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zFqSx_CbQ4#z?O%!qFjF(C?9QVvD`-gd6-XVWfLm%*%4Lfq7?ruihoZ0^aD_F(@Z95 z&tH*lZ)--+0;a-|69^`@#!k+TCI&YDk?agD5uBV29G%@AO-vD3nb=s_m=MUx#pJ~i F{tJ}kn@a!y diff --git a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index f79a928..bdc5c56 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -83,7 +83,7 @@ urlcolor=black]{hyperref} \newpagestyle{miestilo}{ %-- Cabecera par y cabecera impar \sethead[\thepage][][\textsl{\chaptername\ \thechapter. \chaptertitle}] -{\textsl{Una plantilla para el TFG \ - \ Mario Pérez Riera}}{}{\thepage} +{\textsl{Una plantilla para el TFG \ - \ Sixto José Martínez Gan}}{}{\thepage} %-- Pie de página par y pie de página impar %\setfoot[][\textsl{\thesection. \sectiontitle}][] %{}{\textsl{\thesection. \sectiontitle}}{} @@ -122,6 +122,7 @@ urlcolor=black]{hyperref} \usepackage[negrita, parentesis, vertical, punto]{vectores_TFG} % Cosas propias +\newcommand{\lineal}[1]{\nolbreaks{\ensuremath{#1}-lineal}} \newcommand{\modulo}[1]{\nolbreaks{\ensuremath{#1}-módulo}} \newcommand{\submodulo}[1]{\nolbreaks{\ensuremath{#1}-submódulo}} \newcommand{\espaciovectorial}[1]{\nolbreaks{\ensuremath{#1}-espacio} vectorial} @@ -145,6 +146,7 @@ urlcolor=black]{hyperref} \newcommand{\mayorquelexpos}{\mayorquemodular{LEX}{POS}} \newcommand{\mayorqueposdeglex}{\mayorquemodular{POS}{GRAD-LEX}} \newcommand{\mayorquedeglexpos}{\mayorquemodular{GRAD-LEX}{POS}} +\newcommand{\mayorquemodulos}{\mayorque{\mathrm{m\acute{o}d}}} \newcommand{\LT}[1]{\ensuremath{\mathrm{LT}\left(#1\right)}} \newcommand{\LM}[1]{\ensuremath{\mathrm{LM}\left(#1\right)}} @@ -270,23 +272,23 @@ 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\todo[fancyline]{¿Es necesario explicar cómo serían las combinaciones lineales de los módulos o con la analogía a espacios vectoriales hecha antes es suficiente?} de estos vectores, + 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, \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 $\mathbf{v}$. \end{ejemplo} \begin{definicion} - Sea $N$ un \submodulo{A} de $M$ y $G$ un subconjunto cualquiera de $N$. Diremos que $N$ está generado por $G$ y lo denotaremos como $N = \engendrado{G}$ si + Sea $N$ un \submodulo{A} de $M$ y $G$ un subconjunto cualquiera de $N$. Diremos que $N$ está generado por $G$ y lo denotaremos como $N = \engendrado{G}_A$ si \begin{equation*} N = \definicionconjunto{a_1g_1 + a_2g_2 + \dots + a_kg_k}{a_i \in A, g_i \in G, k \in \naturales}. \end{equation*} - En el caso que exista algún $G = \conjunto{g_1, g_2, \dots, g_n}$ de forma que $N = \engendrado{G}$ diremos que $N$ está finitamente generado y lo escribiremos como $N = \engendrado{g_1, g_2, \dots, g_n}$. + En el caso que exista algún $G = \conjunto{g_1, g_2, \dots, g_n}$ de forma que $N = \engendrado{G}_A$ diremos que $N$ está finitamente generado y lo escribiremos como $N = \engendrado{g_1, g_2, \dots, g_n}_A$. Cuando el anillo con el que estamos trabajando esté claro, omitiremos el subíndice. \end{definicion} \begin{definicion} \label{Definición Divisibilidad Módulos} Sean $m$ y $l$ dos elementos de un \modulo{A} $M$. Diremos que $m$ divide a $l$ si existe $a \in A$ de forma que $am = l$. \end{definicion} \begin{definicion} - Sea $N$ un \submodulo{A} de $M$ y $\sim$ la siguiente relación de equivalencia sobre $M$: $m \sim m' \Leftrightarrow m - m' \in N$. Definimos el cociente de un módulo por un submódulo como $\cociente{M}{N} := \cociente{M}{\sim}$. Los elementos de este cociente se denotarán por $m + N$. + Sea $N$ un \submodulo{A} de $M$ y $\sim$ la siguiente relación de equivalencia sobre $M$: \nolbreaks{$m \sim m' \Leftrightarrow m - m' \in N$}. Definimos el cociente de un módulo por un submódulo como $\cociente{M}{N} := \cociente{M}{\sim}$, que es un \modulo{A} con las operaciones heredadas. Los elementos de este cociente se denotarán por $m + N$. \end{definicion} \section{Homomorfismos} @@ -296,7 +298,7 @@ urlcolor=black]{hyperref} \item $\phi\!\left( m + m' \right) = \phi\!\left( m \right) + \phi\!\left( m' \right)$ y \item $\phi\!\left( am \right) = a\phi\!\left( m \right)$ \end{itemize} - para todo $m, m' \in M$ y $a \in A$. + para todo $m, m' \in M$ y $a \in A$. En ocasiones, nos referiremos a los homomorfismos de \modulo{A}s como a aplicaciones \lineal{A}es. \end{definicion} \begin{ejemplo} La inclusión de $\enteros^n$ en $\racionales^n$ es un homomorfismo de \modulo{\enteros}s, sin embargo, no es un homomorfismo de \modulo{\racionales}s ya que $\enteros^n$ no es un \modulo{\racionales}. @@ -316,17 +318,18 @@ urlcolor=black]{hyperref} \begin{equation*} \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 $f$: + 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 de $f$ forman el submódulo de derivaciones tangentes a $f$: \begin{equation} \label{Definicion Modulo Derivaciones de f} \derivaciones{f} := \definicionconjunto{\delta \in \derivaciones{\Kx}}{\delta\!\left( f \right) \in \engendrado{f}}. \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} - Diremos que $A$ es noetheriano si todos sus ideales son finitamente generados. + Diremos que un anillo $A$ es noetheriano si todos sus ideales son finitamente generados. - Diremos también que un \modulo{A} $M$ es noetheriano si todo submódulo $N \subseteq M$ está finitamente generado. Es decir, si existen $n_1, n_2, \dots, n_k$ en $N$ de forma que $N = \engendrado{n_1, n_2, \dots, n_k}_A$. + Diremos también que un \modulo{A} $M$ es noetheriano si todo submódulo $N \subseteq M$ está finitamente generado. Es decir, si existen $n_1, n_2, \dots, n_k$ en $N$ de forma que $N = \engendrado{n_1, n_2, \dots, n_k}$. \end{definicion} + De aquí en adelante, salvo que se indique lo contrario, asumiremos que $A$ siempre es un anillo noetheriano. \begin{lema} Sea $M$ un \modulo{A} noetheriano. Los submódulos y módulos cocientes de $M$ son también noetherianos. \end{lema} @@ -440,9 +443,9 @@ urlcolor=black]{hyperref} 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} \label{Secuencia Exacta Definición Módulo Derivaciones} - 0 \rightarrow \derivaciones{f} \xrightarrow{\Phi} \derivaciones{\Kx} \xrightarrow{\kappa} \Kx^n \xrightarrow{\iota} J \xrightarrow{\pi} \frac{J}{J \cap \engendrado{f}} \rightarrow 0. + 0 \rightarrow \derivaciones{f} \xrightarrow{\Phi} \derivaciones{\Kx} \xrightarrow{\kappa} \Kx^n \xrightarrow{\xi} J \xrightarrow{\pi} \frac{J}{J \cap \engendrado{f}} \rightarrow 0. \end{equation} - La función $\Phi$ es una inclusión, la función $\kappa$ es el isomorfismo que envía $\partial_i$ a \vectorbasecanonica{i}, $\iota$ envía una derivación $\delta$ a su evaluación en $f$ ---$\delta\!