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z@1Iym`{bK8V&XPvmLTN?EZFTAWpNv{fZwGuDXkn0V@BIecZ`Z@HCl^7*L4;-dU|dH=2?1beWPFC%1pNNtmd zPfkb(#+X+|E=XbGqL%TU3^73>GEb!twhG2c>GDL=ZBN hhC`nI0-aGfFOxQjhe(3P#m&#f!;L~qE3GVp@^9t)8N~nq diff --git a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index 430013f..0289c5f 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -45,6 +45,7 @@ \usepackage{diffcoeff} \usepackage{nolbreaks} \usepackage{nicefrac} +\usepackage{stix} % Bibliografía. \usepackage[backend=biber]{biblatex} @@ -820,23 +821,78 @@ urlcolor=black]{hyperref} Si $G$ es una base de \Groebner, entonces, como $\spoly{f}{g} \in \engendrado{G}$, por el teorema \ref{Teorema Buen Funcionamiento División Groebner} tenemos que el resto de su división entre los elementos de $G$ será $0$. Como $\Kx^n$ es noetheriano, podemos asumir desde el principio que $G = \conjunto{g_1, g_2, \dots, g_k}$ + \todo[inline]{Ver cómo lo metemos.} \end{proof} Para poder calcular las sicigias necesitaremos dos resultados. El primero nos permite restringir una base de Gröbner a solamente unas coordenadas. - \begin{lema} - contenidos... + \begin{lema} \label{Lema Restricción Groebner} + Sea $>$ un orden monomial de módulos que satisface + \begin{equation} \label{Condición Orden Eliminación} + x^\mathbf{\alpha}\vectorbasecanonica{i} > x^\mathbf{\beta}\vectorbasecanonica{j} \Leftrightarrow \begin{cases} + i < j& \text{o} \\ + i = j& \text{y} \ x^\mathbf{\alpha} >' x^\mathbf{\beta} + \end{cases} \quad \text{donde $>'$ es un orden monomial.} + \end{equation} + Es decir, un orden que mira primero la posición en el orden usual. Sea $M$ un \submodulo{\Kx} de $\Kx^n$ y $G$ una base de \Groebner{} de este con respecto al orden $>$. Entonces, para el módulo restringido a las componentes a partir de la \nolbreaks{$s$-ésima}, es decir, + \begin{equation*} + M_{s} := \Big\lbrace\left.\vectorhorizontalcuatro{m_s}{m_{s+1}}{\dots}{m_n} \right\rvert (\underbrace{0, 0, \dots, 0}_{s-1}, m_s, m_{s+1}, \dots, m_n) \in M \Big\rbrace, + \end{equation*} + el conjunto $G_s$ es una base de \Groebner{} con respecto al orden $>$. \end{lema} + \begin{proof} + Tomemos un $f_s = \vectorhorizontalcuatro{f_s}{f_{s+1}}{\dots}{f_n}$ en $M_s$. Por definición, este tiene un elemento asociado en $M$, $f = \left( 0, 0, \dots, 0, f_s, f_{s+1}, \dots, f_m \right)$. Al ser $G$ una base de \Groebner{} de $M$, tendremos que habrá algún $g \in G$ de forma que $\LM{g} \vert \LM{f}$. Ahora bien, por la condición que satisface el orden, sabemos que todos los términos de $g$ deben encontrarse en la \nolbreaks{$s$-ésima} componente o por debajo, lo cual nos garantiza que $g = \left( 0, 0, \dots, 0, g_s, g_{s+1}, \dots, g_n \right)$. Gracias a esto, sabemos que también se cumplirá que $\LM{g_s} \vert \LM{f_s}$ y, como $G_s \subseteq M_s$, esto es equivalente a que $\LM{M_s} = \engendrado{\LM{G}}$ lo cual nos garantiza que $G_s$ es base de \Groebner{} con respecto a este orden. + \end{proof} + + Démonos cuenta de que esto nos permite calcular bases de \Groebner{} a $M_s$ con respecto a cualquier orden: escogemos un orden que satisfaga la condición que impone el lema \ref{Lema Restricción Groebner} con el cual $G_s$. Pero démonos cuenta de que $G_s$ es un conjunto generador de $M_s$ independientemente del orden escogido, así que cambiamos el orden y calculamos la base de \Groebner{} a partir de $G_s$ con respecto a este nuevo orden. - Con este resultado, ya podemos calcular sicigias. \begin{teorema} \label{Teorema Calculo Sicigias} - contenidos... + Sean $f_1, f_2, \dots, f_k$ elementos de $\Kx^r$ y sean $f_1^+, f_2^+, \dots, f_k^+$ los correspondientes elementos en $\Kx^{r+k}$ pero en este caso por el principio, es decir, si $f_i = \vectorhorizontalcuatro{a_1}{a_2}{\dots}{a_r}$ entonces + \begin{equation*} + f_i^+ = (a_1, a_2, \dots, a_r, \underbrace{0, 0, \dots, 0}_{k}). + \end{equation*} + Entonces, si $G$ es una base de \Groebner{} de $M := \engendrado{f_1^+ + \vectorbasecanonica{r + 1}, f_2^+ + \vectorbasecanonica{r + 2}, \dots, f_k^+ + \vectorbasecanonica{r + k}}$ con respecto a un orden $>$ que satisfaga la condición \eqref{Condición Orden Eliminación}, se tiene que ---siguiendo la notación introducida en el lema \ref{Lema Restricción Groebner}--- + \begin{equation*} + \syz{f_1, f_2, \dots, f_k} = \engendrado{G_{r+1}}. + \end{equation*} + En particular, $G_{r+1}$ es una base de \Groebner{} de \syz{f_1, f_2, \dots, f_k} con respecto al orden $>$. \end{teorema} + \begin{proof} + Por el lema anterior sabemos que $G_{r+1}$ es una base de \Groebner{} de $M_{r+1}$. Queda pues comprobar que $M_{r+1} = \syz{f_1, f_2, \dots, f_k}$. Supongamos\footnote{Para que el lector sepa en todo momento cuándo un elemento está en $\Kx^{r+k}$ o $\Kx^{k}$, utilizamos el subíndice en el elemento dándole consistencia con la notación utilizada en el lema anterior. Pero este no debe ser interpretado de forma distinta que si pusiéramos $m'$ o $\overline{m}$; es decir, el subíndice sobre un elemento no debe verse como el resultado de una <>.} que $m_{r+1} = \left(\mu_{r+1}, \mu_{r+2}, \dots, \mu_{r+k}\right) \in M_{r+1}$, entonces existe un elemento asociado $m = \left( 0, 0, \dots, 0, \mu_{r+1}, \mu_{r+2}, \dots, \mu_{r+k}\right) \in M$. Podemos expresar $m$ de dos formas, una en términos de los vectores de la base canónica y otra en términos de los $f_i^+ + \vectorbasecanonica{r + i}$ que generan $M$: + \begin{align*} + \mu_{r+1}\vectorbasecanonica{r+1} + \mu_{r+2}\vectorbasecanonica{r+2} + \dots + \mu_{r+k}\vectorbasecanonica{r+k} &= b_1\left( f_1^+ + \vectorbasecanonica{r+1} \right) + b_2\left( f_2^+ + \vectorbasecanonica{r+2} \right) + \dots + b_k\left( f_k^+ + \vectorbasecanonica{r+k} \right), \quad b_j \in \Kx; \\ + &= \left( b_1f_1^+ + b_2f_2^+ + \dots + b_kf_k^+ \right) \ + \ \left( b_{1}\vectorbasecanonica{r+1} + b_{2}\vectorbasecanonica{r+2} + \dots + b_{k}\vectorbasecanonica{r+k} \right). + \end{align*} + Como todas las componentes de $f_i^+$ desde la \nolbreaks{$r+1$-ésima} ---incluida--- son $0$, podemos extraer dos conclusiones. La primera es que $b_i = \mu_i$ ya que son los únicos coeficientes que acompañan a las últimas componentes; y la segunda es que + \begin{equation*} + \mu_1f_1 + \mu_2f_2 + \dots + \mu_kf_k = b_1f_1 + b_2f_2 + \dots + b_kf_k = 0. + \end{equation*} + Por ende, $m_{r+1} \in \syz{f_1, f_2, \dots, f_k}$. En el otro sentido, si tenemos $m_{r+1} := \left( \mu_1, \mu_2, \dots, \mu_k \right) \in \syz{f_1, f_2, \dots, f_k}$, entonces existe + \begin{equation*} + m := \mu_{r+1}\left( f_1^+ + \vectorbasecanonica{r+1} \right) + \mu_{r+2}\left( f_2^+ + \vectorbasecanonica{r+2} \right) + \dots + \mu_{r+k}\left( f_k^+ + \vectorbasecanonica{r+k} \right) \in M + \end{equation*} + 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} \chapter{El Criterio de Saito} - 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. + 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. \begin{ejemplo} - \todo[inline]{Ejemplo de submódulo de $\Kx^r$ que no admite base.