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b/TFG_Sixto_José_Martínez_Gan.tex @@ -306,9 +306,9 @@ urlcolor=black]{hyperref} \derivaciones{\anillodepolinomios{K}{\mathbf{x}}} := \definicionconjunto{a_1\partial_1 + a_2\partial_2 + \dots + a_n\partial_n}{a_i \in \anillodepolinomios{K}{\mathbf{x}} \, \forall i}. \end{equation*} Es un \modulo{\anillodepolinomios{K}{\mathbf{x}}}. Tomando $f \in \anillodepolinomios{K}{\mathbf{x}}$, aquellas derivaciones ---elementos de \derivaciones{\anillodepolinomios{K}{\mathbf{x}}}--- que cuando se aplican a $f$ devuelven un múltiplo escalar forman el submódulo de derivaciones tangentes a $f$: - \begin{equation*} + \begin{equation} \label{Definicion Modulo Derivaciones de f} \derivaciones{f} := \definicionconjunto{\delta \in \derivaciones{\Kx}}{\delta\!\left( f \right) \in \engendrado{f}}. - \end{equation*} + \end{equation} Puede verse que si identificamos $\partial_i$ con \vectorbasecanonica{i}, tenemos que \derivaciones{\Kx} es isomorfo a $\Kx^n$ y, por esta misma razón, \derivaciones{f} puede verse como un \submodulo{\Kx} de $\Kx^n$. \begin{definicion} @@ -373,9 +373,9 @@ urlcolor=black]{hyperref} \end{definicion} Con las secuencias exactas, podemos resumir nuestro problema en una sola expresión. Recordemos que estamos estudiando el \modulo{\Kx} \derivaciones{f}, que es un \submodulo{\Kx} de \derivaciones{\Kx}. Por otro lado, cada derivación $\delta$, cuando se aplica a $f$ nos da lugar a un elemento en el \submodulo{\Kx} $J := \engendrado{\partial_1 f, \dots, \partial_n f}$. Finalmente, a nosotros los elementos que nos interesan son aquellos que sean múltiplos de $f$, es decir, que pertenezcan al \submodulo{\Kx} \engendrado{f}; para poder identificarlo, haremos el cociente de $J$ por \engendrado{f} para que así todas las expresiones de $J$ que sean múltiplos de $f$ vayan al cero. Todo esto lo podemos expresar como sigue: - \begin{equation*} + \begin{equation} \label{Secuencia Exacta Definición Módulo Derivaciones} 0 \rightarrow \derivaciones{f} \xrightarrow{\iota} \derivaciones{\Kx} \xrightarrow{\kappa} \Kx^n \xrightarrow{\Phi} J \xrightarrow{\pi} \frac{J}{J \cap \engendrado{f}} \rightarrow 0. - \end{equation*} + \end{equation} La función $\iota$ es una inclusión, la función $\kappa$ es el homomorfismo que envía $\partial_i$ a \vectorbasecanonica{i}, $\Phi$ envía una derivación $\delta$ a su evaluación en $f$ ---$\delta\!\left( f \right)$--- y $\pi$ es simplemente la aplicación canónica. \chapter{Bases de Gröbner y Syzygias} @@ -647,18 +647,67 @@ urlcolor=black]{hyperref} \State $t \gets t+1$ \EndWhile \end{algorithmic} - Al final del algoritmo, $\left\lbrace d_1, \dots, d_t \right\rbrace$ forma una base de Gröbner de $N$. + Al final del algoritmo, $\left\lbrace d_1, \dots, d_t \right\rbrace$ es una base de Gröbner de $N$. \end{definicion} - \section{Syzygias} + El funcionamiento de este algoritmo se sustenta en dos pilares. El primero es que, como cada vez que añadimos un nuevo elemento este es un resto, tenemos que los \submodulo{\Kx} \engendrado{\LM{d_1}, \dots, \LM{d_t}} forman una cadena ascendente la cual, por ser $\Kx^r$ noetheriano, debe finalizar en algún momento. Por esto mismo, el algoritmo termina. El segundo es el criterio que garantiza que nuestro conjunto generador sea una base de Gröbner. + \begin{teorema}[criterio de Buchberger] + Sea $G$ un subconjunto de $\Kx^r$ equipado con un orden monomial de módulos $>$. $G$ es una base de Gröbner de \engendrado{G} si y solo si para toda pareja de elementos $f$ y $g$ en $G$ se tiene que el resto de $\spoly{f}{g} = 0$ entre $G$ es $0$. + \end{teorema} + \begin{proof} + contenidos... + \end{proof} + + \section{Sicigias} + + En la sección anterior hemos