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módulo de derivaciones polinómicas de un polinomio: qué clase de objeto es, para lo cual necesitaremos utilizar la teoría de módulos; cómo calcularlo, donde explicaremos herramientas como las bases de \Groebner{} y las sicigias; y por último, una demostración del Criterio de Saito, que nos permite saber si el dicho módulo de derivaciones admite una base o no. + + In this paper we answer several questions about the module of polynomial derivations of a polynomial: which kind of object it is, where we will need to use module theory; how to compute it, with an explanation of tools like \Groebner{} bases and syzygies; and lastly, a proof of Saito's Criterion, which allows us to know whether the derivations module admits a basis or not. %--- FIN DEL RESUMEN \clearemptydoublepage @@ -285,6 +286,7 @@ urlcolor=black]{hyperref} \includegraphics[scale=0.3]{primerasuperficie.png}\\ $x^2z + y^3 = 0$ \end{figure} + En este caso, \derivaciones{x^2z + y^3} no es libre. Puede encontrarse el cálculo de todo lo comentado hasta ahora para este caso en el apéndice \ref{Apéndice Cálculo}. Sin embargo, añadiendo un plano que cree una especie de <> a esas esquinas que aparecen, que se consigue multiplicando por $x$, nos queda la siguiente superficie: \begin{figure}[h] \centering @@ -329,7 +331,7 @@ urlcolor=black]{hyperref} Un módulo sobre un cuerpo es un espacio vectorial pero no todas las propiedades de espacios vectoriales se trasladan a módulos. - %La definición de espacio vectorial es más restrictiva que la de módulo, por lo que todo \nolbreaks{$C$-espacio vectorial} donde $C$ es un cuerpo es también un \modulo{C}. Por ejemplo, $\reales^n$ es un \modulo{\reales}. + %La definición de espacio vectorial es más restrictiva que la de módulo, por lo que todo \nolbreaks{$C$-espacio vectorial} donde $C$ es un cuerpo es también un \modulo{C}. Por ejemplo, $\racionales^n$ es un \modulo{\racionales}. A lo largo de este trabajo será de gran importancia el anillo de polinomios $\anillodepolinomios{K}{x_1, x_2, \dots, x_n}$ sobre un cuerpo $K$, al que denotaremos \anillodepolinomios{K}{\mathbf{x}}. Sobre todo porque a partir de este construimos el conjunto de vectores de polinomios $\anillodepolinomios{K}{\mathbf{x}}^n$, que es un \modulo{\anillodepolinomios{K}{\mathbf{x}}}. \end{ejemplo} @@ -341,10 +343,10 @@ urlcolor=black]{hyperref} \begin{ejemplo} Si pensamos en \enteros{} como un \modulo{\enteros}, $2\enteros$ es un \submodulo{\enteros}. Esto debería recordarnos a los ideales de \enteros. De hecho, un ideal de un anillo $A$ es un \submodulo{A} del propio $A$. - Consideremos el \modulo{\anillodepolinomios{\reales}{x, y}} $\anillodepolinomios{\reales}{x, y}^2$ y tomemos un par de elementos, como pueden ser $\formatovector{u} := \vectordos{x^2y + xy^2}{x + y}$ y $\formatovector{v} := \vectordos{y^3}{x^3}$. El conjunto de combinaciones lineales de estos vectores, + Consideremos el \modulo{\anillodepolinomios{\racionales}{x, y}} $\anillodepolinomios{\racionales}{x, y}^2$ y tomemos un par de elementos, como pueden ser $\formatovector{u} := \vectordos{x^2y + xy^2}{x + y}$ y $\formatovector{v} := \vectordos{y^3}{x^3}$. El conjunto de combinaciones lineales de estos vectores, \begin{equation*} - \engendrado{\formatovector{u}, \formatovector{v}}_{\anillodepolinomios{\reales}{x, y}} := \definicionconjunto{a \formatovector{u} + b\formatovector{v}}{a, b \in \anillodepolinomios{\reales}{x, y}} = \definicionconjunto{\vectordos{a\left( x^2y + xy^2 \right) + by^3}{a\left( x + y \right) + bx^3}}{a, b \in \anillodepolinomios{\reales}{x, y}}. - \end{equation*} es un \submodulo{\anillodepolinomios{\reales}{x, y}}. De hecho, a este tipo de módulos los llamaremos submódulos generados por $\mathbf{u}$ y $\mathbf{v}$. + \engendrado{\formatovector{u}, \formatovector{v}}_{\anillodepolinomios{\racionales}{x, y}} := \definicionconjunto{a \formatovector{u} + b\formatovector{v}}{a, b \in \anillodepolinomios{\racionales}{x, y}} = \definicionconjunto{\vectordos{a\left( x^2y + xy^2 \right) + by^3}{a\left( x + y \right) + bx^3}}{a, b \in \anillodepolinomios{\racionales}{x, y}}. + \end{equation*} es un \submodulo{\anillodepolinomios{\racionales}{x, y}}. De hecho, a este tipo de módulos los llamaremos submódulos generados por $\mathbf{u}$ y $\mathbf{v}$. \end{ejemplo} \begin{definicion} Sea $N$ un \submodulo{A} de $M$ y $G$ un subconjunto cualquiera de $N$. Diremos que $N$ \emph{está generado por} $G$ y lo denotaremos como $N = \engendrado{G}_A$ si @@ -441,7 +443,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. Sea $M$ un \modulo{A} noetheriano. Los submódulos y módulos cocientes de $M$ son también noetherianos. \end{lema} \begin{lema} - Que $M$ sea noetheriano es equivalente a que toda cadena ascendente de submódulos $N_1 \subseteq N_2 \subseteq \dots$ sea eventualmente estacionaria. + Que $M$ sea noetheriano es equivalente a que toda cadena ascendente de submódulos $N_1 \subseteq N_2 \subseteq \dots$ llega a ser estacionaria en algún momento. \end{lema} \begin{definicion} @@ -478,7 +480,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \begin{definicion} Diremos que $M$ es un \modulo{A} \emph{reflexivo} si $\Psi$ es un isomorfismo. \end{definicion} - \begin{ejemplo} + \begin{ejemplo} \label{Ejemplo Espacio Vectorial es Reflexivo} Veamos que todo espacio vectorial de dimensión finita es reflexivo. Sea $V$ un $K$-espacio vectorial de dimensión $n$ y con base $\conjunto{v_1, v_2, \dots, v_n}$. El conjunto \dual{V} admite como base a la base dual compuesta por los $\mathbf{e}^*_{i}$ definidos como \begin{equation*} \mathbf{e}^{*}_{i}\!