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a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -50,8 +50,8 @@ \usepackage{enumitem} % Bibliografía. -%\usepackage[backend=biber]{biblatex} -%\addbibresource{bibliografia.bib} +\usepackage[backend=biber]{biblatex} +\addbibresource{bibliografia.bib} \usepackage{algpseudocodex} \algrenewcommand\algorithmicwhile{\textbf{mientras}} @@ -542,7 +542,106 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du Sea $P$ un \modulo{A} libre con base $\left\lbrace p_1, p_2, \dots, p_k \right\rbrace$ y tomemos el resto de premisas como en la Definición~\ref{Definición Módulo Proyectivo}. Al ser $P$ un módulo libre, cualquier homomorfismo de \modulo{A}s $\phi: P \rightarrow N$ está unívocamente determinado por las imágenes de la base de $P$. Sean $m_1, m_2, \dots, m_k$ elementos de $M$ de forma que $e\!\left( m_i \right) = h\!\left( p_i \right)$, cuya existencia está garantizada por ser $e$ un epimorfismo. Definiendo $\phi\!\left(p_i \right) := m_i$ tenemos que $e\!\left( \phi\!\left( p_i \right) \right) = h\!\left( p_i \right)$ por lo que $e \circ \phi = h$. \end{proof} Una introducción a los módulos proyectivos puede encontrarse en \cite[sección 1.2]{LIBROPROYECTIVOS}. - + + \section{Producto Exterior} + \todo{Si haces lo de la bilineal lo puedes usar aquí} + De cara a la demostración del Criterio de Saito, utilizaremos una operación que nos será de gran ayuda: el producto exterior. Para entender esta, necesitaremos entender primero el producto tensorial. En esencia, el producto tensorial aparece en el siguiente contexto: supongamos que $M$, $N$ y $O$ son \modulo{A}s y tenemos una aplicación $b : M \times N \rightarrow O$ que es bilineal, o sea, que $b\left( \bullet, n \right)$ y $b\left( m, \bullet \right)$ son homorfismos de \modulo{A}s para cualquier $m \in M$ y $n \in N$; nos gustaría encontrar un \modulo{A} $E$ a partir de $M \times N$ en el cual exista un homomorfismo de \modulo{A}s $b' : E \rightarrow O$ que se comporte como $b$. En resumen, queremos que para cada $\left( m, n \right)$ en $M \times N$ exista un elemento $m * n$ en $E$ de forma que $b'\left( m*n \right) = b\left( m, n \right)$. Esta es la motivación de la construcción del producto tensorial. + \begin{definicion} + Sea $A$ un anillo y $M$ y $N$ dos \modulo{A}s. Tomemos como $E$ al \modulo{A} definido como $\bigoplus_{\substack{m \in M \\ n \in N}} A \cdot \vectorbasecanonica{m, n}$ donde \vectorbasecanonica{m, n} debe entenderse como la tupla $\left( m, n \right)$ sin ningún tipo de operación asociada, es decir, no debemos pensar que $\vectorbasecanonica{m, n} + \vectorbasecanonica{m', n'} = \vectorbasecanonica{m+m', n+n'}$ al igual que no pensamos que $\euno + \edos = \etres$. Ahora, consideremos dentro de $E$ el \submodulo{A} $S$ definido por + \begin{equation*} + S = \engendrado{ \left.\begin{gathered} + \vectorbasecanonica{m + m', n} - \vectorbasecanonica{m, n} - \vectorbasecanonica{m', n}, \quad + \vectorbasecanonica{m, n + n'} - \vectorbasecanonica{m, n} - \vectorbasecanonica{m, n'}, \\ + \vectorbasecanonica{am, n} - a\vectorbasecanonica{m, n}, \quad + \vectorbasecanonica{m, an} - a\vectorbasecanonica{m, n} + \end{gathered} \right\rvert a \in A, m \in M, n \in N}. + \end{equation*} + Definimos el p\emph{roducto tensorial} de $M$ por $N$ como $M \otimes N := \cociente{E}{S}$ y denotaremos a la clase de equivalencia