corollary 4.4.25. Let \( K/k \) be a function field of genus \( g \) and let \( \tau : K \rightarrow K \) be a \( k \) -embedding. Then for any integer \( n > g \), the number of strong fixed points of \( \tau \) is at most\n\n\[ 1 + \deg \tau + \left( {n + \frac{\deg \tau }{n}}\right) g + \frac{2{g}^{2}\left( {g - 1}\... | Proof. We may as well assume that there is at least one strong fixed point, \( P \), or there is nothing to prove. Let \( d \mathrel{\text{:=}} n + g \geq {2g} + 1 \) . Then \( \dim L\left( {dP}\right) = n + 1 \) by Riemann-Roch (2.2.9). Let \( {\phi }^{\prime } = \left( {1,{\phi }_{1}^{\prime },\ldots ,{\phi }_{n}^{\p... | Yes |
Lemma 4.4.26. Let \( K \) be a function field of genus \( g \) and let \( \sigma \in \operatorname{Gal}\left( {K/k}\right) \) . If \( \sigma \) fixes more than \( {2g} + 2 \) points of \( {\mathbb{P}}_{K} \) then \( \sigma = 1 \) . | Proof. By (4.4.19), \( \sigma \) fixes only finitely many points. Let \( P \) be a point not fixed by \( \sigma \) . By Riemann-Roch, there is a nonconstant function \( x \in L\left( {\left( {g + 1}\right) P}\right) \) . Then \( {\sigma }^{-1}\left( x\right) \in L\left( {\left( {g + 1}\right) {P}^{\sigma }}\right) \), ... | Yes |
Lemma 4.5.1. Suppose \( S \subseteq R \) are \( k \) -algebras, \( R \) is an integral domain, and \( x \in S \) . If \( R/S \) and \( R/{Rx} \) are finite-dimensional, then \( {\dim }_{k}\left( {R/{Rx}}\right) = {\dim }_{k}\left( {S/{Sx}}\right) \) . | Proof. Consider the inclusion diagram\n\n\n\n\n\nMultiplication by \( x \) induces an isomorphism \( R/S \simeq {Rx}/{Sx} \) . It follows that \( R/{Sx} \) is finite-dimensional, and thus\n\n\[ \n{\dim }_{k}\left( {R... | Yes |
Theorem 4.5.2. Let \( V = \mathbf{V}\left( f\right) \) be a plane projective curve with irreducible defining polynomial \( f \), let \( g = g\left( {{X}_{0},{X}_{1},{X}_{2}}\right) \) be any homogeneous polynomial, and let \( a \in V \cap \mathbf{V}\left( g\right) \) . Choosing notation so that \( {X}_{0}\left( a\right... | Proof. Letting \( {g}^{ * } \mathrel{\text{:=}} g\left( {1,{X}_{1},{X}_{2}}\right) \) restricted to \( V \) as usual, we have \( A/\left( f\right) \simeq {\mathcal{O}}_{a} \) and \( A/\left( {f, g}\right) \simeq {\mathcal{O}}_{a}/{\mathcal{O}}_{a}{g}^{ * } \) . Let \( {R}_{a} \) be the integral closure of \( {\mathcal{... | Yes |
Corollary 4.5.4. If \( V = \mathbf{V}\left( f\right) \) is a plane curve, then \( \deg V = \deg f \) . | We can make this more explicit by looking closely at a point \( a \in V \cap L \) . Translate coordinates so that \( a = \left( {1 : 0 : 0}\right) \) and let \( \widehat{f} \mathrel{\text{:=}} f\left( {1, x, y}\right) \) be the dehomogenized defining polynomial. Then \( A \) is the ring of rational functions in \( x \)... | Yes |
Lemma 4.5.6. Let \( \phi = \left( {{x}_{0},{x}_{1},{x}_{2}}\right) \) be any plane map \( {\mathbb{P}}_{K} \rightarrow V \subseteq {\mathbb{P}}^{2} \), and suppose that \( \phi \) is normalized and nonsingular at \( P \) . Let \( {a}_{0},{a}_{1},{a}_{2} \in k \) . Then the equation of the tangent line to \( V \) at \( ... | Proof. Put \( \ell \mathrel{\text{:=}} \mathop{\sum }\limits_{i}{a}_{i}{x}_{i} \) . Then the first condition is equivalent to \( {v}_{P}\left( \ell \right) \geq 1 \) . When this occurs, expanding \( \ell \) in a power series in \( t \) and using (2.5.14) we see that the two conditions together are equivalent to \( {v}_... | Yes |
Lemma 4.5.7. Let \( V = \mathbf{V}\left( f\right) \) and put \( {f}_{i} \mathrel{\text{:=}} \partial f/\partial {X}_{i} \) for \( i = 0,1,2 \) . If \( V \) is nonsingular at a, then the tangent line at a is \( \left( {{f}_{0}\left( a\right) : {f}_{1}\left( a\right) : {f}_{2}\left( a\right) }\right) \) . | Proof. First of all, if \( f \) has degree \( d \), then each \( {f}_{i} \) has degree \( d - 1 \), so that \( \left( {{f}_{0}\left( a\right) : {f}_{1}\left( a\right) : {f}_{2}\left( a\right) }\right) \) is a well-defined point of the dual plane. Renumbering the coordinate axes if necessary, we may assume that \( {X}_{... | Yes |
Lemma 4.5.12. With the above notation, we have\n\n\\[ \n\\frac{dx}{{\\widehat{f}}_{y}} = \\det \\left( A\\right) {z}^{e - 3}\\frac{d{x}^{\prime }}{{\\widehat{{f}^{\prime }}}_{{y}^{\prime }}} \n\\] | Proof. Suppose that \\( {X}^{\prime } = {A}^{\prime }{X}^{\prime \\prime } \\), where \\( {A}^{\prime } \\) is another change of variable. Since \\( \\det \\left( {A{A}^{\prime }}\\right) = \\det \\left( A\\right) \\det \\left( {A}^{\prime }\\right) \\) and \\( {X}_{0}/{X}_{0}^{\prime \\prime } = \\left( {{X}_{0}/{X}_{... | Yes |