\left( f \right)$--- y $\pi$ es simplemente la aplicación canónica. + La función $\Phi$ es una inclusión, la función $\kappa$ es el isomorfismo que envía $\partial_i$ a \vectorbasecanonica{i}, $\xi$ envía una derivación $\delta$ a su evaluación en $f$ ---$\delta\!\left( f \right)$--- y $\pi$ es simplemente la aplicación canónica. Un tipo de módulo muy importante en la demostración del Criterio de Saito es el módulo proyectivo, que nos sirve en muchos casos como tipo de módulo intermedio para algunos razonamientos. \begin{definicion} \label{Definición Módulo Proyectivo} @@ -457,10 +460,10 @@ urlcolor=black]{hyperref} \end{equation*} \begin{ejemplo} - Sean $M = \cociente{\enteros}{4\enteros}$, $N = \cociente{\enteros}{2\enteros}$ y $P = \cociente{\enteros}{2\enteros}$ vistos como \modulo{\enteros}s. Como $e$ es un epimorfismo de \modulo{\enteros}s, necesariamente tendremos que $e\!\left( 0 \right) = e\!\left( 2 \right) = 0$ y $e\!\left( 1 \right) = e\!\left( 3 \right) = 1$. Como $h$ simplemente tomaremos la identidad. Sea ahora $\phi$ un homomorfismo de \modulo{\enteros}s cualquiera; existen dos posibilidades para $\phi\!\left( 1 \right)$: + Sean $M = \cociente{\enteros}{4\enteros}$, $N = \cociente{\enteros}{2\enteros}$ y $P = \cociente{\enteros}{2\enteros}$ vistos como \modulo{\enteros}s. Como $e$ es un epimorfismo de \modulo{\enteros}s, necesariamente tendremos que $e\!\left( 0 \right) = e\!\left( 2 \right) = 0$ y $e\!\left( 1 \right) = e\!\left( 3 \right) = 1$. Como $h$ simplemente tomaremos la identidad. Sea ahora $\phi: P \rightarrow M$ un homomorfismo de \modulo{\enteros}s cualquiera; existen dos posibilidades para $\phi\!\left( 1 \right)$: \begin{itemize} - \item si $\phi\!\left( 1 \right) = 0$, entonces $e\!\left( \phi\!\left( 1 \right) \right) = 0 \neq h\!\left( 1 \right)$; - \item si por el contrario $\phi\!\left( 1 \right) = 1$, entonces $\phi\!\left( 0 \right) = \phi\!\left( 2\cdot 1 \right) = 2\phi\!\left( 1 \right) = 2$ y de nuevo llegamos a que $e\!\left( \phi\!\left( 1 \right) \right) = 0 \neq h\!\left( 1 \right)$. + \item si $\phi\!\left( 1 \right) \in \conjunto{0, 2}$, entonces $e\!\left( \phi\!\left( 1 \right) \right) = 0 \neq h\!\left( 1 \right)$; + \item si por el contrario $\phi\!\left( 1 \right) \in \conjunto{1, 3}$, entonces $\phi\!\left( 0 \right) = \phi\!\left( 2\cdot 1 \right) = 2\phi\!\left( 1 \right) = 2$ y de nuevo llegamos a que $e\!\left( \phi\!\left( 1 \right) \right) = 0 \neq h\!\left( 1 \right)$. \end{itemize} En conclusión, $\cociente{\enteros}{2\enteros}$ no es un \modulo{\enteros} proyectivo. \end{ejemplo} @@ -500,15 +503,26 @@ urlcolor=black]{hyperref} En el mismo anillo que el ejemplo anterior, consideremos el ideal $I := \engendrado{x + y, xyz, x^2 + y^2 + z^2}$. Con este ideal la situación es todavía peor que con el anterior. ¿Nos sirven estos generadores? ¿Y cómo hacemos para dividir con ellos? - Vemos por lo tanto que en el momento en el que introducimos más variables, perdemos la capacidad de resolver el problema de una forma razonable. Para poder solucionar este problema, deberemos generalizar el algoritmo