de cada $\vectorbasecanonica{m, n}$ como $m \otimes n$ + \todo{Propiedad universal} + \end{definicion} + \begin{ejemplo} + Quizás el ejemplo más natural sea el producto escalar. Así pues, tomemos la aplicación $\productoescalarantilambda{\bullet}{\bullet} : \reales^3 \times \reales^3 \rightarrow \reales$, que es bilineal. En este caso, todo elemento de $\reales^3 \otimes \reales^3$ es de la siguiente forma ---entendiendo $S$ como en la definición de antes---: + \begin{equation}\label{Desglose Producto Tensorial} + \begin{aligned} + \vectortres{a}{b}{c} \otimes \vectortres{d}{e}{f} + S &= \left( a\euno + b\edos + c\etres \right) \otimes \left( d\euno + e\edos + f\etres \right) + S \\ + &= ad \euno\otimes\euno + ae\euno\otimes\edos + af\euno\otimes\etres + bd\edos\otimes\euno \\ &\phantom{text} + be\edos\otimes\edos + bf\edos\otimes\etres + cd\etres\otimes\euno + ce\etres\otimes\edos + cf\etres\otimes\etres + S + \end{aligned} + \end{equation} + (De aquí en adelante omitiremos los <<$+S$>> o similares sumas por el \submodulo{A} que estemos haciendo el cociente.) + + Vemos pues que \definicionconjunto{\vectorbasecanonica{i} \otimes \vectorbasecanonica{j}}{ i, j \in \conjunto{1, 2, 3}} es una base de $\reales^3 \otimes \reales^3$. Por lo tanto, para encontrar una aplicación lineal $\phi : \reales^3 \otimes \reales^3 \rightarrow \reales$ que coincida con el producto escalar debemos dar las imágenes de la base. En este caso, + \begin{equation*} + \phi\!\left(\vectorbasecanonica{i} \otimes \vectorbasecanonica{j} \right) = \productoescalarantilambda{\vectorbasecanonica{i}}{\vectorbasecanonica{j}} = \begin{cases} + 1 &\text{si $i = j$;}\\ + 0 &\text{si $i \neq j$.} + \end{cases} + \end{equation*} + Si evaluamos esta aplicación lineal $f$ en el producto anterior veremos que, en efecto, nos devuelve el valor que queremos. Por lo tanto, gracias al producto tensorial hemos convertido una aplicación bilineal en una aplicación lineal a costa de tener un dominio algo más complicado. + \end{ejemplo} + + Ahora supongamos que aquello en lo que queremos trabajar son aplicaciones bilineales alternadas, es decir, $b : M \times M \rightarrow N$ donde $b\!\left(m, m'\right) = -b\!\left( m', m \right)$. En esta situación tendremos que imponer que $m \otimes n = - n\otimes m$. + \begin{definicion} + Sea $A$ un anillo y $M$ un \modulo{A}. Consideremos el \submodulo{A} $T$ de $M \otimes M$ definido como + \begin{equation*} + T = \engendrado{m \otimes m' - m' \otimes m \vert m, m' \in M}. + \end{equation*} + Definiremos el \emph{producto exterior }de $M$ por $M$ como $M \wedge M := \cociente{\left( M \otimes M \right)}{T}$ y denotaremos la clase de equivalencia de cada $\left( m, m' \right)$ como $m \wedge m'$.\todo{Otra propiedad universal} + \end{definicion} + Nótese que ahora $m \wedge m = 0$. + \begin{ejemplo} + El ejemplo natural en el caso de aplicaciones bilineales alternadas es el producto vectorial $\bullet \times \bullet : \reales^3 \times \reales \rightarrow \reales^3$. Podemos aplicar el mismo proceso que en \eqref{Desglose Producto Tensorial}. + \begin{align*} + \vectortres{a}{b}{c} \wedge \vectortres{d}{e}{f} &= \left( a\euno + b\edos + c\etres \right) \wedge \left( d\euno + e\edos + f\etres \right) \\ + &= ad \euno\wedge\euno + ae\euno\wedge\edos + af\euno\wedge\etres + bd\edos\wedge\euno \\ &\phantom{text} + be\edos\wedge\edos + bf\edos\wedge\etres + cd\etres\wedge\euno + ce\etres\wedge\edos + cf\etres\wedge\etres; \\ + \intertext{suprimimos los elementos de la forma $\vectorbasecanonica{i} \wedge \vectorbasecanonica{i}$ y ponemos todas las parejas $\vectorbasecanonica{i} \wedge \vectorbasecanonica{j}$ de forma que $i < j$;} + &= ae\euno\wedge\edos + af\euno\wedge\etres - bd\euno\wedge\edos + bf\edos\wedge\etres - cd\euno\wedge\etres - ce\edos\wedge\etres \\ + &= \left( ae - bd \right)\euno\wedge\edos + \left( af - cd \right)\euno\wedge\etres + \left( bf - ce \right)\edos\wedge\etres. + \end{align*} + Vemos por lo tanto que $\reales^3 \wedge \reales^3$ tiene como base $\left\lbrace \euno\wedge\edos, \euno\wedge\etres, \edos\wedge\etres \right\rbrace$ y las imágenes de estos son + \begin{equation*} + \phi\!\left( \euno\wedge\edos \right) = \euno \times \edos = \etres, \quad \phi\!\left( \euno\wedge\etres \right) = \euno \times \etres = -\edos \ \text{y} \ \phi\!\left( \edos\wedge\etres \right) = \edos \times \etres = \euno. + \end{equation*} + Vemos de nuevo que la aplicación $\phi$ es lineal y nos devuelve el producto vectorial. + \end{ejemplo} + + Estas mismas ideas pueden extenderse a aplicaciones \nolbreaks{$n$-lineales} para construir productos como $\bigotimes^n M = M \otimes M \otimes \dots \otimes M$ y, consecuentemente a aplicaciones \nolbreaks{$n$-lineales} alternadas para construir $\bigwedge^n M := M \wedge M \wedge \dots \wedge M$. + + Aunque pueda parecer que los productos exteriores hacen un espacio <>, esto no tiene por qué ser así. + \begin{ejemplo} + Consideremos $\reales^3 \wedge \reales^3 \wedge \reales^3$. Los elementos son de la forma + \begin{align} + \vectortres{a}{b}{c} \wedge \vectortres{d}{e}{f} \wedge \vectortres{g}{h}{i} &= aei\euno\wedge\edos\wedge\etres + afh\euno\wedge\etres\wedge\edos + bdi\edos\wedge\euno\wedge\etres \nonumber \\ + &\phantom{text} + bfg\edos\wedge\etres\wedge\euno + cdh\etres\wedge\euno\wedge\edos + ceg\etres\wedge\edos\wedge\euno; \nonumber \\ + \intertext{donde podemos ver que, por ejemplo, $\edos\wedge\etres\wedge\euno = -\edos\wedge\euno\wedge\etres = \euno\wedge\edos\wedge\etres$ y así expresamos todos estos productos en términos de $\euno\wedge\edos\wedge\etres$;} + &= \left( aei + bfg + cdh - \left( ceg + afh + bdi \right) \right) \euno\wedge\edos\wedge\etres. \label{Producto Exterior Como Determinante} + \end{align} + Vemos, por lo tanto, que $\reales^3\wedge\reales^3\wedge\reales^3$ es en realidad $\reales\left( \euno\wedge\edos\wedge\etres \right)$, un espacio isomorfo a \reales. + \end{ejemplo} + Lo que ocurre en estos casos es que los distintos órdenes en los que pongamos los elementos de un producto exterior $v_1\wedge v_2\wedge \dots \wedge v_k$ solo lo cambian en signo. Podemos resumir lo que ocurre en el siguiente lema. + \begin{lema} \label{Lema Rango Producto Exterior} + Sea $V$ un espacio vectorial de dimensión $n$. Entonces, $\dim \bigwedge^k V = \binom{n}{k}$. En particular, nótese que si $k > n$, entonces $\bigwedge^k V = \conjunto{0}$. + \end{lema} + + Por otro lado, podemos extender los homomorfismos de \modulo{A}s que ya conocemos a productos exteriores de una forma natural. + \begin{definicion} + Sea $A$ un anillo, $M$ y $N$ dos \modulo{A}s y $\phi : M \rightarrow N$ un homomorfismo de \modulo{A}s. Definimos la \emph{aplicación exterior }$\bigwedge^k \phi: \bigwedge^k M \rightarrow \bigwedge^k N$ como sigue: + \begin{equation*} + m_1\wedge m_2 \wedge \dots \wedge m_k \longmapsto \phi\!