Lemma 4.5.14. Let \( V \) be a plane curve. Then there exists a linear change of coordinates in \( {\mathbb{P}}^{2} \) such that if \( \phi = \left( {1, x, y}\right) \) is the natural map in this coordinate system then:\n\n1. The lines \( \mathbf{V}\left( {X}_{i}\right) \left( {i = 0,1,2}\right) \) are generic.\n\n2. T... | Proof. Choose a point \( {a}_{0} \notin V \) . The set of all lines through \( {a}_{0} \) (usually called the pencil at \( {a}_{0} \) ) is the set of all points on a line in the dual plane. This line meets the dual curve at finitely many points, which means that there are only finitely many tangents to \( V \) through ... | Yes |
Corollary 4.5.16. Let \( V \) be a plane curve of degree \( d \) and genus \( g \) . Then\n\n\[ g = \left( \begin{matrix} d - 1 \\ 2 \end{matrix}\right) - \frac{1}{2}\delta \left( V\right) \]\n\nIn particular, \( g \leq \left( {d - 1}\right) \left( {d - 2}\right) /2 \) with equality if and only if \( V \) is nonsingula... | Proof. The formula is immediate from the definition of \( \Delta \), because \( \left\lbrack {{dx}/{\widehat{f}}_{y}}\right\rbrack \) is canonical of degree \( {2g} - 2 \) . | No |
Lemma 4.5.18. Let \( V \) be a plane curve with natural map \( \left( {1, x, y}\right) \) . Assume that \( y \) is integral over \( k\left\lbrack x\right\rbrack \) . Then\n\n\[ k\left\lbrack {x, y}\right\rbrack = \cap \left\{ {{\mathcal{O}}_{P}\left\lbrack y\right\rbrack \mid P \in {\mathbb{P}}_{k\left( x\right) }\text... | Proof. Obviously, \( k\left\lbrack x\right\rbrack \subseteq {\mathcal{O}}_{P} \) and thus \( k\left\lbrack {x, y}\right\rbrack \subseteq {\mathcal{O}}_{P}\left\lbrack y\right\rbrack \) for every \( P \) containing \( x \) . Conversely, let \( K \mathrel{\text{:=}} k\left( {x, y}\right) \) . Then every element \( u \in ... | Yes |
Corollary 4.5.20. With the above notation, \( C\\left( V\\right) \\subseteq k\\left\\lbrack V\\right\\rbrack \), and\n\n\\[ L\\left( \\left\\lbrack {\\omega \\left( V\\right) }\\right\\rbrack \\right) = C\\left( V\\right) \\cap k{\\left\\lbrack V\\right\\rbrack }_{d - 3}. \\]\n\nIn particular, a differential form \( \\... | Proof. From the definitions, we see that\n\n\\[ C\\left( V\\right) = \\cap \\left\\{ {{C}_{P}\\left( y\\right) \\mid P \\in {\\mathbb{P}}_{k\\left( x\\right) }}\\right. \\text{and}\\left. {{v}_{P}\\left( x\\right) \\geq 0}\\right\\} \\text{.}\\]\n\nThus,(4.5.18) implies that \( C\\left( V\\right) \\subseteq k\\left\\lb... | Yes |
Lemma 5.1.2. Let \( d \) and \( n \) be positive integers. Then\n\n\[{\left( 1 - {t}^{{nd}/\left( {n, d}\right) }\right) }^{\left( n, d\right) } = \mathop{\prod }\limits_{{{\theta }^{n} = 1}}1 - {\left( \theta t\right) }^{d},\] \n\nwhere \( \left( {n, d}\right) \mathrel{\text{:=}} \gcd \left( {n, d}\right) \) and the p... | Proof. The basic identity is\n\n(5.1.3)\n\n\[1 - {t}^{n} = \mathop{\prod }\limits_{{{\theta }^{n} = 1}}1 - {\theta t}\]\n\nIf \( \theta \) is a primitive \( {n}^{\text{th }} \) root of unity, then \( \mu \mathrel{\text{:=}} {\theta }^{d} \) is a primitive \( n/{\left( n, d\right) }^{\text{th }} \) root of unity, and we... | Yes |
Theorem 5.1.8. Let \( K \) be a function field over a finite field \( k \) of order \( q \) . Then the degree map \( \deg : \operatorname{Div}\left( K\right) \rightarrow \mathbb{Z} \) is onto. | Proof. Let \( r \) be the index of the image of the degree map as in (5.1.7) above. We have \( {Z}_{K}\left( {\theta t}\right) = {Z}_{K}\left( t\right) \) for \( \theta \) an \( {r}^{\text{th }} \) root of unity, so (5.1.4) becomes\n\n(*) \[ {Z}_{{K}_{r}}\left( {t}^{r}\right) = {Z}_{K}{\left( t\right) }^{r} \]\n\nThere... | Yes |
Corollary 5.2.4. \( {L}_{K}\left( t\right) \) satisfies the functional equation\n\n\[ \n{L}_{K}\left( t\right) = {q}^{g}{t}^{2g}{L}_{K}\left( \frac{1}{qt}\right) \n\]\n\nIn particular, \( \deg {L}_{K}\left( t\right) = {2g} \) and if \( {L}_{K}\left( t\right) = \mathop{\sum }\limits_{{i = 0}}^{{2g}}{a}_{i}{t}^{i} \), th... | Proof. The functional equation for \( {L}_{K}\left( t\right) \) follows easily from (5.1.9) and the functional equation for \( {Z}_{K}\left( t\right) \) . From it we get\n\n\[ \n\mathop{\sum }\limits_{{i = 0}}^{n}{a}_{i}{t}^{i} = {q}^{g}{t}^{2g}\mathop{\sum }\limits_{{i = 0}}^{n}{a}_{i}{\left( qt\right) }^{-i} = \matho... | Yes |
Corollary 5.2.5. There exist algebraic integers \( \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{2g}}\right\} \) such that\n\n\[ \n{L}_{K}\left( t\right) = \mathop{\prod }\limits_{{i = 1}}^{{2g}}\left( {1 - {\alpha }_{i}t}\right) \n\]\n\nand \( {\alpha }_{i}{\alpha }_{{2g} - i + 1} = q \) for \( 1 \leq i \leq g \) . | Proof. From (5.1.9) we have \( {L}_{K}\left( t\right) = \left( {1 - t}\right) \left( {1 - {qt}}\right) {Z}_{K}\left( t\right) \) . It follows that \( {L}_{K}\left( t\right) \) has integer coefficients and constant term equal to 1 . By (5.2.4) the leading coefficient of \( {L}_{K}\left( t\right) \) is \( {q}^{g} \) . Th... | Yes |