\left( \vectorbasecanonica{j} \right) := \begin{cases} @@ -486,7 +488,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. 0 & \text{si no.} \end{cases} \end{equation*} - Como una base de \dual{V} tiene tantos elementos como una de $V$, $V$ y \dual{V} deben ser isomorfos y, por ende, \dual{V} y \dualdual{V} también. Así pues, $V$ y \dualdual{V} son isomorfos. Pero para verificar que $V$ sea reflexivo debemos cerciorarnos de que la aplicación $\Psi$ sea un isomorfismo. Para ello, supongamos que $\Psi\!\left( v \right) = \Psi\!\left( v' \right)$; entonces, como $\mathrm{id} \in \dual{V}$, $\mathrm{id}\!\left( v \right) = \mathrm{id}\!\left( v' \right)$ por lo que $\Psi$ es inyectiva, lo cual ya nos garantiza que sea un isomorfismo. + Como una base de \dual{V} tiene tantos elementos como una de $V$, $V$ y \dual{V} deben ser isomorfos y, por ende, \dual{V} y \dualdual{V} también. Así pues, $V$ y \dualdual{V} son isomorfos. Pero para verificar que $V$ sea reflexivo debemos cerciorarnos de que la aplicación $\Psi$ sea un isomorfismo. Para ello, supongamos que $\Psi\!\left( v \right) = \Psi\!\left( v' \right)$; entonces, como $\mathrm{id} \in \dual{V}$, $\mathrm{id}\!\left( v \right) = \mathrm{id}\!\left( v' \right)$ por lo que $\Psi$ es inyectiva, lo cual ya nos garantiza que sea un isomorfismo. Este ejemplo aplica también a módulos libres de rango finito. \end{ejemplo} \begin{ejemplo} Consideremos ahora el \modulo{\enteros} \cociente{\enteros}{p\enteros} donde $p$ es un número primo. Sabemos que $\cociente{\enteros}{p\enteros} = \engendrado{1}$; por lo tanto, cualquier $\phi \in \dual{\left( \cociente{\enteros}{p\enteros} \right)}$ está unívocamente determinado por la imagen de $1$. Supongamos que $\phi\!\left( 1 \right) = x \in \enteros$, se debe cumplir que @@ -495,6 +497,39 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \end{equation*} Es decir, que $\dual{\left( \cociente{\enteros}{p\enteros} \right)} = \conjunto{\dual{\mathbf{0}}}$ donde $\dual{\mathbf{0}}$ envía cualquier elemento de \cociente{\enteros}{p\enteros} al $0$ de \enteros. Al estar \dual{\left( \cociente{\enteros}{p\enteros} \right)} solo compuesto por la aplicación que todo lo lleva al cero, su dual, $\dualdual{\left( \cociente{\enteros}{p\enteros} \right)}$, solo estará compuesto por la aplicación \dualdual{\mathbf{0}} que envía todo al $0$ de \enteros{} también. Al ser todos estos conjuntos finitos, podemos ver que la cardinalidad de \dualdual{\left( \cociente{\enteros}{p\enteros} \right)} y \cociente{\enteros}{p\enteros} no coinciden y por lo tanto no pueden ser isomorfos, impidiendo esto a su vez que \cociente{\enteros}{p\enteros} sea un \modulo{\enteros} reflexivo. \end{ejemplo} + + + \begin{wrapfigure}[7]{r}{0.3\textwidth} + \vspace{-1.2cm} + \begin{equation*} + \begin{tikzcd} + M_\mathrm{Q} \times \left( \dual{M} \right)_\mathrm{Q} \arrow[r, "\Phi^{M_\mathrm{Q}}"] & \cuerpofracciones{\Kx} & \\ + M \hspace{0.05cm}\times \hspace{0.4cm}\dual{M} \arrow[u, shift left=6mm] \arrow[u, shift right=6mm] \arrow[r, "\Phi^{M}"] & \Kx \arrow[u] & \\ + 0 \hspace{17pt} \phantom{. \times } 0\arrow[u, shift left=6mm] \arrow[u, shift right=6mm] & 0 \arrow[u] & + \end{tikzcd} + \end{equation*} + \caption{Diagrama de aplicaciones duales.} \label{Diagrama Dual Reflexivo} + \end{wrapfigure} + + \begin{lema} + Si $M$ es un \modulo{\Kx} finitamente generado, entonces \dual{M} es reflexivo. + \end{lema} + \begin{proof} + Para cualquier \modulo{A} $B$, denotaremos $B_\mathrm{Q} := B \otimes \cuerpofracciones{A}$ (véase la Definición \ref{Definición Extensión de Escalares}). Consideremos el diagrama de homomorfismos de \modulo{\Kx} de la figura \ref{Diagrama Dual Reflexivo}. Las inyecciones que aquí mostramos son las naturales: en el caso de las de la izquierda, mandan los elementos $m$ bien de $M$ o \dual{M} a los correspondientes $m \otimes 1$; y en el de la derecha se manda $p$ a $\frac{p}{1}$. Estas son inyecciones por el Lema \ref{Lema Extensión es Inyección}. + + Podemos ver los conjuntos de la parte baja del diagrama como subconjuntos de los de arriba. Así, pensaremos en $\dual{M}$ como en + \begin{equation*} + \definicionconjunto{h \in \left( \dual{M} \right)_\mathrm{Q}}{h\!