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zQjaeiAZXeS3qm<)5pzh-VpoZcgP&tvxiBwLc za^g{-HsY{lcm6|np^)MzpP}%CW%u-GG-mk;wuh`wuc3N3Gt18WTDc<7FS~x!^H2tV zJ3zbC4CsPtmMEHEOWpFTdBQ(Q>&K^&k8&5o5mpK1*;==ia@7GOtf?lyz0;xJPpN?MP?I!cQKYLcjvlO1C@GOOA zDSQc(0*m6OKsihWTGz9np1(_;zZ9MqD$kO6R?D+mp4IZKmS?p*tL0fO&uV!u7Rs|w zo`v!(lxLwl3+0zU9q x^\mathbf{\beta} \implies x^{\mathbf{\gamma}} x^\mathbf{\alpha} > x^{\mathbf{\gamma}}x^\mathbf{\beta} \end{equation*} para todo $\mathbf{\alpha}, \mathbf{\beta}$ y $\mathbf{\gamma}$ en $\naturales^{n}$. - \todo{Decir que el orden es bueno o hablar de órdenes locales y globales y mixtos.} + \end{definicion} + Sin embargo esta propiedad deja en el aire el papel del $1$. ¿Cómo debemos ordenarlo? + \begin{definicion} + Sea $>$ un orden monomial. Si para todo $\mathbf{\alpha} \in \naturales^n$ + \begin{itemize} + \item $x^{\mathbf{\alpha}} > 1$, diremos que el orden es global; + \item $x^{\mathbf{\alpha}} < 1$, diremos que el orden es local; + \item si no son ninguna de las dos anteriores, diremos que el orden es mixto. + \end{itemize} \end{definicion} \begin{ejemplo}[Orden lexicográfico] Una posibilidad de ordenar los monomios es como si estos fueran palabras en el diccionario. Por ejemplo, podemos ver $x^3y^2z^4$ como $xxxyyzzzz$ y en $x^2y^4z^4$ como $xxyyyyzzzz$. Así, $x^3y^2z^4 > x^2y^4z^4$. De manera más formal, @@ -623,17 +636,16 @@ urlcolor=black]{hyperref} Con todo esto, ya podemos definir un algoritmo de la división de un elemento por otro en $\Kx^r$. En esencia, imitaremos al algoritmo tradicional solo que, en este caso, nos guiaremos por los monomios líderes y no por el grado. \begin{definicion} \label{Algoritmo de la División Simple} - \todo{diferenciar vectores y polinomios. Por ejemplo $f$ es un vector pero cociente $c$ es un polinomio.} - Supongamos que tenemos dos polinomios $f$ y $g$ en $\Kx^n$ junto con un orden monomial sobre este. + Supongamos que tenemos dos polinomios $f$ y $g$ en $\Kx^r$ 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. + \State Definimos $D \gets f$, $d \gets g$, $c \gets 0$ y $r \gets \mathbf{0}$ como las variables que representan el dividendo, divisor, cociente y resto, respectivamente. Nótese que $D$, $d$ y $r$ son vectores de $\Kx^r$ mientras que $c$ es un polinomio de \Kx. \Repeat \While{$\LM{d} \vert \LM{D}$} \State Sea $m \in \Kx$ de forma que $\LM{D} = m\LM{d}$. \State $D \gets D - m d$, $c \gets c + m$ \EndWhile \State $D \gets D - \LM{D}$, $r \gets r + \LM{D}$ - \Until{$D = 0$} + \Until{$D = \mathbf{0}$} \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} @@ -958,8 +970,6 @@ urlcolor=black]{hyperref} \end{ejemplo} Estas mismas ideas pueden extenderse a aplicaciones \nolbreaks{$n$-lineales} para construir productos como $\bigotimes^n M = M \otimes M \otimes \dots \otimes M$ y, consecuentemente a aplicaciones \nolbreaks{$n$-lineales} alternadas para construir $\bigwedge^n M := M \wedge M \wedge \dots \wedge M$. - \todo{Añadir la relación con el determinante y decir que si $n$ es mayor que el número de generadores del módulo entonces el producto exterior es cero} - Aunque pueda parecer que los productos exteriores hacen un espacio <>, esto no tiene por qué ser así. \begin{ejemplo} @@ -974,7 +984,7 @@ urlcolor=black]{hyperref} \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} - Sea $V$ un espacio vectorial de dimensión $n$. Entonces, $\dim \bigwedge^k V = \binom{n}{k}$. + Sea $V$ un espacio vectorial de dimensión $n$. Entonces, $\dim \bigwedge^k V = \binom{n}{k}$. En particular, nótese que si $k > n$, entonces $\bigwedge^k V = \conjunto{0}$. \end{lema} Por otro lado, podemos extender los homomorfismos de \modulo{A}s que ya conocemos a productos exteriores de una forma natural. @@ -984,10 +994,19 @@ urlcolor=black]{hyperref} m_1\wedge m_2 \wedge \dots \wedge m_k \longmapsto \phi\!