Corollary 5.2.6. Let \( {L}_{K}\left( t\right) = \mathop{\prod }\limits_{{i = 1}}^{{2g}}\left( {1 - {\alpha }_{i}t}\right) \) . Then \( {L}_{{K}_{n}}\left( t\right) = \mathop{\prod }\limits_{{i = 1}}^{{2g}}\left( {1 - {\alpha }_{i}^{n}t}\right) \) . | Proof. This is straightforward using (5.1.4) and the identity (5.1.3). | No |
Corollary 5.3.1. The Riemann Hypothesis holds for \( K \) if and only if it holds for some scalar extension \( {K}_{n} \) . | Proof. Since the zeros of \( {Z}_{{K}_{n}}\left( t\right) \) are just the \( {n}^{\text{th }} \) powers of the zeros of \( {Z}_{K}\left( t\right) \) by (5.2.6), we have \( \left| \alpha \right| = {q}^{1/2} \) if and only if \( \left| {\alpha }^{n}\right| = {q}^{n/2} \) . | Yes |
Theorem 5.3.4. Let \( K \) be a function field over a finite field \( k \) of order \( q \), and let\n\n\[ \n{L}_{K}\left( t\right) = \mathop{\prod }\limits_{{i = 1}}^{{2g}}\left( {1 - {\alpha }_{i}t}\right) \n\]\n\nThen\n\n\[ \n\frac{{Z}_{K}^{\prime }\left( t\right) }{{Z}_{K}\left( t\right) } = \frac{1}{1 - t} + \frac... | Define \( {\mathcal{L}}_{K}\left( t\right) \mathrel{\text{:=}} {Z}_{K}^{\prime }\left( t\right) /{Z}_{K}\left( t\right) \) . Then it follows that the Riemann Hypothesis is equivalent to an apparently weaker inequality:\n\nCorollary 5.3.5. With notation as above, the following statements are equivalent:\n\n1. \( \left| ... | Yes |
Corollary 5.3.5. With notation as above, the following statements are equivalent:\n\n1. \( \\left| {\\alpha }_{i}\\right| = {q}^{1/2} \) for all \( i \) .\n\n2. There exist constants \( {C}_{0},{C}_{1} \) such that \( \\left| {{a}_{{K}_{n}}\\left( 1\\right) - {q}^{n}}\\right| \\leq {C}_{0} + {C}_{1}{q}^{n/2} \) for alm... | Proof. It is obvious that 1) implies 2). Assuming 2) and using \( {a}_{{K}_{n}}\\left( 1\\right) = {b}_{K}\\left( n\\right) \) , we get\n\n\\[ \n\\left| {{\\mathcal{L}}_{K}\\left( t\\right) - \\frac{q}{1 - {qt}}}\\right| \\leq f\\left( t\\right) + \\frac{{C}_{0}}{1 - t} + \\frac{{C}_{1}}{1 - {q}^{1/2}t}\n\\]\nfor some ... | Yes |
Lemma 5.3.6. Let \( \mathfrak{f} = {\mathfrak{f}}_{K} \) be the Frobenius map and let \( Q \in {\mathbb{P}}_{K} \) . Then \( {Q}^{\mathfrak{f}} = Q \) if and only if \( Q \) is defined over \( k \) . | Proof. Let \( x \in {\mathcal{O}}_{Q} \cap K \) . By definition, we have \( x\left( {Q}^{\mathfrak{f}}\right) = \mathfrak{f}\left( x\right) \left( Q\right) = x{\left( Q\right) }^{q} \) . It follows that if \( {Q}^{ \dagger } = Q \), then \( x\left( Q\right) \in k \) for all \( x \in {\mathcal{O}}_{Q} \cap K \) . Puttin... | Yes |
Lemma 5.3.8. With the above notation, suppose that\n\n\\[ \n\\left| k\\right| = q = {p}^{2r} > 4{g}_{{K}^{\prime }}^{4}{\\left( {g}_{{K}^{\prime }} - 1\\right) }^{2}.\n\\]\n\nThen \\( {p}_{m}\\left( {{K}^{\prime }/K,\\sigma }\\right) \\leq 1 + {q}^{m} + 2{g}_{{K}^{\prime }}{q}^{m/2} \\) for all \\( \\sigma \\in G \\) a... | Proof. Fix \\( \\sigma \\in G \\) and a positive integer \\( m \\), and put \\( \\tau \\mathrel{\\text{:=}} {\\mathfrak{f}}^{m}{\\sigma }^{-1} \\), where \\( \\mathfrak{f} = {\\mathfrak{f}}_{K} \\) is the Frobenius map. Then \\( {\\mathbb{P}}_{m}\\left( {{K}^{\prime }/K,\\sigma }\\right) \\) is just the set of fixed po... | Yes |
Theorem 1.1 Every K-vector space admits at least one basis. Every free family is contained in a basis. Every generating family contains a basis. All the bases of \( E \) have the same cardinality (which is called the dimension of \( E \) ). | For general vector spaces, this statement is a consequence of the axiom of choice. As such, it is overwhelmingly (but not universally) accepted by mathematicians. Because we are interested throughout this book in finite-dimensional spaces, for which the existence of bases follows from elementary considerations, we pref... | Yes |
Corollary 1.1 In particular,\n\n\[ \dim F \cap G \geq \dim F + \dim G - \dim E. \] | If \( F \cap G = \{ 0\} \), one writes \( F \oplus G \) instead of \( F + G \), and one says that the sum of \( F \) and \( G \) is direct. Proposition 1.2 gives\n\n\[ \dim F \oplus G = \dim F + \dim G. \] | No |
Proposition 1.3 If \( F \) is a subspace of \( E \), then\n\n\[ \n\dim F + \dim {F}^{0} = \dim E.\n\] | Proof. Let \( \left\{ {{v}^{1},\ldots ,{v}^{r}}\right\} \) be a basis of \( F \), which we extend as a basis \( \mathcal{B} \) of \( E \) . Let \( \left\{ {{\ell }^{1},\ldots ,{\ell }^{n}}\right\} \) be the dual basis of \( {E}^{\prime } \) . Then \( {F}^{0} \) equals \( \operatorname{Span}\left( {{\ell }^{r + 1},\ldot... | Yes |