\left( m \right) \in A \ \forall m \in M}, + \end{equation*} + o en otras palabras, como todos aquellos homomorfismos en el conjunto ampliado a las fracciones que <> a $M$ funcionan como un elemento de \dual{M}. + Ahora recordemos por el \ref{Ejemplo Espacio Vectorial es Reflexivo} que $M_\mathrm{Q}$ es reflexivo así que $\dualdual{M_Q} \simeq M_Q$. Esto nos permite ver \dualdual{M} como + \begin{equation*} + \definicionconjunto{m \in M_\mathrm{Q}}{ \phi(m) \in A \ \forall \phi \in \dual{M}}. + \end{equation*} + + \todo[inline]{Esto terminarlo.} + \end{proof} + La reflexividad jugará un papel importante en la demostración del Criterio de Saito que ilustramos en el esquema de \eqref{Esquema Demostración}. Más propiedades acerca de los módulos reflexivos pueden encontrarse en \cite[\href{https://stacks.math.columbia.edu/tag/0AUY}{Tag 0AUY}]{STACKS}. En ocasiones, encadenaremos homomorfismos y podemos necesitar expresar las relaciones entre las imágenes y los núcleos de las distintas aplicaciones. Para expresar esto, utilizaremos las sucesiones exactas. @@ -600,8 +635,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. 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. + De cara a la demostración del Criterio de Saito, utilizaremos una operación que nos será de gran ayuda: el producto exterior. Para comprenderlo, necesitaremos entender primero el producto tensorial. Este aparece en el siguiente contexto: supongamos que $M$, $N$ y $P$ son \modulo{A}s y tenemos una aplicación $b : M \times N \rightarrow O$ que es bilineal; 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 convertir una aplicación bilineal en una lineal. 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*} @@ -612,11 +646,20 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \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} + 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 $m \otimes n$. \end{definicion} + Ahora sí, tenemos la siguiente propiedad universal que buscábamos al principio: para todo mapa bilineal $b$ podemos encontrar otro mapa lineal $l: M \otimes N \to P$ de forma que $b\!\left( m, n \right) = l\!\left( m\otimes n \right)$. + \begin{figure}[h] + \begin{equation*} + \begin{tikzcd} + M \otimes N \arrow[rd, "l"]& & \\ + M \times N \arrow[r, "b"] \arrow[u]& P + \end{tikzcd} + \end{equation*} + \end{figure} + \vspace{-0.7cm} \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---: + Quizás el ejemplo más natural sea el producto escalar. Así pues, tomemos la aplicación $\productoescalarantilambda{\bullet}{\bullet} : \racionales^3 \times \racionales^3 \rightarrow \racionales$, que es bilineal. En este caso, todo elemento de $\racionales^3 \otimes \racionales^3$ 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 \\ @@ -625,7 +668,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \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, + Vemos pues que \definicionconjunto{\vectorbasecanonica{i} \otimes \vectorbasecanonica{j}}{ i, j \in \conjunto{1, 2, 3}} es una base de $\racionales^3 \otimes \racionales^3$. Por lo tanto, para encontrar una aplicación lineal $\phi : \racionales^3 \otimes \racionales^3 \rightarrow \racionales$ 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 $i = j$;}\\ @@ -641,11 +684,11 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \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} + 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 $m \wedge m'$. \end{definicion} - Nótese que ahora $m \wedge m = 0$. + Nótese que ahora $m \wedge m = 0$ y que en este caso tenemos una propiedad universal similar a la del producto tensorial pero con aplicaciones bilineales alternadas. \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}. + El ejemplo natural en el caso de aplicaciones bilineales alternadas es el producto vectorial $\bullet \times \bullet : \racionales^3 \times \racionales \rightarrow \racionales^3$. 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; \\ @@ -653,7 +696,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. &= 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 + Vemos por lo tanto que $\racionales^3 \wedge \racionales^3$ 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*} @@ -664,14 +707,14 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. 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 + Consideremos $\racionales^3 \wedge \racionales^3 \wedge \racionales^3$. 