de la división siguiendo las claves del ejercicio \eqref{Ejemplo Pertenencia Ideal Dos Variables} para que este pueda aplicarse a estos nuevos objetos y, asimismo, deberemos trabajar en una representación del ideal que también nos permita que esto funcione como es debido, para evitar las ambigüedades vistas en este mismo ejercicio. + Vemos por lo tanto que en el momento en el que introducimos más variables, perdemos la capacidad de resolver el problema de una forma razonable. Para poder solucionar este problema, deberemos generalizar el algoritmo de la división para que este pueda aplicarse a estos nuevos objetos y, asimismo, deberemos trabajar en una representación del ideal que sea compatible con nuestro nuevo algoritmo. \section{Graduación de Módulos} - Pensemos con detalle en el algoritmo de la división en un anillo de polinomios familiar como es \anillodepolinomios{\reales}{x}. Al dividir $p = x^3 + 2x^2 - x + 1$ entre $q = x^2 - 1$, buscamos anular el término de mayor grado de $p$ ---$x^3$--- haciendo al mayor término de $q$ ---$x^2$--- alcanzar a este. Es decir, lo que hacemos puede expresarse como - \begin{align*} - x^3 + 2x^2 - x + 1 &= 0q + x^3 + 2x^2 - x + 1 \\ - x^3 + 2x^2 - x + 1 &= xq + - \end{align*} \todo[inline]{Completar.} - Como puede apreciarse, el funcionamiento de este algoritmo depende muy fuertemente de la noción de grado de un polinomio. Sin embargo, en el caso de módulos el grado no siempre está claro. Por lo tanto, introducimos el concepto de que un anillo o un módulo esté <>. + Pensemos con detalle en el algoritmo de la división en un anillo de polinomios familiar como es \anillodepolinomios{\reales}{x}. Al dividir $p = x^3 + 2x^2 - x + 1$ entre $q = x^2 - 1$, buscamos anular el término de mayor grado de $p$ ---$x^3$--- haciendo al mayor término de $q$ ---$x^2$--- alcanzar a este. Esto es básicamente en lo que pensamos en todo momento mientras realizamos el algoritmo de la división <>: + \begin{center} + \begin{blockarray}{rrrrcl} + \begin{block}{rrrrc|l} + $x^3$ & $+2x^2$ & $-x\phantom{)}$ & $+1\phantom{)}$ & & \cocientecaja{x^2 - 1} \\ + $-(x^3$ & & $-x)$ & & & $x + 2$ \\ + \end{block} + \begin{block}{rrrrcl} + \cmidrule{1-4} + \tiny 1\textsuperscript{a} iteración: &$+2x^2$ & & $+1\phantom{)}$ & & \\ + & $-(2x^2$ & & $-2)$ & & \\ + \cmidrule{2-4} + & & \tiny 2\textsuperscript{a} iteración: & $+3\phantom{)}$ & & \\ + \end{block} + \end{blockarray} + \end{center} + Como puede apreciarse, el funcionamiento de este algoritmo depende muy fuertemente de la noción de grado de un polinomio. Al estar generalizando todo, es oportuno que generalicemos también la noción de grado a anillos y módulos. \begin{definicion} Sea $A$ un anillo. Se dice que es un anillo graduado si existe una descomposición $\bigoplus_{g=0}^{\infty} A_g$ donde $A_g$ es un grupo abeliano para todo $g$ y se cumple que $A_g A_h \subseteq A_{g+h}$ para todo $g$ y $h$. @@ -525,7 +539,7 @@ urlcolor=black]{hyperref} \begin{equation*} x_1^{\alpha_1}x_2^{\alpha_2}\dots x_n^{\alpha_n} \longleftrightarrow \left( \alpha_1, \alpha_2, \dots, \alpha_n \right). \end{equation*} - Tomando $\mathbf{\alpha} \in \naturales^n$, notaremos $x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_n^{\alpha_n}$ como $x^\mathbf{\alpha}$. Así pues, podemos tratar + Tomando $\mathbf{\alpha} \in \naturales^n$, notaremos $x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_n^{\alpha_n}$ como $x^\mathbf{\alpha}$. Así pues, un orden monomial se corresponde con un orden total en $\naturales^n$ de forma que sea compatible con la estructura de monoide que posee el conjunto de monomios. \begin{definicion} Un orden monomial $>$ es un orden total sobre los monomios en $n$ variables que respeta la multiplicación, es decir, que satisface \begin{equation*} @@ -552,17 +566,20 @@ urlcolor=black]{hyperref} \end{ejemplo} Para poder generalizar este concepto 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 $\begin{psmallmatrix} 2xy + y^2 \\ x^3 \\ y - \end{psmallmatrix}$ puede escribirse como $2xy\euno + y^2\euno + x^3\edos + y\etres$. Así que a las expresiones de la forma $x^\mathbf{\alpha}\vectorbasecanonica{i}$ con $i \in \left\lbrace 1, \dots, n \right\rbrace$ las llamaremos monomios de \Kx. + \end{psmallmatrix}$ puede escribirse como $2xy\euno + y^2\euno + x^3\edos + y\etres$. \begin{definicion} - Sea $>$ un orden monomial en \Kx. Un orden monomial de módulos $>_\text{mód}$ es un orden total sobre los monomios de $\Kx^r$ que es compatible con la estructura de \modulo{\Kx} de $\Kx^r$ y con el orden $>$, es decir, que satisface + A las expresiones de la forma $x^\mathbf{\alpha}\vectorbasecanonica{i}$ con $i \in \left\lbrace 1, \dots, n \right\rbrace$ las llamaremos monomios de \Kx. + \end{definicion} + \begin{definicion} + Sea $>$ un orden monomial en \Kx. Un orden monomial de módulos \mayorquemodulos{} es un orden total sobre los monomios de $\Kx^r$ que es compatible con la estructura de \modulo{\Kx} de $\Kx^r$ y con el orden $>$, es decir, que satisface \begin{itemize} - \item $x^{\mathbf{\alpha}}\vectorbasecanonica{i} >_\text{mód} x^{\mathbf{\beta}}\vectorbasecanonica{j} \implies x^{\mathbf{\alpha}}x^\mathbf{\gamma}\vectorbasecanonica{i} >_\text{mód} x^{\mathbf{\beta}}x^\mathbf{\gamma}\vectorbasecanonica{j}\quad$ y - \item $x^\mathbf{\alpha} > x^\mathbf{\beta} \implies x^\mathbf{\alpha}\vectorbasecanonica{i} > x^\mathbf{\beta}\vectorbasecanonica{i}$ + \item $x^{\mathbf{\alpha}}\vectorbasecanonica{i} \mayorquemodulos x^{\mathbf{\beta}}\vectorbasecanonica{j} \implies x^{\mathbf{\alpha}}x^\mathbf{\gamma}\vectorbasecanonica{i} \mayorquemodulos x^{\mathbf{\beta}}x^\mathbf{\gamma}\vectorbasecanonica{j}\quad$ y + \item $x^\mathbf{\alpha} > x^\mathbf{\beta} \implies x^\mathbf{\alpha}\vectorbasecanonica{i} \mayorquemodulos x^\mathbf{\beta}\vectorbasecanonica{i}$ \end{itemize} - para todo $\mathbf{\alpha}, \mathbf{\beta}$ y $\mathbf{\gamma}$ en $\naturales^n$ e $i$ y $j$ en $\left\lbrace 1, \dots, r \right\rbrace$. + para todo $\mathbf{\alpha}, \mathbf{\beta}$ y $\mathbf{\gamma}$ en $\naturales^n$ e $i$ y $j$ en $\left\lbrace 1, \dots, r \right\rbrace$. Por lo general, omitiremos el subíndice <<$\mathrm{m\acute{o}d}$>> ya que se sobreentenderá que estamos tratando con módulos. \end{definicion} \begin{ejemplo} - La extensión natural del orden monomial \mayorquelex{} a los monomios de $\Kx^n$ es ordenar primero en función de la posición, haciendo que las primeras posiciones vayan antes que las últimas ---por