Corollary 1.2 A subspace \( F \) of \( E \) equals its bipolar \( {\left( {F}^{0}\right) }^{0} \) . Rigorously speaking, \( {\left( {F}^{0}\right) }^{0} = \delta \left( F\right) . | Proof. Obviously, \( {\left( {F}^{0}\right) }^{0} \subset F \) . In addition, their dimensions are equal to \( n - \dim {F}^{0} \) because of Proposition 1.3. Therefore they coincide. | No |
Proposition 1.4 We recall that \( E \) is a finite-dimensional vector space. Then\n\n- \( {\left( {u}^{ * }\right) }^{ * } = u \) .\n\n- \( {\left( \ker u\right) }^{0} = R\left( {u}^{ * }\right) \) and \( R{\left( u\right) }^{0} = \ker {u}^{ * } \) .\n\n- \( u \) is injective if and only if \( {u}^{ * } \) is surjectiv... | Proof. Let \( x \in E \) and \( L \in {F}^{\prime } \) be given. Then\n\n\[ \n\left( {{u}^{* * } \circ {\delta }_{E}\left( x\right) }\right) \left( L\right) = \left( {{\delta }_{x} \circ {u}^{ * }}\right) \left( L\right) = {\delta }_{x}\left( {{u}^{ * }\left( L\right) }\right) = {\delta }_{x}\left( {L \circ u}\right) \... | Yes |
Proposition 1.5 A scalar product satisfies the Cauchy-Schwarz inequality\n\n\\[ b{\\left( x, y\\right) }^{2} \\leq q\\left( x\\right) q\\left( y\\right) ,\\;\\forall x, y \\in E. \\]\n\nThe equality holds true if and only if \\( x \\) and \\( y \\) are colinear. | Proof. The polynomial\n\n\\[ t \\mapsto q\\left( {{tx} + y}\\right) = q\\left( x\\right) {t}^{2} + {2b}\\left( {x, y}\\right) t + q\\left( y\\right) \\]\n\n\ntakes nonnegative values for \\( t \\in \\mathbb{R} \\) . Hence its discriminant \\( 4\\left( {b{\\left( x, y\\right) }^{2} - q\\left( x\\right) q\\left( y\\right... | Yes |
Proposition 2.1 Let \( M \) be the matrix associated with \( u \) in the bases \( {\mathcal{B}}_{E} \) and \( {\mathcal{B}}_{F} \). Then the matrix of the adjoint \( {u}^{ * } \) in the dual bases is \( {M}^{T} \). | Proof. Let \( {v}^{j} \) be the elements of \( {\mathcal{B}}_{E},{w}^{k} \) those of \( {\mathcal{B}}_{F} \), and \( {\alpha }^{j},{\beta }^{k} \) those of the dual bases. We have \( {\alpha }^{j}\left( {v}^{i}\right) = {\delta }_{i}^{j} \) and \( {\beta }^{\ell }\left( {w}^{k}\right) = {\delta }_{\ell }^{k} \). Let \(... | Yes |
Proposition 2.2 The rank of a submatrix of \( M \) is not larger than that of \( M \) . | Proof. Just remark that the submatrix \( {M}^{\prime } \) formed by retaining only the rows of indices \( {i}_{1} < \cdots < {i}_{r} \) and the columns of indices \( {j}_{1} < \cdots < {j}_{r} \) is given by a formula \( {M}^{\prime } = {PMQ} \) where \( P \) is the matrix of projection from \( {K}^{n} \) over the spac... | Yes |
Proposition 2.5 Given a subset \( A \) of \( {K}^{n} \), its biorthogonal is the subspace spanned by \( A \) : | \[ {\left( {A}^{ \bot }\right) }^{ \bot } = \operatorname{Span}\left( A\right) \] | Yes |
Corollary 2.2 If \( M \in {\mathbf{M}}_{n}\left( K\right) \), then\n\n\[ \left( {M : {K}^{n} \rightarrow {K}^{n}\text{ is bijective }}\right) \Leftrightarrow \ker M = \{ 0\} \Leftrightarrow \operatorname{rk}M = n. \] | In particular, there is a well-defined notion of inverse in \( {\mathbf{M}}_{n}\left( K\right) \) : a left-inverse exists if and only if a right-inverse exists, and then they are equal to each other. In particular, this inverse is unique. | No |
If \( M \in {\mathbf{M}}_{n}\left( A\right) \) is triangular, then 3.1 Determinant | \[ \det M = {m}_{11}\cdots {m}_{nn} \] | Yes |
Proposition 3.2 If two columns of \( M \) are equal, then \( \det M = 0 \) . | Proof. Let us assume that the \( k \) th and the \( \ell \) th columns are equal, with \( k < \ell \) . The symmetric group \( {S}_{n} \) is the disjoint union of the alternate group \( {A}_{n} \), made of even permutations (those with \( \varepsilon \left( \sigma \right) = + 1 \) ) and of \( \tau {A}_{n} \), where \( ... | Yes |
For every \( 1 \leq i, j \leq n \), \[ \frac{\partial \text{ Det }}{\partial {m}_{ij}}\left( M\right) = {\widehat{m}}_{ij} \] | Proof. Let \( N \) be the matrix obtained from \( M \) by removing the \( i \) th row and the \( j \) th column. The partial derivative of \( {\pi }_{\sigma } \) is zero, unless \( \sigma \left( i\right) = j \). In the latter case, we may write \[ {\pi }_{\sigma }\left( M\right) = {m}_{ij}{\pi }_{\rho }\left( N\right) ... | Yes |
Proposition 3.4 Let \( B \in {\mathbf{M}}_{n \times m}\left( A\right), C \in {\mathbf{M}}_{m \times l}\left( A\right) \), and an integer \( p \leq n, l \) be given. Let \( 1 \leq {i}_{1} < \cdots < {i}_{p} \leq n \) and \( 1 \leq {k}_{1} < \cdots < {k}_{p} \leq l \) be indices. Then the corresponding minor in the produ... | Proof. The corollaries are trivial. We only prove the Cauchy-Binet formula. The calculation of the \( i \) th row (respectively, the \( j \) th column) of \( {BC} \) involves only the \( i \) th row of \( B \) (respectively, the \( j \) th column of \( C \) ), thus one may assume that \( p = n = l \) . The minor to be ... | Yes |
