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resolver?}¿Cuáles son estas? ¿Qué estructura siguen? ¿Y cómo podemos calcularlas? - -\chapter{Módulos} + Es decir, con estas derivaciones de \derivaciones{f} estamos obteniendo campos de vectores tangentes a la hipersuperficie $f=0$. Este conjunto será el principal objeto de estudio de este documento. + + A primera vista, es apreciable la relación con los vectores, pero no se tratará de espacios vectoriales, sino de una generalización de estos a coeficientes en un anillo: los módulos. Desarrollaremos una descripción de este tipo de objeto en el capítulo \ref{Capítulo de Módulos}. A continuación, notemos que la condición $\delta\!\left( f \right) = hf$ puede reescribirse como + \begin{equation*} + \sum_{i = 1}^{n}a_i \diffp{f}{x_i} = hf \iff \sum_{i = 1}^{n}a_i \diffp{f}{x_i} - hf = 0 \iff \vectorhorizontal{\diffp{f}{x_1}, \diffp{f}{x_2}, \dots, \diffp{f}{x_n}, f} \vectorvertical{a_1 \\ a_2 \\ \vdots \\ a_n \\ -h} = 0. + \end{equation*} + El vector de coeficientes expresa una <> entre los elementos de \vectorhorizontal{\diffp{f}{x_1}, \diffp{f}{x_2}, \dots, \diffp{f}{x_n}, f} al igual que una solución de un sistema lineal homogéneo expresa una relación entre las columnas de una matriz. El conjunto de todas estas relaciones o soluciones del sistema se conoce como el módulo de sicigias. Para poder calcularlo, necesitaremos las Bases de \Groebner: una herramienta que nos ayuda a la hora de determinar si un elemento pertenece o no a un ideal de \Kx, pero en este caso generalizada también a módulos. Todo esto lo desarrollaremos en el capítulo \ref{Capítulo de Bases de Groebner y Sicigias}. + + Por último, una de las principales dificultades que presentan los módulos son las bases. Así como en cualquier espacio vectorial sabemos que siempre hay una base y que todo subespacio también la tiene, no tiene por qué ser así en el caso de módulos. Para poder discernir en particular si \derivaciones{f} admite una base, la herramienta a utilizar es el Criterio de Saito. Este resultado afirma que \derivaciones{f} admite una base si y solo si se pueden encontrar $\delta_1, \delta_2, \dots, \delta_n \in \derivaciones{f}$ de forma que ---viéndolos como vectores de $\Kx^n$--- se cumpla + \begin{equation*} + \det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf\quad \text{donde $u$ es una unidad de \Kx}. + \end{equation*} + En el capítulo \ref{Capítulo del Criterio de Saito} desarrollaremos la demostración de este criterio basados en \cite{VALLES}. + + De hecho, este concepto de que \derivaciones{f} admita una base no se comprende del todo bien actualmente. Por ejemplo, si tomamos $f := x^2z + y^3 \in \anillodepolinomios{K}{x,y,z}$ nos encontramos con la superficie + \begin{figure}[h] + \centering + \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 + \includegraphics[scale=0.3]{segundasuperficie.png}\\ + $x\left( x^2z + y^3 \right) = 0$ + \end{figure} + + Y en este caso, \derivaciones{f} sí que admite una base. Para garantizarlo, hacemos uso del Criterio de Saito con + \begin{align*} + \delta_1 &:= \left( x^2z + y^3 \right)\diffp{}{z},\\ + \delta_2 &:= y\diffp{}{y} + 3z\diffp{}{z} \quad \text{y} \\ + \delta_3 &:= x\diffp{}{x} - 2z\diffp{}{z}. + \end{align*} + Viendo estos como vectores de $\Kx^r$, hacemos el determinante: + \begin{equation*} + \det\!