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. + Vemos, por lo tanto, que $\racionales^3\wedge\racionales^3\wedge\racionales^3$ es en realidad $\racionales\left( \euno\wedge\edos\wedge\etres \right)$, un espacio isomorfo a \racionales. \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} @@ -714,10 +757,22 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \end{definicion} El cuerpo de fracciones de un anillo nos permite llevar un \modulo{A} a un \espaciovectorial{\cuerpofracciones{A}}. - \begin{definicion} + \begin{definicion} \label{Definición Extensión de Escalares} Llamaremos extensión de escalares del \modulo{A} $M$ al conjunto $M \otimes \cuerpofracciones{A}$. \end{definicion} - \begin{lema} + \begin{definicion} + Sea $S$ un subconjunto de $A$ multiplicativamente cerrado. Definimos la \emph{localización de $M$ con respecto a $S$} como el \modulo{A} + \begin{equation*} + S^{-1}M := \definicionconjunto{\frac{m}{s}}{m \in M, s \in S}, + \end{equation*} + donde $\frac{m}{s}$ denota la clase de equivalencia de $\left( m, s \right) \in M \times S$ con respecto a la relación de equivalencia\footnote{La definición de localización es, en realidad, más general: \begin{equation*} + \left( m, s \right) \sim \left( m', s' \right) \iff \exists\sigma \in S \ \text{tal que} \ \sigma\left( ms' - m's \right) = 0. + \end{equation*} Pero en para nosotros, como asumimos que no hay divisores del $0$, nuestra definición es suficiente.} + \begin{equation*} + \left( m, s \right) \sim \left( m', s' \right) \iff ms' = m's. + \end{equation*} + \end{definicion} + \begin{lema} \label{Lema Extensión es Inyección} La extensión de escalares de un \modulo{A} $M$ es un \espaciovectorial{\cuerpofracciones{A}}. Además, la aplicación $\varphi: M \rightarrow M \otimes \cuerpofracciones{A}$ dada por $m \mapsto m \otimes 1$ es inyectiva si y solo si $M$ es libre de torsión. \end{lema} \begin{proof} @@ -731,7 +786,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. &\implies& am \neq 0 & \end{array} \end{equation*} - \todo{La otra implicación.} + Vemos también gracias a esto que todo elemento de $M \otimes \cuerpofracciones{A}$ puede escribirse como $\mu \otimes \frac{1}{q}$. Consideremos el homomorfismo de $M \otimes \cuerpofracciones{A}$ en $\left( A \setminus \left\lbrace 0 \right\rbrace \right)$ dado por $\mu \otimes \frac{1}{q} \mapsto \frac{\mu}{q}$. Vemos por ella que $\frac{\mu}{q} = 0$ si y solo si $\mu = 0$; es decir, que $\mu \otimes \frac{1}{q} = 0$ si y solo si $\mu = 0$. Esto demuestra que la última implicación de la cadena anterior funciona también en el sentido contrario, completando la demostración. \end{proof} \begin{definicion} Sea $M$ un \modulo{A} finitamente generado. Diremos que $M$ tiene rango $r$ si $M \otimes \cuerpofracciones{A}$ es un \espaciovectorial{\cuerpofracciones{A}} de dimensión $r$. @@ -776,8 +831,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. Veamos cómo se afronta este problema a distintos niveles. \begin{ejemplo} \label{Ejemplo Pertenencia Ideal Una Variable} -\todo{Es una tontería, pondría los racionales en lugar de los reales, también para otros ejemplos.} - Consideremos el \modulo{\reales} \anillodepolinomios{\reales}{x} ---solo una variable--- y a \nolbreaks{$f := x^{3} - 6x^{2} + 11x - 6$} y \nolbreaks{$g := x^{3} - 9x^{2} + 26x - 24$} dos elementos de este. Queremos determinar si un polinomio cualquiera, por ejemplo \nolbreaks{$p = 3x^7 + 4x^5 - 3x^2 + x + 1$}, pertenece al submódulo ---o en este caso, ideal--- $\engendrado{f, g}$. Como \anillodepolinomios{\reales}{x} es un dominio euclídeo, es dominio de ideales principales y, por lo tanto, + Consideremos el \modulo{\racionales} \anillodepolinomios{\racionales}{x} ---solo una variable--- y a \nolbreaks{$f := x^{3} - 6x^{2} + 11x - 6$} y \nolbreaks{$g := x^{3} - 9x^{2} + 26x - 24$} dos elementos de este. Queremos determinar si un polinomio cualquiera, por ejemplo \nolbreaks{$p = 3x^7 + 4x^5 - 3x^2 + x + 1$}, pertenece al submódulo ---o en este caso, ideal--- $\engendrado{f, g}$. Como \anillodepolinomios{\racionales}{x} es un dominio euclídeo, es dominio de ideales principales y, por lo tanto, \begin{equation*} \engendrado{f, g} = \engendrado{\mcd{f, g}} = \engendrado{x^2 - 5x + 6}. \end{equation*} @@ -785,16 +839,16 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \begin{equation*} p = \underbrace{\left( 3x^5 + 15x^4 + 61x^3 + 215x^2 + 709x + 2252 \right)}_{\text{cociente $q$}} \underbrace{\left( x^2 - 5x + 6 \right)}_{\text{divisor $d$}} + \underbrace{7007x - 13511}_{\text{resto $r$}}. \end{equation*} - Por el Teorema de la División, sabemos que este resto es el único polinomio de \anillodepolinomios{\reales}{x} que satisface $p = q d + r$ con grado menor que $d$. Por lo tanto, al no ser $0$, podemos afirmar que $p$ no está en \engendrado{f,g}. + Por el Teorema de la División, sabemos que este resto es el único polinomio de \anillodepolinomios{\racionales}{x} que satisface $p = q d + r$ con grado menor que $d$. Por lo tanto, al no ser $0$, podemos afirmar que $p$ no está en \engendrado{f,g}. \end{ejemplo} - Este ejemplo también ilustra un cambio de perspectiva que será muy importante a lo largo de toda esta sección: la división deja de ser importante por los valores exactos que arroja y su relevancia es más bien cualitativa. Con esto nos referimos a que, como tal, los valores concretos del cociente y el resto no tienen ninguna relevancia a la hora de determinar si $p$ pertenece o no a \engendrado{f, g}. Conocidas las propiedades de este algoritmo de la división en \anillodepolinomios{\reales}{x}, solo nos interesa si el resto es $0$ o no, puesto que es lo que determina si $p$ pertenece o no a \engendrado{f, g}. En base a esto, nuestra motivación para la división debe cambiar de ser <> a un más abstracto <>. \todo{En realidad con este lenguaje, $r$ es importante ya que determina la clase de $p$ en el cociente.