\left( m_1 \right)\wedge \phi\!\left( m_2 \right) \wedge \dots \wedge \phi\!\left( m_k \right). + \end{equation*} + \end{definicion} + Un fenómeno relevante ocurre cuando tenemos dos espacios vectoriales $V$ y $W$ de la misma dimensión cada uno con una base \conjunto{v_1, v_2, \dots, v_n} y \conjunto{w_1, w_2, \dots, w_n} y una aplicación lineal $\phi : V \rightarrow W$ entre ellos. Escribiendo las coordenadas en función de estas bases, la aplicación lineal $\phi$ tiene asociada una matriz $P$ de forma que las coordenadas de la imagen de $v_i$ a través de $\phi$ son la \nolbreaks{$i$-ésima} columna de $P$. Ahora bien, $\bigwedge^n V$ y $\bigwedge^n W$ son espacios vectoriales de dimensión $1$ con base $v_1\wedge v_2\wedge \dots \wedge v_n$ y $w_1\wedge w_2\wedge \dots \wedge w_n$, respectivamente. Por lo tanto, la aplicación $\nombrefuncionexterior{n}{\phi} : \bigwedge^n V \rightarrow \bigwedge^n W$ simplemente será multiplicar por un escalar. Qué escalar es este nos lo dice el siguiente lema. + \begin{proposicion} \label{Aparición Determinante Función Exterior} + Sea $\phi : V \rightarrow W$ como en el párrafo anterior. Entonces, la aplicación lineal $\bigwedge^n \phi : \bigwedge^n V \rightarrow \bigwedge^n W$ envía + \begin{equation*} + v_1\wedge v_2\wedge \dots \wedge v_n \mapsto \det\Phi w_1\wedge w_2\wedge \dots \wedge w_n. + \end{equation*} + \end{proposicion} + \begin{proof} + Es suficiente con el razonamiento del párrafo anterior y un desarrollo similar al de \eqref{Producto Exterior Como Determinante}, que como puede verse, es el determinante de una matriz $3\times 3$: + \begin{equation} + \funcionexterior{n}{\phi}{v_1\wedge v_2\wedge \dots \wedge v_n} = P_1 \wedge P_2 \wedge \dots \wedge P_n = \left( \det P \right) w_1\wedge w_2\wedge \dots \wedge w_n. + \end{equation} + \end{proof} \chapter{Bases de Gröbner y Sicigias} A partir de este momento solo trabajaremos con módulos $\Kx^n$ ya que son los involucrados en el módulo de derivaciones. @@ -932,105 +1031,6 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{teorema} En este capítulo explicaremos la demostración de este resultado que aparecen en \cite{VALLES}. Pero primero, deberemos ilustrar algunas construcciones que se utilizan en dicha demostración. - \section{Producto Exterior} - \todo{Si haces lo de la bilineal lo puedes usar aquí} - De cara a la demostración del Criterio de Saito, utilizaremos una operación que nos será de gran ayuda: el producto exterior. Para entender esta, necesitaremos entender primero el producto tensorial. En esencia, el producto tensorial aparece en el siguiente contexto: supongamos que $M$, $N$ y $O$ son \modulo{A}s y tenemos una aplicación $b : M \times N \rightarrow O$ que es bilineal, o sea, que $b\left( \bullet, n \right)$ y $b\left( m, \bullet \right)$ son homorfismos de \modulo{A}s para cualquier $m \in M$ y $n \in N$; nos gustaría encontrar un \modulo{A} $E$ a partir de $M \times N$ en el cual exista un homomorfismo de \modulo{A}s $b' : E \rightarrow O$ que se comporte como $b$. En resumen, queremos que para cada $\left( m, n \right)$ en $M \times N$ exista un elemento $m * n$ en $E$ de forma que $b'\left( m*n \right) = b\left( m, n \right)$. Esta es la motivación de la construcción del producto tensorial. - \begin{definicion} - Sea $A$ un anillo y $M$ y $N$ dos \modulo{A}s. Tomemos como $E$ al \modulo{A} definido como $\bigoplus_{\substack{m \in M \\ n \in N}} A \cdot \vectorbasecanonica{m, n}$ donde \vectorbasecanonica{m, n} debe entenderse como la tupla $\left( m, n \right)$ sin ningún tipo de operación asociada, es decir, no debemos pensar que $\vectorbasecanonica{m, n} + \vectorbasecanonica{m', n'} = \vectorbasecanonica{m+m', n+n'}$ al igual que no pensamos que $\euno + \edos = \etres$. Ahora, consideremos dentro de $E$ el \submodulo{A} $S$ definido por - \begin{equation*} - S = \engendrado{ \left.\begin{gathered} - \vectorbasecanonica{m + m', n} - \vectorbasecanonica{m, n} - \vectorbasecanonica{m', n}, \quad - \vectorbasecanonica{m, n + n'} - \vectorbasecanonica{m, n} - \vectorbasecanonica{m, n'}, \\ - \vectorbasecanonica{am, n} - a\vectorbasecanonica{m, n}, \quad - \vectorbasecanonica{m, an} - a\vectorbasecanonica{m, n} - \end{gathered} \right\rvert a \in A, m \in M, n \in N}. - \end{equation*} - Definimos el p\emph{roducto tensorial} de $M$ por $N$ como $M \otimes N := \cociente{E}{S}$ y denotaremos a la clase de equivalencia de cada $\vectorbasecanonica{m, n}$ como $m \otimes n$ - \todo{Propiedad universal} - \end{definicion} - \begin{ejemplo} - Quizás el ejemplo más natural sea el producto escalar. Así pues, tomemos la aplicación $\productoescalarantilambda{\bullet}{\bullet} : \reales^3 \times \reales^3 \rightarrow \reales$, que es bilineal. En este caso, todo elemento de $\reales^3 \otimes \reales^3$ es de la siguiente forma ---entendiendo $S$ como en la definición de antes---: - \begin{equation}\label{Desglose Producto Tensorial} - \begin{aligned} - \vectortres{a}{b}{c} \otimes \vectortres{d}{e}{f} + S &= \left( a\euno + b\edos + c\etres \right) \otimes \left( d\euno + e\edos + f\etres \right) + S \\ - &= ad \euno\otimes\euno + ae\euno\otimes\edos + af\euno\otimes\etres + bd\edos\otimes\euno \\ &\phantom{text} + be\edos\otimes\edos + bf\edos\otimes\etres + cd\etres\otimes\euno + ce\etres\otimes\edos + cf\etres\otimes\etres + S - \end{aligned} - \end{equation} - (De aquí en adelante omitiremos los <<$+S$>> o similares sumas por el \submodulo{A} que estemos haciendo el cociente.) - - Vemos pues que \definicionconjunto{\vectorbasecanonica{i} \otimes \vectorbasecanonica{j}}{ i, j \in \conjunto{1, 2, 3}} es una base de $\reales^3 \otimes \reales^3$. Por lo tanto, para encontrar una aplicación lineal $\phi : \reales^3 \otimes \reales^3 \rightarrow \reales$ que coincida con el producto escalar debemos dar las imágenes de la base. En este caso, - \begin{equation*} - \phi\!\left(\vectorbasecanonica{i} \otimes \vectorbasecanonica{j} \right) = \productoescalarantilambda{\vectorbasecanonica{i}}{\vectorbasecanonica{j}} = \begin{cases} - 1 &\text{si $i = j$;}\\ - 0 &\text{si $i \neq j$.} - \end{cases} - \end{equation*} - Si evaluamos esta aplicación lineal $f$ en el producto anterior veremos que, en efecto, nos devuelve el valor que queremos. Por lo tanto, gracias al producto tensorial hemos convertido una aplicación bilineal en una aplicación lineal a costa de tener un dominio algo más complicado. - \end{ejemplo} - - Ahora supongamos que aquello en lo que queremos trabajar son aplicaciones bilineales alternadas, es decir, $b : M \times M \rightarrow N$ donde $b\!\left(m, m'\right) = -b\!