\begin{pmatrix} + 0 & 0 & x \\ + 0 & y & 0 \\ + x^2z + y^3 & 3z & -2z + \end{pmatrix} = -y\left[ x\left( x^2z + y^3 \right) \right] = -yf. + \end{equation*} + El criterio de Saito se cumple, y por lo tanto, $\delta_1$, $\delta_2$ y $\delta_3$ forman una base de \derivaciones{f}. + +\chapter{Módulos} \label{Capítulo de Módulos} \section{Conceptos básicos} Los conjuntos con los que trabajaremos son una generalización de los espacios vectoriales donde los coeficientes pertenecen a un anillo en vez de a un cuerpo. A esta generalización la llamamos módulo. En este capítulo veremos solo algunas de las propiedades y definiciones necesarias para nuestros propósitos. Sin embargo, la teoría de módulos se extiende mucho más allá de esta exigua presentación, por lo que recomendamos que se consulte un texto introductorio como \cite{SINGULARLIBRO} o \cite{COCOA}. @@ -642,7 +683,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du \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} +\chapter{Bases de Gröbner y Sicigias} \label{Capítulo de Bases de Groebner y Sicigias} A partir de este momento solo trabajaremos con módulos $\Kx^n$ ya que son los involucrados en el módulo de derivaciones. @@ -993,7 +1034,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 con sus primeras $r$ componentes nulas y las últimas $k$ coinciden con las de $m_{r+1}$. Por lo tanto, $m_{r+1} \in M_{r+1}$. Concluimos, pues, que $M_{r+1} = \syz{f_1, f_2, \dots, f_k}$. \end{proof} Por lo tanto, para calcular \syz{f_1, f_2, \dots, f_k}, primero calcularemos una base de \Groebner{} $G$ de $$\engendrado{f_1^+ + \vectorbasecanonica{r + 1}, f_2^+ + \vectorbasecanonica{r + 2}, \dots, f_k^+ + \vectorbasecanonica{r + k}}.$$ Por este último teorema, sabemos que $G_{k+1}$ es un conjunto generador de \syz{f_1, f_2, \dots, f_k} y en particular una base de \Groebner{} con respecto al orden que hayamos escogido. Si deseamos una base de \Groebner{} con respecto a otro orden, como decíamos antes de este teorema, basta con aplicar el algoritmo de Buchberger sobre $G_{k+1}$. -\chapter{El Criterio de Saito} +\chapter{El Criterio de Saito} \label{Capítulo del Criterio de Saito} En el capítulo anterior concluíamos con un método para calcular \derivaciones{f} utilizando las sicigias, una herramienta que hemos derivado de las bases de Gröbner. Con estas, podemos obtener un conjunto generador de \derivaciones{f} e incluso una base de Gröbner de este mismo que nos permita determinar si una derivación está o no en el conjunto. Pero otra pregunta de gran importancia aparece cuando consideramos una gran diferencia entre módulos y espacios vectoriales: así como para estos últimos siempre tenemos que los subespacios admiten una base, en el caso de los módulos no tiene por qué ser así. @@ -1254,7 +1295,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \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} + \chapter{Un ejemplo completo de cálculo}\label{Apéndice Cálculo} El algoritmo para calcular las sicigias que nos permite obtener \derivaciones{f}, en esencia el algoritmo de Buchberger, tienen un alto coste computacional que hace impracticable ejecutarlos <>. En la práctica, se utilizan sistemas de Álgebra Computacional que ya vienen preparados con estos algoritmos en sus versiones más óptimas. Sin embargo, como ocurre con muchas otras operaciones a lo largo de nuestra experiencia matemática, conviene saber cómo hacerlas a mano para tener una mayor intuición. En muchos textos que tratan sobre estos asuntos se suelen omitir los ejemplos a este tipo de operaciones o se dan algunos en casos demasiado sencillos ---ideales de \Kx{} en vez de \modulo{\Kx} de $\Kx^n$ donde da la casualidad que todo se anula a la primera--- por lo que creemos conveniente desarrollar en este apéndice por completo un caso no trivial para que el lector se lleve una imagen realista de qué ocurre detrás de la pantalla. Además, ilustraremos también después cómo se resuelve utilizando un sistema dedicado a este tipo de cálculos.