} + Este ejemplo también ilustra un cambio de perspectiva que será muy importante a lo largo de toda esta sección: la división deja de ser importante por los valores exactos que arroja y su relevancia es más bien cualitativa. Con esto nos referimos a que, como tal, el valor concreto del cociente no tienen ninguna relevancia a la hora de determinar si $p$ pertenece o no a \engendrado{f, g} y el del resto nos importa ya que nos da la clase de equivalencia, pero si hubiese más de un posible representante, daría igual cuál nos fuese devuelto. En concreto, en \anillodepolinomios{\racionales}{x}, solo nos interesa si el resto es $0$ o no, puesto que es lo que determina si $p$ pertenece o no a \engendrado{f, g}. En base a esto, nuestra motivación para la división debe cambiar de ser <> a un más abstracto <>. Visto así, el Ejemplo \ref{Ejemplo Pertenencia Ideal Una Variable} supone una situación idílica: por ser un dominio de ideales principales, sabemos que vamos a poder representar a cualquier conjunto generador con un solo elemento; por lo tanto, sobre cualquier conjunto generador siempre habrá definida una división, que consistirá en dividir como ya conocemos por el elemento generador. Además, por el Teorema de la División, conocemos que el representante $r$ que satisfaga $\grado r < \grado d$ es único, lo cual nos permite discernir si $p$ está en \engendrado{f, g} viendo si $r = 0$ o no. En espacios más complicados, muchas de estas buenas propiedades desaparecen e impiden extender el proceso anterior. \begin{description}[wide, leftmargin=0pt,itemindent=6pt] - \item[Más variables:] Consideremos ahora el anillo de polinomios $\anillodepolinomios{\reales}{x, y, z}$, un elemento $f: = 4x^2y+ 12xy^2 + \frac{5}{2}xz^3 + x^2yz$ y el ideal $I := \engendrado{f}$. Si bien este espacio no tiene por qué ser un dominio de ideales principales, al solo haber un generador podemos obviar de momento esa parte. Siguiendo la línea del Ejemplo~\ref{Ejemplo Pertenencia Ideal Una Variable}, para ver si $p \in \anillodepolinomios{\reales}{x, y, z}$ deberíamos dividir por $f$ y ver si el resto es $0$ o no. Pero llegados a este punto, ¿cómo deberíamos dividir? Podemos intentar imitar el algoritmo usual de la división, pero este necesita siempre una noción de orden entre los términos del divisor y también del dividendo. En este caso, ¿deberíamos tomar $4x^2y$ como <> de $f$ o igual sería mejor tomar a uno de mayor grado como $\frac{5}{2}xz^3$? Pero $\frac{5}{2}xz^3$ y $x^2yz$ tienen el mismo grado, ¿cuál escoger, entonces? La elección de este <> en dividendo y divisor cambiará el funcionamiento del algoritmo, así que debemos cerciorarnos de cuáles de estas opciones son válidas y también revisar si el algoritmo tradicional adaptado de esta manera sirve su propósito. - \item[Más generadores:] Sin cambiar de espacio, consideremos ahora el ideal $I := \engendrado{x + y, xyz, x^2 + y^2 + z^2}$. En este caso, el que \anillodepolinomios{\reales}{x, y, z} no sea un dominio de ideales principales es un problema. ¿Cómo podemos dividir siquiera? Otra dificultad a la que nos enfrentamos aquí una vez hayamos entendido cómo dividir será saber si nuestro sistema de generadores es compatible con la división ya que, por ejemplo, a $x^2 + y^2 + z^2$ se le puede restar $\left( x+y \right)^2$ e $I$ podría expresarse como $\engendrado{x+y, xyz, -2xy + z^2}$. ¿Sirven ambas representaciones? + \item[Más variables:] Consideremos ahora el anillo de polinomios $\anillodepolinomios{\racionales}{x, y, z}$, un elemento $f: = 4x^2y+ 12xy^2 + \frac{5}{2}xz^3 + x^2yz$ y el ideal $I := \engendrado{f}$. Si bien este espacio no tiene por qué ser un dominio de ideales principales, al solo haber un generador podemos obviar de momento esa parte. Siguiendo la línea del Ejemplo~\ref{Ejemplo Pertenencia Ideal Una Variable}, para ver si $p \in \anillodepolinomios{\racionales}{x, y, z}$ deberíamos dividir por $f$ y ver si el resto es $0$ o no. Pero llegados a este punto, ¿cómo deberíamos dividir? Podemos intentar imitar el algoritmo usual de la división, pero este necesita siempre una noción de orden entre los términos del divisor y también del dividendo. En este caso, ¿deberíamos tomar $4x^2y$ como <> de $f$ o igual sería mejor tomar a uno de mayor grado como $\frac{5}{2}xz^3$? Pero $\frac{5}{2}xz^3$ y $x^2yz$ tienen el mismo grado, ¿cuál escoger, entonces? La elección de este <> en dividendo y divisor cambiará el funcionamiento del algoritmo, así que debemos cerciorarnos de cuáles de estas opciones