\left( m_1 \right)\wedge \phi\!\left( m_2 \right) \wedge \dots \wedge \phi\!\left( m_k \right). \end{equation*} \end{definicion} - Un fenómeno relevante ocurre cuando tenemos dos espacios vectoriales $V$ y $W$ de la misma dimensión cada uno con una base \conjunto{v_1, v_2, \dots, v_n} y \conjunto{w_1, w_2, \dots, w_n} y una aplicación lineal $\phi : V \rightarrow W$ entre ellos. Escribiendo las coordenadas en función de estas bases, la aplicación lineal $\phi$ tiene asociada una matriz $P$ de forma que las coordenadas de la imagen de $v_i$ a través de $\phi$ son la \nolbreaks{$i$-ésima} columna de $P$. Ahora bien, $\bigwedge^n V$ y $\bigwedge^n W$ son espacios vectoriales de dimensión $1$ con base $v_1\wedge v_2\wedge \dots \wedge v_n$ y $w_1\wedge w_2\wedge \dots \wedge w_n$, respectivamente. Por lo tanto, la aplicación $\nombrefuncionexterior{n}{\phi} : \bigwedge^n V \rightarrow \bigwedge^n W$ simplemente será multiplicar por un escalar. Evaluémosla en $v_1\wedge v_2\wedge \dots \wedge v_n$ para ver exactamente cuál es. Pero antes, démonos cuenta de que en \eqref{Producto Exterior Como Determinante} la forma de simplificar construía el determinate, lo cual es cierto en general: - \begin{equation} \label{Aparición Determinante Función Exterior} - \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} + Un fenómeno relevante ocurre cuando tenemos dos espacios vectoriales $V$ y $W$ de la misma dimensión cada uno con una base \conjunto{v_1, v_2, \dots, v_n} y \conjunto{w_1, w_2, \dots, w_n} y una aplicación lineal $\phi : V \rightarrow W$ entre ellos. Escribiendo las coordenadas en función de estas bases, la aplicación lineal $\phi$ tiene asociada una matriz $P$ de forma que las coordenadas de la imagen de $v_i$ a través de $\phi$ son la \nolbreaks{$i$-ésima} columna de $P$. Ahora bien, $\bigwedge^n V$ y $\bigwedge^n W$ son espacios vectoriales de dimensión $1$ con base $v_1\wedge v_2\wedge \dots \wedge v_n$ y $w_1\wedge w_2\wedge \dots \wedge w_n$, respectivamente. Por lo tanto, la aplicación $\nombrefuncionexterior{n}{\phi} : \bigwedge^n V \rightarrow \bigwedge^n W$ simplemente será multiplicar por un escalar. Qué escalar es este nos lo dice el siguiente lema. + \begin{proposicion} \label{Aparición Determinante Función Exterior} + Sea $\phi : V \rightarrow W$ como en el párrafo anterior. Entonces, la aplicación lineal $\bigwedge^n \phi : \bigwedge^n V \rightarrow \bigwedge^n W$ envía + \begin{equation*} + v_1\wedge v_2\wedge \dots \wedge v_n \mapsto \det\Phi w_1\wedge w_2\wedge \dots \wedge w_n. + \end{equation*} + \end{proposicion} + \begin{proof} + Es suficiente con el razonamiento del párrafo anterior y un desarrollo similar al de \eqref{Producto Exterior Como Determinante}, que como puede verse, es el determinante de una matriz $3\times 3$: + \begin{equation} + \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} \section{Demostración del Criterio de Saito} @@ -1011,9 +1030,29 @@ urlcolor=black]{hyperref} \end{equation*} forma un cuerpo al cual llamaremos cuerpo de fracciones de $A$. El homomorfismo natural $A \hookrightarrow \cuerpofracciones{A}$ es inyectivo. \end{definicion} - \todo{Definir la extensión de escalares como $M \otimes \cuerpofracciones{A}$ y decir cuándo es inyectiva.