Theorem 3.1 If \( B, C \in {\mathbf{M}}_{n}\left( A\right) \), then \( \det \left( {BC}\right) = \det B \cdot \det C \) . | Proof. The corollaries are trivial. We only prove the Cauchy-Binet formula. The calculation of the \( i \) th row (respectively, the \( j \) th column) of \( {BC} \) involves only the \( i \) th row of \( B \) (respectively, the \( j \) th column of \( C \) ), thus one may assume that \( p = n = l \) . The minor to be ... | Yes |
Theorem 3.1 If \( B, C \in {\mathbf{M}}_{n}\left( A\right) \), then \( \det \left( {BC}\right) = \det B \cdot \det C \) . | Proof. The corollaries are trivial. We only prove the Cauchy-Binet formula. The calculation of the \( i \) th row (respectively, the \( j \) th column) of \( {BC} \) involves only the \( i \) th row of \( B \) (respectively, the \( j \) th column of \( C \) ), thus one may assume that \( p = n = l \) . The minor to be ... | Yes |
Theorem 3.2 The polynomial Det is irreducible in \( A\left\lbrack {{x}_{11},\ldots ,{x}_{nn}}\right\rbrack \) . | Proof. We proceed by induction on the size \( n \) . If \( n = 1 \), there is nothing to prove. Thus let us assume that \( n \geq 2 \) . We denote by \( D \) the ring of polynomials in the \( {x}_{ij} \) with \( \left( {i, j}\right) \neq \left( {1,1}\right) \), so that \( A\left\lbrack {{x}_{11},\ldots ,{x}_{nn}}\right... | Yes |
Proposition 3.5 Given \( M \in {\mathbf{M}}_{n}\left( A\right) \), the following assertions are equivalent.\n\n1. There exists \( N \in {\mathbf{M}}_{n}\left( A\right) \) such that \( {MN} = {I}_{n} \).\n\n2. There exists \( {N}^{\prime } \in {\mathbf{M}}_{n}\left( A\right) \) such that \( {N}^{\prime }M = {I}_{n} \).\... | Proof. Let us show that (1) is equivalent to (3). If \( {MN} = {I}_{n} \), then \( \det M \cdot \det N = \) 1; hence \( \det M \in {A}^{ * } \) . Conversely, if \( \det M \) is invertible, \( {\left( \det M\right) }^{-1}{\widehat{M}}^{T} \) is an inverse of \( M \) by (3.5). Analogously,(2) is equivalent to (3). The th... | Yes |
Proposition 3.6 Let \( M, N \) be nonsingular \( n \times n \) matrices. Then we have\n\n\[{\left( MN\right) }^{-1} = {N}^{-1}{M}^{-1},\;{\left( {M}^{k}\right) }^{-1} = {\left( {M}^{-1}\right) }^{k},\;{\left( {M}^{T}\right) }^{-1} = {\left( {M}^{-1}\right) }^{T}.\] | Proof. We calculate\n\n\[ \left( {{N}^{-1}{M}^{-1}}\right) \left( {MN}\right) = {N}^{-1}\left( {{M}^{-1}M}\right) N = {N}^{-1}{I}_{n}N = {N}^{-1}N = {I}_{n},\]\n\nwhence the first identity. The second one is standard. Finally\n\n\[{\left( {M}^{-1}\right) }^{T}{M}^{T} = {\left( M{M}^{-1}\right) }^{T} = {I}_{n}^{T} = {I}... | Yes |
A triangular matrix is invertible if and only if its diagonal entries are invertible; its inverse is then triangular of the same type, upper or lower. | Proof. Let us write the proof for upper-triangular matrices. Because of Propositions 3.1 and 3.5 , invertibility of the matrix amounts to that of its diagonal entries.\n\nWe thus assume that each \( {m}_{ii} \) is nonzero. Let \( T \) be the inverse of \( M \) and denote its columns by \( {T}^{\left( 1\right) },\ldots ... | Yes |
Proposition 3.9 (Schur complement formula.) Let \( M \in {\mathbf{M}}_{n}\left( K\right) \) read blockwise 3.3 Invertibility\n\n\[ M = \left( \begin{array}{ll} A & B \\ C & D \end{array}\right) \]\n\nwhere the diagonal blocks are square and \( A \) is invertible. Then\n\n\[ \det M = \det A\det \left( {D - C{A}^{-1}B}\r... | Proof. We use a trick that is developed in Chapter 11. Because \( A \) is invertible, \( M \) factorizes as a product \( {LU} \) of block-triangular matrices, with\n\n\[ L = \left( \begin{matrix} I & 0 \\ C{A}^{-1} & I \end{matrix}\right) ,\;U = \left( \begin{matrix} A & B \\ 0 & D - C{A}^{-1}B \end{matrix}\right) . \]... | Yes |
Corollary 3.3 Let \( M \in {\mathbf{{GL}}}_{n}\left( k\right) \), with \( n = {2m} \), read blockwise\n\n\[ M = \left( \begin{array}{ll} A & B \\ C & D \end{array}\right) ,\;A, B, C, D \in {\mathbf{{GL}}}_{m}\left( k\right) .\n\]\n\nThen\n\n\[ {M}^{-1} = \left( \begin{array}{ll} {\left( A - B{D}^{-1}C\right) }^{-1} & {... | Proof. We can verify the formula by multiplying by \( M \) . The only point to show is that the inverses are meaningful; that is, \( A - B{D}^{-1}C,\ldots \) are invertible. Because of the symmetry of the formulæ, it is enough to check it for a single term, namely \( D - C{A}^{-1}B \) . However, \( \det \left( {D - C{A... | Yes |
Proposition 3.10 If \( M \) is invertible, the coordinates of the solution of equation (3.7) are given by\n\n\[ \n{x}_{i} = \frac{\det M\left( {i;b}\right) }{\det M}\n\]\n\n(3.8)\n\nwhere \( M\left( {i;b}\right) \) is the matrix formed by replacing in \( M \) the ith column by the vector \( b \) . | Proof. Let us denote by \( {X}_{i}\left( b\right) \) the expression in the right-hand side of (3.8), and \( X\left( b\right) \) the vector whose coordinates are \( {X}_{i}\left( b\right) \) for \( i = 1,\ldots, n \) . The map \( b \mapsto X\left( b\right) \) is linear, and thus corresponds to a matrix \( N \in {\mathbf... | Yes |