son válidas y también revisar si el algoritmo tradicional adaptado de esta manera sirve su propósito. + \item[Más generadores:] Sin cambiar de espacio, consideremos ahora el ideal $I := \engendrado{x + y, xyz, x^2 + y^2 + z^2}$. En este caso, el que \anillodepolinomios{\racionales}{x, y, z} no sea un dominio de ideales principales es un problema. ¿Cómo podemos dividir siquiera? Otra dificultad a la que nos enfrentamos aquí una vez hayamos entendido cómo dividir será saber si nuestro sistema de generadores es compatible con la división ya que, por ejemplo, a $x^2 + y^2 + z^2$ se le puede restar $\left( x+y \right)^2$ e $I$ podría expresarse como $\engendrado{x+y, xyz, -2xy + z^2}$. ¿Sirven ambas representaciones? \item[Más <>:] Por último, también debemos tener en cuenta que por lo general trabajaremos en $\Kx^r$, lo cual hará que los elementos tengan más de una componente. ¿Cómo integrar las componentes en las divisiones? \end{description} @@ -839,7 +893,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \end{equation*} Este orden monomial es el \nolbreaks{graduado-lexicográfico} y se denota por \mayorquedeglex. \end{ejemplo} - Para poder generalizar estos órdenes a un módulo como $\Kx^r$ primero tenemos que aclarar qué es un monomio en este anillo. La elección es bastante natural cuando consideramos que un elemento de, por ejemplo, $\anillodepolinomios{\reales}{x, y}^3$ como puede ser $\begin{psmallmatrix} + Para poder generalizar estos órdenes a un módulo como $\Kx^r$ primero tenemos que aclarar qué es un monomio en este anillo. La elección es bastante natural cuando consideramos que un elemento de, por ejemplo, $\anillodepolinomios{\racionales}{x, y}^3$ como puede ser $\begin{psmallmatrix} 2xy + y^2 \\ x^3 \\ y \end{psmallmatrix}$ puede escribirse como $2xy\euno + y^2\euno + x^3\edos + y\etres$. \begin{definicion} @@ -897,7 +951,7 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición \end{definicion} La única novedad de este algoritmo con respecto al que ya conocemos de polinomios en una sola variable es la instrucción de la línea 6. En ella, le sustraemos a $D$ su término líder, lo añadimos al resto y repetimos todo el proceso otra vez con el resto de términos de $D$. Esto lo hacemos para reducir todo lo posible el resto, garantizando que ninguno de sus términos sea divisible por \LM{g}. De no hacer esto, solo podríamos garantizar que, al final de la ejecución, $\LT{D}$ no es divisible por $\LT{g}$. \begin{ejemplo} - Sean $f := 3x^4\euno + 2x^3\euno + 2x^2y\edos + x^2\euno$ y $g := x^2\euno + 2y\etres$ dos elementos de $\anillodepolinomios{\reales}{x, y}^3$ con el orden monomial \mayorquedeglexpos. Aunque hayamos expresado en la definición todo de una forma muy <>, en esencia, lo que hacemos es bastante similar a una división usual: + Sean $f := 3x^4\euno + 2x^3\euno + 2x^2y\edos + x^2\euno$ y $g := x^2\euno + 2y\etres$ dos elementos de $\anillodepolinomios{\racionales}{x, y}^3$ con el orden monomial \mayorquedeglexpos. Aunque hayamos expresado en la definición todo de una forma muy <>, en esencia, lo que hacemos es bastante similar a una división usual: \begin{center} \begin{blockarray}{rrrrrrrcl} \begin{block}{rrrrrrrc|l} @@ -943,9 +997,9 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición \Until{$D = 0$} \end{algorithmic} \end{definicion} - Un ejemplo como el de \eqref{Ejemplo División Compuesta} muestra cómo funciona este algoritmo. Ahora, haría falta generalizar todavía más para ser capaces también de dividir por un número infinito de divisores, pero los siguientes resultado nos permitirán ver que esto no es necesario. + Un ejemplo como el de \eqref{Ejemplo División Compuesta} muestra cómo funciona este algoritmo. Ahora, haría falta generalizar todavía más para ser capaces también de dividir por un número infinito de divisores, pero los siguientes resultado nos permitirán ver que esto no es necesario. El primero de estos, el Teorema de la base de Hilbert, nos garantiza el buen comportamiento de los anillos de polinomios en el sentido de ser noetherianos, lo cual jugará un papel fundamental en el funcionamiento de las bases de \Groebner. \begin{teorema}[de la base de Hilbert] - Sea $A$ un anillo noetheriano. Entonces el anillo de polinomios \anillodepolinomios{A}{x_1, \dots, x_n} es también noetheriano. \todo{Di algo más.