ejemplo $\vectorbasecanonica{2} > \vectorbasecanonica{3}$---, y si hubiese un empate, entonces se utilizase el orden \mayorquelex. Así definimos el siguiente orden monomial de módulos: + Una manera de extender el orden monomial \mayorquelex{} a los monomios de $\Kx^n$ es ordenar primero en función de la posición, haciendo que las primeras posiciones vayan antes que las últimas ---por ejemplo $\vectorbasecanonica{2} > \vectorbasecanonica{3}$---, y si hubiese un empate, entonces se utilizase el orden \mayorquelex. Así definimos el siguiente orden monomial de módulos: \begin{equation} \label{Definicion Orden Lexicográfico Módulos} x^\mathbf{\alpha}\vectorbasecanonica{i} > x^\mathbf{\beta}\vectorbasecanonica{j} \Longleftrightarrow \begin{cases} i < j \quad \text{o} \\ @@ -570,16 +587,12 @@ urlcolor=black]{hyperref} \end{cases} \end{equation} Sin embargo, podríamos haber tomado la decisión de que primero se comparasen $x^\mathbf{\alpha}$ y $x^\mathbf{\beta}$ y, en caso de empate, se hubiesen mirado las posiciones. Ambas posibilidades dan lugar a órdenes monomiales de módulos distintos: %Poner otra ecuación de definición con el otro orden. - Vemos pues que es necesario aclarar si la posición de los monomios debe ser evaluada antes o después, así que denotamos a estos órdenes como \mayorquemodular{POS}{LEX} para el correspondiente a la definición \eqref{Definicion Orden Lexicográfico Módulos} y \mayorquemodular{LEX}{POS} para aquel que primero compara según el orden lexicográfico y después según la posición. - - - Por ver un caso concreto, $xy\edos \mayorquelexpos y^3\euno$ pero $y^3\euno \mayorqueposlex xy\edos$. + Vemos pues que es necesario aclarar si la posición de los monomios debe ser evaluada antes o después, así que denotamos a estos órdenes como \mayorquemodular{POS}{LEX} para el correspondiente a la definición \eqref{Definicion Orden Lexicográfico Módulos} y \mayorquemodular{LEX}{POS} para aquel que primero compara según el orden lexicográfico y después según la posición. Por ver un caso concreto, $xy\edos \mayorquelexpos y^3\euno$ pero $y^3\euno \mayorqueposlex xy\edos$. Esta generalización que hemos hecho no está limitada a \mayorquelex, también se podría realizar con otros órdenes como \mayorquedeglex. Incluso se podrían permutar las posiciones para obtener todavía más órdenes. \end{ejemplo} - Con el concepto de orden monomial de módulos damos el primer paso en dirección a la respuesta de las preguntas con las que abríamos el capítulo. Ahora, podemos dar algunas respuestas a las preguntas que nos aparecían en los ejemplos \eqref{Ejemplo Ideal Principal Tres Variables} y \eqref{Ejemplo Ideal Tres Variables Varios Generadores}. - + Con el concepto de orden monomial de módulos damos el primer paso en dirección a la respuesta de las preguntas con las que abríamos el capítulo. \begin{definicion} Sea $p := c_1x^\mathbf{s_1}\vectorbasecanonica{i_1} + c_2x^\mathbf{s_2}\vectorbasecanonica{i_2} + \dots + c_mx^\mathbf{s_m}\vectorbasecanonica{i_m}$ un polinomio de $\Kx^r$ y $>$ un orden monomial de módulos. Sea $k \in \left\lbrace 1, \dots, m \right\rbrace$ de forma que $x^\mathbf{s_k}$ sea el máximo del conjunto de monomios de $p$ según el orden $>$, entonces \begin{itemize} @@ -589,9 +602,9 @@ urlcolor=black]{hyperref} \end{itemize} \end{definicion} - Con este aparataje ya tenemos una noción de cuáles son los términos <> dentro de un polinomio, lo cual nos hace empezar a vislumbrar cómo dividir uno por otro. Lo primero es entender cuándo pueden dividirse dos monomios de $\Kx^r$. Según la definición \ref{Definición Divisibilidad Módulos}, para $x^\mathbf{\alpha}\vectorbasecanonica{i}$ y $x^\mathbf{\beta}\vectorbasecanonica{j}$ dos monomios de \Kx{} tenemos que $x^\mathbf{\alpha}\vectorbasecanonica{i}$ divide a $x^\mathbf{\beta}\vectorbasecanonica{j}$ si $i = j$ y $x^\mathbf{\alpha} \vert x^\mathbf{\beta}$. Ahora sí, podemos dar un algoritmo de la división en $\Kx^r$. Este consiste en imitar el algoritmo usual para dividir un polinomio por otro en una variable; pero en este caso, en vez de guiarnos por el grado para encontrar los términos <> del cociente y el dividendo, utilizamos el término líder. + Con este aparataje ya tenemos una noción de cuáles son los términos <> dentro de un polinomio, lo cual nos hace empezar a vislumbrar cómo dividir uno por otro. Lo primero es entender cuándo pueden dividirse dos monomios de $\Kx^r$. Según la definición \ref{Definición Divisibilidad Módulos}, para $x^\mathbf{\alpha}\vectorbasecanonica{i}$ y $x^\mathbf{\beta}\vectorbasecanonica{j}$ dos monomios de \Kx{} tenemos que $x^\mathbf{\alpha}\vectorbasecanonica{i}$ divide a $x^\mathbf{\beta}\vectorbasecanonica{j}$ si $i = j$ y $x^\mathbf{\alpha} \vert x^\mathbf{\beta}$. Ahora sí, podemos dar un algoritmo de la división en $\Kx^r$. Este consiste en imitar el algoritmo usual para dividir un polinomio por otro en una variable; pero en este caso, en vez de guiarnos por el grado para construir el cociente, utilizamos los términos líderes. \begin{definicion} \label{Algoritmo de la División Simple} - Supongamos que tenemos dos polinomios $f$ y $g$ en $\Kx^n$ junto con un orden monomial sobre este. Para dividir $f$ entre $g$ aplicaremos un algoritmo que consiste en replicar el algoritmo que ya conocemos para los polinomios en una variable. Sin embargo, ahora los términos que guían los coeficientes que vamos poniendo son $\LT{f}$ y $\LT{g}$: + Supongamos que tenemos dos polinomios $f$ y $g$ en $\Kx^n$ junto con un orden monomial sobre este. \begin{algorithmic}[1] \State Definimos $D \gets f$, $d \gets g$, $c \gets 0$ y $r \gets 0$ como las variables que representan el dividendo, divisor, cociente y resto, respectivamente. \Repeat @@ -604,7 +617,7 @@ urlcolor=black]{hyperref} \end{algorithmic} Al finalizar este algoritmo, tendremos que $f = c g + r$ donde ninguno de los términos de $r$ es divisible por \LM{g}. \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 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 $\LT{g}$. + 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: \begin{center} @@ -979,6 +992,9 @@ urlcolor=black]{hyperref} Ahora asumamos que existen $\delta_1, \delta_2, \dots, \delta_n$ de forma que $\det\left( \delta_1, \delta_2, \dots, \delta_n \right) = fu$ con $u$ una unidad de \Kx. \todo[inline]{Terminar la otra implicación de Saito.} + \appendix + \chapter{Un ejemplo completo de cálculo} + \nocite{*} \printbibliography \end{document} \ No newline at end of file