} + + El cuerpo de fracciones de un anillo nos permite llevar un \modulo{A} a un \espaciovectorial{\cuerpofracciones{A}}. \begin{definicion} - Sea $M$ un \modulo{A} finitamente generado. Diremos que $M$ tiene rango $r$ si $M \otimes \cuerpofracciones{A}$ es un \modulo{\cuerpofracciones{A}} libre de rango $r$. + Llamaremos extensión de escalares del \modulo{A} $M$ al conjunto $M \otimes \cuerpofracciones{A}$. + \end{definicion} + \begin{lema} + 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} + Para la primera afirmación, vemos que $M \otimes \cuerpofracciones{A}$ es un \modulo{\cuerpofracciones{A}} y, como \cuerpofracciones{A} es un cuerpo, tenemos un espacio vectorial. + + Para la segunda, véase que + \begin{equation*} + \begin{array}{rcll} + \ker\varphi = 0 &\iff& m \otimes 1 \neq 0 &\forall m \in M\setminus\conjunto{0} \\ + &\iff & am \otimes \displaystyle\frac{1}{a} \neq 0 &\forall a \in A \setminus \conjunto{0} \\ + &\implies& am \neq 0 & + \end{array} + \end{equation*} + \todo{La otra implicación.} + \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 $r$. \end{definicion} Recordemos que ya en \ref{Lema Rango Producto Exterior} había un resultado sobre el rango del producto exterior de los espacios vectoriales. Generalizamos esto a los módulos. @@ -1021,7 +1060,7 @@ urlcolor=black]{hyperref} 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} \begin{proof} - Consideremos $\left( \bigwedge^s M \right) \otimes \cuerpofracciones{A}$. Hay un isomorfismo de \modulo{\cuerpofracciones{A}}s entre $\bigwedge^s \otimes \cuerpofracciones{A}$ y $\bigwedge^s\left( M \otimes \cuerpofracciones{A} \right)$: + Consideremos $\left( \bigwedge^s M \right) \otimes \cuerpofracciones{A}$. Hay un isomorfismo de \modulo{\cuerpofracciones{A}}s entre $\bigwedge^s M \otimes \cuerpofracciones{A}$ y $\bigwedge^s\left( M \otimes \cuerpofracciones{A} \right)$: \begin{align*} \bigwedge^sM \otimes \cuerpofracciones{A} &\longleftrightarrow \bigwedge^s\left( M \otimes \cuerpofracciones{A} \right) \\ \left( m_1 \wedge m_2 \wedge \dots \wedge m_s \right) \otimes \frac{a}{b} &\longrightarrow \left( m_1 \otimes \frac{a}{b} \right) \wedge \left( m_2 \otimes 1 \right) \wedge \dots \wedge \left( m_s \otimes 1 \right) \\ @@ -1348,7 +1387,7 @@ urlcolor=black]{hyperref} \end{equation*} \section{Con ordenador} - Con el ejemplo anterior queda suficientemente ilustrado que realizar este procedimiento a mano puede ser tedioso y en casos con expresiones más enrevesadas y con más términos puede llegar a ser impracticable. Como ya decíamos antes, se suele utilizar software especializado para realizar estas operaciones. En nuestro caso, utilizamos el programa \cite{SINGULAR} el cual entre sus muchas funciones, tiene algunas dedicadas exclusivamente al cómputo de sicigias y bases de Gröbner. Mostramos aquí cómo realizar el mismo procedimiento que hemos visto antes a través de esta herramienta. + Con el ejemplo anterior queda suficientemente ilustrado que realizar este procedimiento a mano puede ser tedioso y en casos con expresiones más enrevesadas y con más términos puede llegar a ser impracticable. Como ya decíamos antes, se suele utilizar software especializado para realizar estas operaciones. En nuestro caso, utilizamos el programa \cite{SINGULARPROGRAMA} el cual entre sus muchas funciones, tiene algunas dedicadas exclusivamente al cómputo de sicigias y bases de Gröbner. Mostramos aquí cómo realizar el mismo procedimiento que hemos visto antes a través de