} + Sea $A$ un anillo noetheriano. Entonces el anillo de polinomios \anillodepolinomios{A}{x_1, \dots, x_n} es también noetheriano. \end{teorema} \begin{proposicion} \label{Proposición Kx^r noetheriano}%Proposición 2.1.29 de Singular. Sea $A$ un anillo noetheriano y $M$ un \modulo{A} finitamente generado. Entonces, $M$ es un \modulo{A} noetheriano. @@ -954,7 +1008,7 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición Con respecto a los problemas que se presentaban al principio, ya hemos resuelto varios: tenemos un orden para los monomios que nos permite tener un convenio de cuáles son los términos principales de los polinomios, hemos integrado las distintas componentes de un vector de $\Kx^r$ al concepto de monomio y sabemos cómo dividir por ellas, y también sabemos cómo afrontar la división con más de un divisor. Sin embargo, todavía podemos encontrarnos con casos como el siguiente. \begin{ejemplo} \label{Ejemplo Fallo División} - Supongamos que $N := \engendrado{x\euno + y\edos, y\euno + x\edos}$ es un \submodulo{\anillodepolinomios{\reales}{x,y}} de $\anillodepolinomios{\reales}{x, y}^2$ y queremos saber si $f := x^2\edos - y^2\edos$ pertenece a $N$. Al aplicar el algoritmo de la división siguiendo el orden \mayorqueposlex{} de \ref{Algoritmo de la División Compuesta} nos encontramos que ninguno de los dos generadores de $N$ puede reducir a $f$, por lo tanto el resto es el propio $f$. Como el resto es distinto de $0$, estaríamos tentados a decir que $f$ no pertenece a $N$, pero démonos cuenta de que $f = -y\left( x\euno + y\edos \right) + x\left( y\euno + x\edos \right)$ y, por ende, sí que pertenece. + Supongamos que $N := \engendrado{x\euno + y\edos, y\euno + x\edos}$ es un \submodulo{\anillodepolinomios{\racionales}{x,y}} de $\anillodepolinomios{\racionales}{x, y}^2$ y queremos saber si $f := x^2\edos - y^2\edos$ pertenece a $N$. Al aplicar el algoritmo de la división siguiendo el orden \mayorqueposlex{} de \ref{Algoritmo de la División Compuesta} nos encontramos que ninguno de los dos generadores de $N$ puede reducir a $f$, por lo tanto el resto es el propio $f$. Como el resto es distinto de $0$, estaríamos tentados a decir que $f$ no pertenece a $N$, pero démonos cuenta de que $f = -y\left( x\euno + y\edos \right) + x\left( y\euno + x\edos \right)$ y, por ende, sí que pertenece. \end{ejemplo} Para entender bien lo que ocurre en este ejemplo, necesitamos el siguiente objeto. \begin{definicion} @@ -1004,7 +1058,7 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición Con todo esto, ya tenemos una solución al problema de la pertenencia de un elemento a un submódulo: obtener la base de Gröbner del submódulo y dividir por ella para ver si el resto es nulo o no. Sin embargo, no tenemos garantías de que existan bases de Gröbner para todos los submódulos y, en caso de que sí las haya, tampoco sabemos cómo calcularlas. - Recordemos el problema que nos habíamos encontrado en el Ejemplo~\ref{Ejemplo Fallo División}. En este ejemplo, había monomios líderes ocultos en los términos de los generadores. Para desenmascararlos, lo que haremos será lo que vimos en ese ejemplo: anular los términos libres para forzar a que estos aparezcan. Para eso, dados $f = x^\mathbf{\alpha}\vectorbasecanonica{i}$ y $g = x^\mathbf{\alpha}\vectorbasecanonica{j}$ elementos de $\Kx^r$ y $a$ y $b$ elementos de \Kx{} de forma que $a\LM{f} = b\LM{g} = \mathrm{mcm}\left(\LM{f}, \LM{g}\right)$\footnote{Aquí podría haber un pequeño malentendido con la noción de mínimo común múltiplo: por ejemplo, ¿cuál es el mínimo común múltiplo de $xy^2$ y $x^2y$ en \anillodepolinomios{\reales}{x, y}? Lo natural sería decir $x^2y^2$, que es nuestra convención, pero también sería válido decir $3x^2y^2$ ya que ambos tienen el mismo orden monomial. Por simplicidad, siempre nos referiremos a aquel que no está multiplicado por constante.}, definimos la siguiente expresión: + Recordemos el problema que nos habíamos encontrado en el Ejemplo~\ref{Ejemplo Fallo División}. En este ejemplo, había monomios líderes ocultos en los términos de los generadores. Para desenmascararlos, lo que haremos será lo que vimos en ese ejemplo: anular los términos libres para forzar a que estos aparezcan. Para eso, dados $f = x^\mathbf{\alpha}\vectorbasecanonica{i}$ y $g = x^\mathbf{\alpha}\vectorbasecanonica{j}$ elementos de $\Kx^r$ y $a$ y $b$ elementos de \Kx{} de forma que $a\LM{f} = b\LM{g} = \mathrm{mcm}\left(\LM{f}, \LM{g}\right)$\footnote{Aquí podría haber un pequeño malentendido con la noción de mínimo común múltiplo: por ejemplo, ¿cuál es el mínimo común múltiplo de $xy^2$ y $x^2y$ en \anillodepolinomios{\racionales}{x, y}? Lo natural sería decir $x^2y^2$, que es nuestra convención, pero también sería válido decir $3x^2y^2$ ya que ambos tienen el mismo orden monomial. Por simplicidad, siempre nos referiremos a aquel que no está multiplicado por constante.