\left( m', m \right)$. En esta situación tendremos que imponer que $m \otimes n = - n\otimes m$. - \begin{definicion} - Sea $A$ un anillo y $M$ un \modulo{A}. Consideremos el \submodulo{A} $T$ de $M \otimes M$ definido como - \begin{equation*} - T = \engendrado{m \otimes m' - m' \otimes m \vert m, m' \in M}. - \end{equation*} - Definiremos el \emph{producto exterior }de $M$ por $M$ como $M \wedge M := \cociente{\left( M \otimes M \right)}{T}$ y denotaremos la clase de equivalencia de cada $\left( m, m' \right)$ como $m \wedge m'$.\todo{Otra propiedad universal} - \end{definicion} - Nótese que ahora $m \wedge m = 0$. - \begin{ejemplo} - El ejemplo natural en el caso de aplicaciones bilineales alternadas es el producto vectorial $\bullet \times \bullet : \reales^3 \times \reales \rightarrow \reales^3$. Podemos aplicar el mismo proceso que en \eqref{Desglose Producto Tensorial}. - \begin{align*} - \vectortres{a}{b}{c} \wedge \vectortres{d}{e}{f} &= \left( a\euno + b\edos + c\etres \right) \wedge \left( d\euno + e\edos + f\etres \right) \\ - &= ad \euno\wedge\euno + ae\euno\wedge\edos + af\euno\wedge\etres + bd\edos\wedge\euno \\ &\phantom{text} + be\edos\wedge\edos + bf\edos\wedge\etres + cd\etres\wedge\euno + ce\etres\wedge\edos + cf\etres\wedge\etres; \\ - \intertext{suprimimos los elementos de la forma $\vectorbasecanonica{i} \wedge \vectorbasecanonica{i}$ y ponemos todas las parejas $\vectorbasecanonica{i} \wedge \vectorbasecanonica{j}$ de forma que $i < j$;} - &= ae\euno\wedge\edos + af\euno\wedge\etres - bd\euno\wedge\edos + bf\edos\wedge\etres - cd\euno\wedge\etres - ce\edos\wedge\etres \\ - &= \left( ae - bd \right)\euno\wedge\edos + \left( af - cd \right)\euno\wedge\etres + \left( bf - ce \right)\edos\wedge\etres. - \end{align*} - Vemos por lo tanto que $\reales^3 \wedge \reales^3$ tiene como base $\left\lbrace \euno\wedge\edos, \euno\wedge\etres, \edos\wedge\etres \right\rbrace$ y las imágenes de estos son - \begin{equation*} - \phi\!\left( \euno\wedge\edos \right) = \euno \times \edos = \etres, \quad \phi\!\left( \euno\wedge\etres \right) = \euno \times \etres = -\edos \ \text{y} \ \phi\!\left( \edos\wedge\etres \right) = \edos \times \etres = \euno. - \end{equation*} - Vemos de nuevo que la aplicación $\phi$ es lineal y nos devuelve el producto vectorial. - \end{ejemplo} - - Estas mismas ideas pueden extenderse a aplicaciones \nolbreaks{$n$-lineales} para construir productos como $\bigotimes^n M = M \otimes M \otimes \dots \otimes M$ y, consecuentemente a aplicaciones \nolbreaks{$n$-lineales} alternadas para construir $\bigwedge^n M := M \wedge M \wedge \dots \wedge M$. - - Aunque pueda parecer que los productos exteriores hacen un espacio <>, esto no tiene por qué ser así. - \begin{ejemplo} - Consideremos $\reales^3 \wedge \reales^3 \wedge \reales^3$. Los elementos son de la forma - \begin{align} - \vectortres{a}{b}{c} \wedge \vectortres{d}{e}{f} \wedge \vectortres{g}{h}{i} &= aei\euno\wedge\edos\wedge\etres + afh\euno\wedge\etres\wedge\edos + bdi\edos\wedge\euno\wedge\etres \nonumber \\ - &\phantom{text} + bfg\edos\wedge\etres\wedge\euno + cdh\etres\wedge\euno\wedge\edos + ceg\etres\wedge\edos\wedge\euno; \nonumber \\ - \intertext{donde podemos ver que, por ejemplo, $\edos\wedge\etres\wedge\euno = -\edos\wedge\euno\wedge\etres = \euno\wedge\edos\wedge\etres$ y así expresamos todos estos productos en términos de $\euno\wedge\edos\wedge\etres$;} - &= \left( aei + bfg + cdh - \left( ceg + afh + bdi \right) \right) \euno\wedge\edos\wedge\etres. \label{Producto