esta herramienta. Lo primero de todo es definir el anillo sobre el que estamos trabajando: \begin{lstlisting} @@ -1406,11 +1445,37 @@ x2z+y3,3z,-3y2,-2z,0, \chapter{Demostración del Criterio de Buchberger} \label{Apéndice Demostración del Criterio de Buchberger} 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]{Terminar} + La implicación contraria es la que conlleva más trabajo. Lo primero que debemos hacer es introducir un orden monomial nuevo en función del conjunto generador. Como $\Kx^r$ es noetheriano, podemos asumir desde el principio que $G = \conjunto{g_1, g_2, \dots, g_k}$. Definimos, por tanto, el siguiente orden monomial de módulos en $\Kx^k$: + \begin{equation*} + x^\mathbf{\alpha}\vectorbasecanonica{i} >_\mathrm{Schreyer} x^\mathbf{\beta}\vectorbasecanonica{j} \iff \begin{cases} + \LM{x^\mathbf{\alpha}g_i} > \LM{x^\mathbf{\beta}} \quad \text{o} \\ + \LM{x^\mathbf{\alpha}g_i} = \LM{x^\mathbf{\beta}} \quad \text{e} \quad $i < j$. + \end{cases} + \end{equation*} + Ahora, para dos generadores $g_i$ y $g_j$ de forma que $\LM{g_i}$ y $\LM{g_j}$ estén en la misma componente, tomamos $c_jm_{ji}$ y $c_im_{ij}$ elementos de \Kx{} de forma que $c_i, c_j \in K$ y $m_{ji}, m_{ij}$ son monomios de \Kx y cumplen + \begin{equation*} + \spoly{g_i}{g_j} = c_jm_{ji}g_i - c_im_{ij}g_j. + \end{equation*} + Esto último lo podemos expresar en términos de los elementos de $G$ como + \begin{equation*} + c_jm_{ji}g_i - c_im_{ij}g_j = \sum_{\nu = 1}^{k} a_{\nu}^{(ij)} g_{\nu}. + \end{equation*} + Con todo esto, definimos la expresión + \begin{equation*} + s_{ij} := c_jm_{ji}\vectorbasecanonica{i} - c_im_{ij}\vectorbasecanonica{j} - \sum_{\nu = 1}^{k} a_{\nu}^{(ij)} \vectorbasecanonica{\nu} \in \Kx^k. + \end{equation*} + Que puede entenderse como un vector que registra una combinación lineal que anula los generadores, es decir, $s_{ij} \in \syz{g_1, g_2, \dots, g_k}$. Además, con respecto al orden de Schreyer, $\LM{s_{ij}} = m_{ji}\vectorbasecanonica{i}$. Esto se debe por un lado a que $\LM{m_{ij}g_j} = \LM{m_{ji}g_i}$ pero $i < j$; y por otro a que + \begin{equation*} + \LM{a_{\nu}^{(ij)} g_{\nu}} \leq \LM{c_jm_{ji}g_i - c_im_{ij}g_j} < \LM{m_{ji}g_i}. + \end{equation*} - \bibliography{bibliografia} - \bibliographystyle{plain} + Con los preliminares terminados, pasamos a demostrar que si el resto de \spoly{g_i}{g_j} entre $G$ es $0$ para todo $i$ y $j$, entonces tenemos una base de \Groebner. Lo primero es tomar un $f \in \engendrado{G}$ y un $a = \left( a_1, a_2, \dots, a_k \right) \in \Kx^k$ de forma que $f = \sum_{i = 1}^{k} a_i g_i$. Ahora consideremos la expresión + \begin{equation*} + a = h + \sum + \end{equation*} + \todo{Completar la demostración.} + %\bibliography{bibliografia} + %\bibliographystyle{plain} \nocite{*} - %\printbibliography + \printbibliography \end{document} \ No newline at end of file diff --git a/bibliografia.bib b/bibliografia.bib index e72e0e1..17b0b70 100644 --- a/bibliografia.bib +++ b/bibliografia.bib @@ -1,5 +1,5 @@ -@Book{SINGULAR, - shorthand = "Singular", +@Book{SINGULARLIBRO, + shorthand = "Greuel", author="Gert-Martin Greuel and Gerhard Pfister", title="A Singular Introduction to Commutative Algebra", publisher = "Springer", @@ -83,7 +83,7 @@ urldate = {2025-05-28}, year = {2025} } -@misc {DGPS, +@misc {SINGULARPROGRAMA, shorthand = {Singular}, title = {{\sc Singular} {4-4-0} --- {A} computer algebra system for polynomial computations}, author = {Decker, Wolfram and Greuel, Gert-Martin and Pfister, Gerhard and Sch\"onemann, Hans},