Proposition 3.11 If \( M \) and \( {M}^{\prime } \) are similar, then \( {P}_{M} = {P}_{{M}^{\prime }} \) . In particular, \( \det M = \) \( \det {M}^{\prime } \) and \( \operatorname{Tr}M = \operatorname{Tr}{M}^{\prime } \) . | The proof is immediate. One deduces that the eigenvalues and their algebraic multiplicities are similarity invariants. This is also true for the geometric multiplicities, by a direct comparison of the kernel of \( \lambda {I}_{n} - M \) and of \( \lambda {I}_{n} - {M}^{\prime } \) . Furthermore, the expression (3.10) p... | No |
Theorem 3.3 (Cayley-Hamilton) Let \( M \in {\mathbf{M}}_{n}\left( K\right) \). Let\n\n\[ \n{P}_{M}\left( X\right) = {X}^{n} + {a}_{1}{X}^{n - 1} + \cdots + {a}_{n} \n\] \n\nbe its characteristic polynomial. Then the matrix \n\n\[ \n{M}^{n} + {a}_{1}{M}^{n - 1} + \cdots + {a}_{n}{I}_{n} \n\] \n\nequals \( {0}_{n} \). | Proof. Let \( R \in {\mathbf{M}}_{n}\left( {K\left( X\right) }\right) \) be the matrix \( X{I}_{n} - M \), and let \( S \) be the adjugate of \( R \). Each \( {s}_{ij} \) is a polynomial of degree less than or equal to \( n - 1 \), because the products arising in the calculation of the cofactors involve \( n - 1 \) lin... | Yes |
Theorem 3.4 An eigenvalue \( \\lambda \\in K \) of \( M \) is semisimple if and only if\n\n\[ \n{K}^{n} = R\\left( {M - \\lambda {I}_{n}}\\right) \\oplus \\ker \\left( {M - \\lambda {I}_{n}}\\right) .\n\] | Proof. We may assume that \( M \) is in block-triangular form as above. We decompose the vectors blockwise accordingly:\n\n\[ \nx = \\left( \\begin{array}{l} {x}_{ + } \\\\ {x}_{ - } \\end{array}\\right) .\n\]\n\nThe eigenspace associated with \( \\lambda \) is that spanned by \( {\\mathbf{e}}^{1},\\ldots ,{\\mathbf{e}... | Yes |
Theorem 3.5 A square matrix \( M \) is trigonalizable over \( {\mathbf{M}}_{n}\left( K\right) \) if and only if its characteristic polynomial splits over \( K \) . | Proof. We proceed by induction over \( n \) . The sufficiency is obvious for \( n = 1 \) .\n\nLet us assume that \( n \geq 2 \) and that \( {P}_{M} \) splits. Thus \( M \) has an eigenvalue \( \lambda \) . Let \( x \) be a corresponding eigenvector. We form a basis \( \mathcal{B} \), whose first element is \( x \) . Le... | Yes |
Proposition 3.19 Let \( M \in {\mathbf{M}}_{n}\left( K\right) \) and let \( {P}_{M} \) be its characteristic polynomial. If \( {P}_{M} = {QR} \) with coprime factors \( Q, R \in K\left\lbrack X\right\rbrack \), then \( {K}^{n} = E \oplus F \), where \( E, F \) are the ranges of \( Q\left( M\right) \) and \( R\left( M\r... | Proof. It is sufficient to prove the first assertion and then to work by induction over the number of factors \( s \). From Bézout’s theorem, there exist \( T, S \in K\left\lbrack X\right\rbrack \) such that \( {RT} + {QS} = 1 \). Hence, every \( x \in {K}^{n} \) can be written as a sum \( y + z \) with \( y = Q\left( ... | Yes |
Lemma 2. Given vectors \( x, y \in {K}^{n} \), the spectrum of \( x{y}^{T} \) is \( \left( {0,\ldots ,0, x \cdot y}\right) \) . | Notice that 0 might have algebraic multiplicity \( n \), if \( x \cdot y = 0 \) . We apply this lemma to the calculation of \( \det \left( {{I}_{n} + x{y}^{T}}\right) \) . The spectrum of \( {I}_{n} + x{y}^{T} \) is just shifted from that of \( x{y}^{T} \) and equals \( \left( {1,\ldots ,1,1 + x \cdot {y}^{T}}\right) \... | No |
Proposition 3.21 Given a matrix \( M \in {\mathbf{M}}_{n}\left( K\right) \) and vectors \( x, y \in {K}^{n} \), we have\n\n\[ \det \left( {M + x{y}^{T}}\right) = \det M + {x}^{T}\widehat{M}y \]\n\nwhere \( \widehat{M} \) is the cofactor matrix. | Proof. Let us begin with the case where \( M \) is nonsingular. Then \( M + x{y}^{T} = M\left( {{I}_{n} + }\right. \) \( \left. {{M}^{-1}x{y}^{T}}\right) \) gives\n\n\[ \det \left( {M + x{y}^{T}}\right) = \left( {\det M}\right) \det \left( {{I}_{n} + {M}^{-1}x{y}^{T}}\right) = \left( {1 + {y}^{T}{M}^{-1}x}\right) \det ... | Yes |
Proposition 3.22 Let \( B \) be an alternate bilinear form on a vector space \( E \), of dimension \( n \) . Then there exists a basis \( \left\{ {{x}_{1},{y}_{1},\ldots ,{x}_{k},{y}_{k},{z}_{1},\ldots ,{z}_{n - {2k}}}\right\} \) such that the matrix of \( B \) in this basis is block-diagonal, equal to \( \operatorname... | Proof. We proceed by induction on the dimension \( n \) . If \( B = 0 \), there is nothing to prove. If \( B \) is nonzero, there exist two vectors \( {x}_{1},{y}_{1} \) such that \( B\left( {{x}_{1},{y}_{1}}\right) \neq 0 \) . Multiplying one of them by \( B{\left( {x}_{1},{y}_{1}\right) }^{-1} \), one may assume that... | Yes |