Exterior Como Determinante} - \end{align} - Vemos, por lo tanto, que $\reales^3\wedge\reales^3\wedge\reales^3$ es en realidad $\reales\left( \euno\wedge\edos\wedge\etres \right)$, un espacio isomorfo a \reales. - \end{ejemplo} - Lo que ocurre en estos casos es que los distintos órdenes en los que pongamos los elementos de un producto exterior $v_1\wedge v_2\wedge \dots \wedge v_k$ solo lo cambian en signo. Podemos resumir lo que ocurre en el siguiente lema. - \begin{lema} \label{Lema Rango Producto Exterior} - Sea $V$ un espacio vectorial de dimensión $n$. Entonces, $\dim \bigwedge^k V = \binom{n}{k}$. En particular, nótese que si $k > n$, entonces $\bigwedge^k V = \conjunto{0}$. - \end{lema} - - Por otro lado, podemos extender los homomorfismos de \modulo{A}s que ya conocemos a productos exteriores de una forma natural. - \begin{definicion} - Sea $A$ un anillo, $M$ y $N$ dos \modulo{A}s y $\phi : M \rightarrow N$ un homomorfismo de \modulo{A}s. Definimos la \emph{aplicación exterior }$\bigwedge^k \phi: \bigwedge^k M \rightarrow \bigwedge^k N$ como sigue: - \begin{equation*} - m_1\wedge m_2 \wedge \dots \wedge m_k \longmapsto \phi\!\left( m_1 \right)\wedge \phi\!\left( m_2 \right) \wedge \dots \wedge \phi\!\left( m_k \right). - \end{equation*} - \end{definicion} - Un fenómeno relevante ocurre cuando tenemos dos espacios vectoriales $V$ y $W$ de la misma dimensión cada uno con una base \conjunto{v_1, v_2, \dots, v_n} y \conjunto{w_1, w_2, \dots, w_n} y una aplicación lineal $\phi : V \rightarrow W$ entre ellos. Escribiendo las coordenadas en función de estas bases, la aplicación lineal $\phi$ tiene asociada una matriz $P$ de forma que las coordenadas de la imagen de $v_i$ a través de $\phi$ son la \nolbreaks{$i$-ésima} columna de $P$. Ahora bien, $\bigwedge^n V$ y $\bigwedge^n W$ son espacios vectoriales de dimensión $1$ con base $v_1\wedge v_2\wedge \dots \wedge v_n$ y $w_1\wedge w_2\wedge \dots \wedge w_n$, respectivamente. Por lo tanto, la aplicación $\nombrefuncionexterior{n}{\phi} : \bigwedge^n V \rightarrow \bigwedge^n W$ simplemente será multiplicar por un escalar. Qué escalar es este nos lo dice el siguiente lema. - \begin{proposicion} \label{Aparición Determinante Función Exterior} - Sea $\phi : V \rightarrow W$ como en el párrafo anterior. Entonces, la aplicación lineal $\bigwedge^n \phi : \bigwedge^n V \rightarrow \bigwedge^n W$ envía - \begin{equation*} - v_1\wedge v_2\wedge \dots \wedge v_n \mapsto \det\Phi w_1\wedge w_2\wedge \dots \wedge w_n. - \end{equation*} - \end{proposicion} - \begin{proof} - Es suficiente con el razonamiento del párrafo anterior y un desarrollo similar al de \eqref{Producto Exterior Como Determinante}, que como puede verse, es el determinante de una matriz $3\times 3$: - \begin{equation} - \funcionexterior{n}{\phi}{v_1\wedge v_2\wedge \dots \wedge v_n} = P_1 \wedge P_2 \wedge \dots \wedge P_n = \left( \det P \right) w_1\wedge w_2\wedge \dots \wedge w_n. - \end{equation} - \end{proof} \section{Demostración del Criterio de Saito} @@ -1498,8 +1498,8 @@ x2z+y3,3z,-3y2,-2z,0, a = h + \sum \end{equation*} \todo{Completar la demostración.} - \bibliography{bibliografia} - \bibliographystyle{plain} -% \nocite{*} -% \printbibliography + %\bibliography{bibliografia} + %\bibliographystyle{plain} + \nocite{*} + \printbibliography \end{document}

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