solucionado el problema de pertenencia a un \submodulo{\Kx} de $\Kx^r$, pero necesitamos conocer los generadores de este submódulo. Por lo tanto, hasta el momento solo hemos resuelto la mitad del problema ya que para $\derivaciones{f}$ todavía no sabemos calcular todavía un conjunto de generadores. Para poder hacer esto, introducimos el concepto de sicigia, una noción que extiende el concepto de solución de un sistema lineal en el caso de un espacio vectorial. + + \begin{definicion} + Sean $A$ un anillo, $M$ un \modulo{A} y $f_1, f_2, \dots, f_k$ elementos de $M$. Una sicigia de los elementos $f_1, f_2, \dots, f_k$ es una \nolbreaks{$k$-tupla} $\left( g_1, g_2, \dots, g_k \right)$ de elementos de $A$ de forma que $f_1g_1 + f_2g_2 + \dots + f_kg_k = 0$. + \end{definicion} + Nótese que el conjunto de sicigias de $f_1, f_2, \dots, f_k$ forma un \submodulo{A} de $A^k$. + \begin{definicion} + Sean $A$ un anillo, $M$ un \modulo{A} y $f_1, f_2, \dots, f_k$ elementos de $M$. Denotaremos por $\syz{f_1, f_2, \dots, f_k}$ al \submodulo{A} de $A^k$ formado por todas las sicigias de $f_1, f_2, \dots, f_k$. + \end{definicion} + + Véase que \derivaciones{f}, tal y como está definido en \eqref{Definicion Modulo Derivaciones de f} puede expresarse en términos 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} a través del isomorfismo $\kappa$ de \eqref{Secuencia Exacta Definición Módulo Derivaciones};} + &\simeq \definicionconjunto{\left( a_1, a_2, \dots, a_n \right) \in \Kx^n}{a_1\partial_1f + a_2\partial_2f + \dots + a_n\partial_nf - pf = 0, p \in \Kx} \\ + &= \definicionconjunto{\left( a_1, a_2, \dots, a_n \right) \in \Kx^n}{\left( a_1, a_2, \dots, a_n, p \right) \in \syz{\partial_1f, \partial_2f, \dots, \partial_nf, f}, p \in \Kx}. + \end{align*} + + Para poder calcular las sicigias necesitaremos dos resultados. El primero nos permite restringir una base de Gröbner a solamente unas coordenadas. + \begin{lema} + contenidos... + \end{lema} + + Con este resultado, ya podemos calcular sicigias. + \begin{teorema} + contenidos... + \end{teorema} \chapter{El Criterio de Saito} - % Contarlo de primeras. + En 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. + + \begin{ejemplo} + \todo[inline]{Ejemplo de submódulo de $\Kx^r$ que no admite base.} + \end{ejemplo} + + \begin{definicion} + Sea $A$ un anillo, $M$ un \modulo{A} y $N$ un \submodulo{A} de $M$. Diremos que $N$ es libre si admite una base. + \end{definicion} + + El criterio de Saito nos da una forma bastante rápida de determinar si \derivaciones{f} es o no libre. + \begin{teorema}[Criterio de Saito] + Sea $f$ un polinomio de \Kx{} y $\delta_1, \delta_2, \dots, \delta_n$ derivaciones de \derivaciones{f}. Entonces, \derivaciones{f} es libre si y solo si $\det\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$ donde $u$ es una unidad de $K$, es decir, un elemento de $K \setminus \left\lbrace 0 \right\rbrace$. + \end{teorema} + En este capítulo explicaremos la demostración de este resultado que aparecen en [\textsc{Referencia al artículo de Vallés}]. Pero primero, deberemos ilustrar algunas construcciones que se utilizan en dicha demostración. \section{Producto Exterior} De cara a la demostración del Criterio de Saito, utilizaremos una operación que nos - % Demostrar que Der R es reflexivo. - % Propiedades de los modulos reflexivos (libre implica proyectivo implica reflexivo y los conversos condicionados), reflexivo implica libre de torsión, - % + \todo[inline]{Demostrar que Der R es reflexivo. Propiedades de los modulos reflexivos (libre implica proyectivo implica reflexivo y los conversos condicionados), reflexivo implica libre de torsión, etc como esquema.} + \end{document} \ No newline at end of file