Corollary 3.5 Given an alternate matrix \( M \in {M}_{n}\left( K\right) \), there exists a matrix \( Q \in \) \( {\mathbf{{GL}}}_{n}\left( K\right) \) such that \[ M = {Q}^{T}\operatorname{diag}\left( {J,\ldots, J,0,\ldots ,0}\right) Q. \] | Obviously, the rank of \( M \), being the same as that of the block-diagonal matrix, equals twice the number of \( J \) blocks. Finally, because \( \det J = 1 \), we have \( \det M = \) \( \varepsilon {\left( \det Q\right) }^{2} \), where \( \varepsilon = 0 \) if there is a zero diagonal block in the decomposition, and... | No |
The rank of an alternate matrix \( M \) is even. The number of \( J \) blocks in the identity (3.17) is the half of that rank. In particular, it does not depend on the decomposition. Finally, the determinant of an alternate matrix is a square in \( K \) . | A very important application of Proposition 3.23 concerns the Pfaffian, whose crude definition is a polynomial whose square is the determinant of the general alternate matrix. First of all, because the rank of an alternate matrix is even, \( \det M = 0 \) whenever \( n \) is odd. Therefore, we restrict our attention fr... | Yes |
Proposition 3.24 If \( K \) is of characteristic 0 and \( A \in {\mathbf{M}}_{n}\left( K\right) \), then \( A \) is nilpotent; that is, \( {A}^{n} = {0}_{n} \) if and only if\n\n\[ \operatorname{Tr}\left( {A}^{k}\right) = 0,\;\forall 1 \leq k \leq n. \] | Proof. If \( A \) is nilpotent, its only eigenvalue is 0, as well as for \( {A}^{k} \), whence \( \operatorname{Tr}\left( {A}^{k}\right) = \) 0 . Conversely, if all these traces vanish, then \( {s}_{1} = \cdots = {s}_{n} = 0 \) and (3.22) yields \( {\sum }_{1} = \cdots {\sum }_{n} = 0 \), whence \( {P}_{A} = {X}^{n} \)... | Yes |
Proposition 3.26 Let \( M \in {\mathbf{M}}_{n}\left( K\right) \) be an irreducible Hessenberg matrix. Then the eigenvalues of \( M \) are geometrically simple. | Proof. The hypothesis implies that all entries \( {m}_{j + 1, j} \) are nonzero. If \( \lambda \) is an eigenvalue, let us consider the matrix \( N \in {\mathbf{M}}_{n - 1}\left( \bar{K}\right) \), obtained from \( M - \lambda {I}_{n} \) by deleting the first row and the last column. It is a triangular matrix, whose di... | Yes |
Theorem 4.1 Let \( E, F, H \) be \( K \) -vector spaces. Then \( \operatorname{Bil}\left( {E \times F;H}\right) \) is isomorphic to \( \mathcal{L}\left( {E \otimes F;H}\right) \) through the formula\n\n\[ b\left( {x, y}\right) = u\left( {x \otimes y}\right) . \] | Proof. If \( u \in \mathcal{L}\left( {E \otimes F;H}\right) \) is given, then \( b\left( {x, y}\right) \mathrel{\text{:=}} u\left( {x \otimes y}\right) \) clearly defines a bilinear map.\n\nConversely, let \( b \in \operatorname{Bil}\left( {E \times F;H}\right) \) be given. Then \( \beta : G \mapsto H \), defined by\n\... | Yes |
Corollary 4.1 Let \( x \in E \) and \( y \in F \) be given vectors. If \( x \neq 0 \) and \( y \neq 0 \), then \( x \otimes y \neq \) 0 . | Proof. There exist linear forms \( \ell \in {E}^{\prime } \) and \( m \in {F}^{\prime } \) such that \( \ell \left( x\right) = m\left( y\right) = 1 \) . Then the map\n\n\[ \left( {w, z}\right) \mapsto b\left( {w, z}\right) \mathrel{\text{:=}} \ell \left( w\right) m\left( z\right) \]\n\nis bilinear over \( E \times F \)... | Yes |
Proposition 4.1 A basis of \( {\operatorname{Sym}}^{2}\left( E\right) \) is provided by the tensors\n\n\[ \n{\mathbf{e}}^{j}{\mathbf{e}}^{k} \mathrel{\text{:=}} \frac{1}{2}\left( {{\mathbf{e}}^{j} \otimes {\mathbf{e}}^{k} + {\mathbf{e}}^{k} \otimes {\mathbf{e}}^{j}}\right) ,\;1 \leq j \leq k \leq n, \n\]\n\nand a basis... | Proof. Clearly, the elements \( {\mathbf{e}}^{j}{\mathbf{e}}^{k} \) span a subspace \( S \) of \( {\operatorname{Sym}}^{2}\left( E\right) \), whereas the elements \( {\mathbf{e}}^{j} \land {\mathbf{e}}^{k} \) span a subspace \( A \) of \( {\Lambda }^{2}\left( E\right) \) . We thus have \( S \cap A = \{ 0\} \) . Also, t... | Yes |
Theorem 4.3 If \( \dim E = n \), then\n\n\[ \dim {\Lambda }^{k}\left( E\right) = \left( \begin{array}{l} n \\ k \end{array}\right) . \]\n\nA basis of \( {\Lambda }^{k}\left( E\right) \) is given by the set of tensors \( {\mathbf{e}}^{{i}_{1}} \land \cdots \land {\mathbf{e}}^{{i}_{k}} \) with \( {i}_{1} < \cdots < {i}_{... | Proof. Because \( {T}^{k}\left( E\right) \) is spanned by the vectors of the form \( {\mathbf{e}}^{{i}_{1}} \otimes \cdots \otimes {\mathbf{e}}^{{i}_{k}},{\Lambda }^{k}\left( E\right) \) is spanned by elements of the form \( {\mathbf{e}}^{{i}_{1}} \land \cdots \land {\mathbf{e}}^{{i}_{k}} \) . However, this vector vani... | Yes |