} + Consideremos el \modulo{\anillodepolinomios{K}{x, y}} \engendrado{x, y}. Por ser noetheriano, sabemos que, de existir alguna base para \engendrado{x, y}, esta debe de ser finita. Supongamos que existe y es $\conjunto{a_1, a_2, \dots, a_k}$. Entonces, existirán unos únicos $p_1, p_2, \dots, p_k$ y $q_1, q_2, \dots, q_k$ en \anillodepolinomios{K}{x, y} de forma que + \begin{equation*} + p_1a_1 + p_2a_2 + \dots + p_ka_k = x \quad \text{y} \quad q_1a_1 + q_2a_2 + \dots + q_ka_k = y. + \end{equation*} + Esto nos permite llegar a la siguiente igualdad: + \begin{equation*} + yp_1a_1 + yp_2a_2 + \dots + yp_ka_k = xy = xq_1a_1 + xq_2a_2 + \dots + xq_ka_k. + \end{equation*} + Ahora bien, por ser $\conjunto{a_1, a_2, \dots, a_k}$ una base, necesariamente tendremos que $yp_i = xq_i$ para todo $i$. Como $\mcd{x, y} = 1$, es necesario que $p_i = xp_i'$. Esto nos lleva a que + \begin{align*} + xp_1'a_1 + xp_2'a_2 + \dots + xp_k'a_k &= x \\ + x\left( p_1'a_1 + p_2a_2' + \dots + p_k'a_k \right) &= x, \\ + \intertext{$\Kx$ es un dominio de integridad por lo que podemos simplificar,} + p_1'a_1 + p_2a_2' + \dots + p_k'a_k &= 1. + \end{align*} + Esto último es una contradicción, ya que $1 \notin \engendrado{x, y}$. \end{ejemplo} \begin{definicion} @@ -982,7 +1038,7 @@ urlcolor=black]{hyperref} \end{aligned}\right\rbrace \implies \text{$M$ es un módulo libre} \end{gather} - Puede encontrarse en \cite[corolario 1.7]{SAITO} una demostración de que \derivaciones{f} es un módulo reflexivo. Siguiendo el esquema, tenemos que \derivaciones{f} es libre de torsión, pero necesitaríamos que tuviese rango $1$ para continuar hacia delante. Lo que haremos será estudiar el módulo $\bigwedge^n \derivaciones{f}$ para reducir este rango y poder continuar. + Puede encontrarse en \cite[corolario 1.7]{SAITO} una demostración de que \derivaciones{f} es un módulo reflexivo de rango $n$. Siguiendo el esquema, tenemos que \derivaciones{f} es libre de torsión, pero necesitaríamos que tuviese rango $1$ para continuar hacia delante. Lo que haremos será estudiar el módulo $\bigwedge^n \derivaciones{f}$ para reducir este rango y poder continuar. \begin{lema} Sea $M$ un \modulo{A} de rango $n$ y $s \in \naturales$. Entonces, $\bigwedge^s M$ es un \modulo{A} de rango $\binom{n}{s}$. \end{lema} @@ -1083,12 +1139,13 @@ urlcolor=black]{hyperref} 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 siguiente diagrama de homomorfismos de \modulo{\Kx}: - - - 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, es directamente la inclusión. Estas son inyecciones si y solo si $M$ es libre de torsión. on el mapa $\mathrm{eval}$ cometemos un abuso de notación y lo reutilizamos en ambos casos aunque no sería igual. Esta aplicación 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. - + 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, es directamente la inclusión. 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 sería igual. Esta aplicación 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}. \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*}