}, definimos la siguiente expresión: \begin{equation*} \spoly{f}{g} = \begin{cases} \frac{a}{\LC{f}}f - \frac{b}{\LC{g}}g &\text{si $i = j$;} \\ @@ -1012,7 +1066,7 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición \end{cases} \end{equation*} \begin{ejemplo} - Sean $f = 3x^2y\euno + 2x\edos$ y $g = 13x\euno + y^5\edos$ elementos de $\anillodepolinomios{\reales}{x, y}^2$ con el orden \mayorquelexpos. Como $\LM{f} = x^2y\euno$ y $\LM{g} = x\euno$ están en la misma componente ---\euno---, y $\mathrm{mcm}\left(\LM{f}, \LM{g}\right) = x^2y = \LM{f} = xy\LM{g}$, tenemos que + Sean $f = 3x^2y\euno + 2x\edos$ y $g = 13x\euno + y^5\edos$ elementos de $\anillodepolinomios{\racionales}{x, y}^2$ con el orden \mayorquelexpos. Como $\LM{f} = x^2y\euno$ y $\LM{g} = x\euno$ están en la misma componente ---\euno---, y $\mathrm{mcm}\left(\LM{f}, \LM{g}\right) = x^2y = \LM{f} = xy\LM{g}$, tenemos que \begin{equation*} \spoly{f}{g} = \frac{1}{3}\left( 3x^2y\euno + 2x\edos \right) + \frac{xy}{13} \left( 13x\euno + y^5\edos \right) = \frac{2}{3}x\edos + \frac{1}{13}xy^6\edos. \end{equation*} @@ -1057,13 +1111,12 @@ A^k\rar["\sigma"]&M\\ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2f_2 + \dots + a_kf_k \end{tikzcd} \end{equation} - + + Con la definición, ya podemos dar por respondida la primera pregunta que nos hacíamos en la introducción: ¿qué es \derivaciones{f}? \begin{definicion} Sean $A$ un anillo, $M$ un \modulo{A} y $f_1, f_2, \dots, f_k$ elementos de $M$. Denotaremos por $\syz{f_1, f_2, \dots, f_k}$ \emph{al \submodulo{A} de $A^k$ formado por todas las sicigias de} $f_1, f_2, \dots, f_k$. \end{definicion} - -\todo{Dale un poco más de bombo a este resultado} - Véase que \derivaciones{f}, tal y como está definido en \eqref{Definicion Modulo Derivaciones de f} puede expresarse en términos de sicigias: + Vemos que \derivaciones{f}, tal y como está definido en \eqref{Definicion Modulo Derivaciones de f} puede expresarse prácticamente como un módulo de sicigias: \begin{align*} \derivaciones{f} &= \definicionconjunto{a_1\partial_1 + a_2\partial_2 + \dots + a_n\partial_n}{a_1\partial_1f + a_2\partial_2f + \dots + a_n\partial_nf = pf, p \in \Kx}; \\ \intertext{recordemos que podemos ver $\partial_i$ como \vectorbasecanonica{i};} @@ -1253,31 +1306,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{equation*} \subsection{$\det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$ implica que \derivaciones{f} es libre} - - \begin{wrapfigure}[7]{r}{0.3\textwidth} - \begin{equation*} - \begin{tikzcd} - \left( \dual{M} \right)_\mathrm{Q} \times M_\mathrm{Q} \arrow[r, "\mathrm{eval}"] & \cuerpofracciones{\Kx} & \\ - \phantom{(}\dual{M}\phantom{)_\mathrm{Q}} \times M\hspace{5pt} \arrow[u, shift left=6mm] \arrow[u, shift right=6mm] \arrow[r, "\mathrm{eval}"] & \Kx \arrow[u] & \\ - 0 \hspace{17pt} \phantom{. \times } 0\arrow[u, shift left=6mm] \arrow[u, shift right=6mm] & 0 \arrow[u] & - \end{tikzcd} - \end{equation*} - \end{wrapfigure} - - Antes de embarcarnos en la otra implicación, necesitaremos un resultado auxiliar. - \begin{lema} - Si $M$ es un \modulo{\Kx} finitamente generado, entonces \dual{M} es reflexivo. - \end{lema} - \begin{proof} - Para cualquier \modulo{A} $B$, denotaremos $B_\mathrm{Q} := B \otimes \cuerpofracciones{A}$. Consideremos el diagrama de homomorfismos de \modulo{\Kx} que aparece a la derecha. Las inyecciones que aquí mostramos son las naturales: en el caso de las de la izquierda, mandan los elementos $m$ bien de $M$ o \dual{M} a los correspondientes $m \otimes 1$; y en el de la derecha se manda $p$ a $\frac{p}{1}$. Estas son inyecciones si y solo si $M$ es libre de torsión ya que en caso de que no, un elemento como $m \otimes 1 = fm\otimes \frac{1}{f}$ podría ser $0$ si $fm = 0$. Con el mapa $\mathrm{eval}$ cometemos un abuso de notación y lo reutilizamos en ambos casos aunque no sea la misma aplicación. Esta envía un par $\left( f, m \right)$ a $f\!\left( m \right)$ en ambos casos, es decir, evalúa la aplicación en el punto que se le dé. - - Podemos ver los conjuntos de la parte baja del diagrama como subconjuntos de los de arriba. Así, pensaremos en $\dual{M}$ como en - \begin{equation*} - \definicionconjunto{h \in \left( \dual{M} \right)_\mathrm{Q}}{\mathrm{eval}\!\left( h, m \right) \in \Kx \ \forall m \in M}, - \end{equation*} - o en otras palabras, como todos aquellos homomorfismos en el conjunto ampliado a las fracciones que <> a $M$ funcionan como un elemento de \dual{M}. - \todo[inline]{Esto terminarlo.} - \end{proof} + Ahora asumamos que existen $\delta_1, \delta_2, \dots, \delta_n$ de forma que $\det\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$ con $u$ una unidad de \Kx{} y volvamos al esquema de \ref{Secuencia Demostración Saito}. A partir de este, sabemos que tenemos \begin{equation*} \begin{array}{rcccl} @@ -1309,8 +1338,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \delta_1 \wedge \delta_2 \wedge \dots \wedge \delta_{i-1} \wedge \delta \wedge \delta_{i+1} \wedge \dots \wedge \delta_{n-1} \wedge \delta_n \in \bigwedge^n N. \end{equation*} Percatémonos de que esta expresión, una vez eliminamos todos los productos exteriores nulos, queda como $\overline{b}_i \delta_1 \wedge \delta_2 \wedge \dots \wedge \delta_n \in \bigwedge^n N$, lo cual es solo posible si $\overline{b}_i \in \Kx$. Con esto concluimos que $\delta \in N$ y, por tanto, que $N = \derivaciones{f}$. - - \todo[inline]{Para poner un ejemplo espero a la introducción, para que case con el ejemplo que allí pongamos.} + \appendix \chapter{Un ejemplo completo de cálculo}\label{Apéndice Cálculo}