Proposition 4.3 If \( A \) is the matrix of \( u \in \mathbf{{End}}\left( E\right) \) in a basis \( \left\{ {{\mathbf{e}}^{1},\ldots ,{\mathbf{e}}^{n}}\right\} \), then the entries of the matrix \( {A}^{\left( k\right) } \) of \( {\Lambda }^{k}\left( u\right) \) in the basis of vectors \( {\mathbf{e}}^{{j}_{1}} \land \... | Proof. This is essentially the same line as in the proof of the Cauchy-Binet formula (Proposition 3.4). | No |
Lemma 5. If \( \ell \) is even and \( {A}_{1},\ldots ,{A}_{\ell } \in {\mathbf{M}}_{n}\left( R\right) \) ( \( R \) a commutative ring), then\n\n\[ \operatorname{Tr}{\mathcal{S}}_{\ell }\left( {{A}_{1},\ldots ,{A}_{\ell }}\right) = 0 \] | Proof. If \( \ell \) is even, we have\n\n\[ \operatorname{Tr}{\mathcal{S}}_{\ell }\left( {{A}_{1},\ldots ,{A}_{\ell }}\right) = - \operatorname{Tr}{\mathcal{S}}_{\ell }\left( {{A}_{2},\ldots ,{A}_{\ell },{A}_{1}}\right) = - \operatorname{Tr}{\mathcal{S}}_{\ell }\left( {{A}_{1},\ldots ,{A}_{\ell }}\right) ,\]\n\nthe fir... | Yes |
Theorem 4.4 (A. Amitsur and J. Levitzki) The standard polynomial in 2n noncommutative indeterminates vanishes over \( {\mathbf{M}}_{n}\left( K\right) \) : for every \( {A}_{1},\ldots ,{A}_{2n} \in {\mathbf{M}}_{n}\left( K\right) \) , we have\n\n\[ \n{\mathcal{S}}_{2n}\left( {{A}_{1},\ldots ,{A}_{2n}}\right) = {0}_{n} \... | Proof. (Taken from Rosset [32].)\n\nWe thus assume that \( K = \mathbb{C} \) . As above, we form the matrix \( A \in {\mathbf{M}}_{n}\left( {\Lambda \left( {\mathbb{C}}^{2n}\right) }\right) \) by\n\n\[ \nA = {A}_{1}{\mathbf{e}}^{1} + \cdots {A}_{2n}{\mathbf{e}}^{2n}. \n\]\n\nBecause of (4.4), what we have to prove is t... | Yes |
Proposition 5.1 The following matrices \( M \in {\mathbf{M}}_{n}\left( \mathbb{C}\right) \) are normal. | - Hermitian matrices, meaning that \( {M}^{ * } = M \)\n- Skew-Hermitian matrices, meaning that \( {M}^{ * } = - M \)\n- Unitary matrices, meaning that \( {M}^{ * } = {M}^{-1} \)\n\nThe Hermitian, skew-Hermitian, and unitary matrices are thus normal. One verifies easily that \( H \) is Hermitian (respectively, skew-Her... | Yes |
Lemma 6. Let \( H \) be Hermitian. If \( {x}^{ * }{Hx} = 0 \) for every \( x \in {\mathbb{C}}^{n} \), then \( H = {0}_{n} \) . | Proof. Using (1.1), we have \( {y}^{ * }{Hx} = 0 \) for every \( x, y \in {\mathbb{C}}^{n} \) . Therefore \( {Hx} = 0 \) for every \( x \), which means \( H = {0}_{n} \) . | Yes |
Proposition 5.3 Let \( M \in {\mathbf{M}}_{n}\left( \mathbb{C}\right) \) be given. If \( {x}^{ * }{Mx} = 0 \) for every \( x \in {\mathbb{C}}^{n} \), then \( M = \) \( {0}_{n} \) . | Proof. We decompose \( M = H + \mathrm{i}K \) into its real and imaginary parts. Recall that \( H, K \) are Hermitian. Then\n\n\[ \n{x}^{ * }{Mx} = {x}^{ * }{Hx} + \mathrm{i}{x}^{ * }{Kx} \n\]\n\nis the decomposition of a complex number into real and imaginary parts. From the assumption, we therefore have \( {x}^{ * }{... | Yes |
Theorem 5.1 (Schur) If \( M \in {\mathbf{M}}_{n}\left( \mathbb{C}\right) \), there exists a unitary matrix \( U \) such that \( {U}^{ * }{MU} \) is upper-triangular. | Proof. We proceed by induction over the size \( n \) of the matrices. The statement is trivial if \( n = 1 \) . Let us assume that it is true in \( {\mathbf{M}}_{n - 1}\left( \mathbb{C}\right) \), with \( n \geq 2 \) . Let \( M \in {\mathbf{M}}_{n}\left( \mathbb{C}\right) \) be a matrix. Because \( \mathbb{C} \) is alg... | Yes |
Corollary 5.1 The set of diagonalizable matrices is a dense subset in \( {\mathbf{M}}_{n}\left( \mathbb{C}\right) \) . | Proof. The triangular matrices with pairwise distinct diagonal entries are diagonalizable, because of Proposition 3.17, and form a dense subset of the triangular matrices. Conjugation preserves diagonalizability and is a continuous operation. Thus the closure of the diagonalizable matrices contains the matrices conjuga... | Yes |
Proposition 5.4 The eigenvalues of Hermitian matrices, as well as those of real symmetric matrices, are real. | Proof. Let \( M \in {\mathbf{M}}_{n}\left( \mathbb{C}\right) \) be an Hermitian matrix and let \( \lambda \) be one of its eigenvalues. Let us choose an eigenvector \( X : {MX} = {\lambda X} \) . Taking the Hermitian adjoint, we obtain \( {X}^{ * }M = \bar{\lambda }{X}^{ * } \) . Hence,\n\n\[ \lambda {X}^{ * }X = {X}^{... | Yes |
The eigenvalues of the unitary matrices, as well as those of real orthogonal matrices, are complex numbers of modulus one. | As before, if \( X \) is an eigenvector associated with \( \lambda \), one has\n\n\[ \n{\left| \lambda \right| }^{2}\parallel X{\parallel }^{2} = {\left( \lambda X\right) }^{ * }\left( {\lambda X}\right) = {\left( MX\right) }^{ * }{MX} = {X}^{ * }{M}^{ * }{MX} = {X}^{ * }X = \parallel X{\parallel }^{2}, \n\] \n\nand th... | Yes |
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