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zV|m{EAyL5dJReh$@B^YGS2%hT%CF-6cMtdqPaYMc*2$pJt(Y(_VRToWngA&{FRM8f z?OdaKz<;m3OEz32T696`%C+mYaLQ zG=&O&40YE*Xa&%jkKnUtLvFygh(MUG0ReCdKw0m)0FCl%zqDz)-#~R0(@F@~E0=B2 z+!4cwR($ZZ zT05-x*j;MvwBozx)EcnjN0-#vWu?ZCXJNOM2HMmbw9=?kYmb#~dM~oltzQq^wkzf zmOg6q)u9(IGm|qjIWv=!lezZ%?A-Dtz&kHz-N;*D88E*&A9gt(cKIq`-g2faXUcMY zCFrw&Da)C%oGHtht(@7)nXR1J%9*WPyCr5TDQC9wEufcNH}t?((02u!qFi5D+o2zJ nz)l#j(vMGv+rK&*{U5KPMfR7T=Lj7PH#IXdHVP#rMNdWwn~id_ diff --git a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index d939365..2a8f3d4 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -648,7 +648,26 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. Sea $P$ un \modulo{A} libre con base $\left\lbrace p_1, p_2, \dots, p_k \right\rbrace$ y tomemos el resto de premisas como en la Definición~\ref{Definición Módulo Proyectivo}. Al ser $P$ un módulo libre, cualquier homomorfismo de \modulo{A}s $\phi: P \rightarrow N$ está unívocamente determinado por las imágenes de la base de $P$. Sean $m_1, m_2, \dots, m_k$ elementos de $M$ de forma que $e\!\left( m_i \right) = h\!\left( p_i \right)$, cuya existencia está garantizada por ser $e$ un epimorfismo. Definiendo $\phi\!\left(p_i \right) := m_i$ tenemos que $e\!\left( \phi\!\left( p_i \right) \right) = h\!\left( p_i \right)$ por lo que $e \circ \phi = h$. \end{proof} Una introducción a los módulos proyectivos puede encontrarse en \cite[sección 1.2]{LIBROPROYECTIVOS}. - + + Una construcción que en este trabajo no trataremos en profundidad pero que sí es relevante para estudiar la libertad de los módulos es la resolución libre, que nos permite expresar por completo un módulo como submódulo de un módulo libre y las relaciones entre sus elementos. + \begin{definicion} + Una resolución libre de $M$ es una sucesión exacta de la forma + \begin{equation*} + \dots \longrightarrow F_{k+1} \longrightarrow F_k \longrightarrow \dots \longrightarrow F_1 \longrightarrow F_0 \longrightarrow M \longrightarrow 0 + \end{equation*} + donde los $F_i$ son \modulo{A}s libres finitamente generados. + \end{definicion} + \begin{lema} \label{Lema Longitud Mínima Resolución} + Todo \modulo{A} ---recuérdese que $A$ es noetheriano--- finitamente generado que admita una resolución de longitud finita + \begin{equation*} + 0 \longrightarrow F_k \longrightarrow \dots \longrightarrow F_1 \longrightarrow F_0 \longrightarrow M \longrightarrow 0 + \end{equation*}tiene una resolución de longitud mínima. + \end{lema} + Dándonos cuenta de que si $M$ es libre, entonces debe admitir una resolución de la forma + \begin{equation*} + 0 \longrightarrow F_0 \longrightarrow M \longrightarrow 0, + \end{equation*} + encontramos en este resultado un criterio de libertad más práctico sobre todo a la hora de demostrar que un módulo no es libre: intentar calcular su resolución mínima y, si tiene, ver si es de la forma anterior. Sin embargo, explicar el cómputo de estas resoluciones mínimas, si bien está al alcance de la teoría presentada, aumentaría significativamente la longitud de este texto, por lo que nos limitamos a indicar hacia un texto que sí las trata, \cite{SINGULARLIBRO}, y a ilustrar su uso en la sección \ref{Apéndice a ordenador}. \section{Producto Exterior} De cara a la demostración del Criterio de Saito, utilizaremos una operación que nos será de gran ayuda: el producto exterior. Para comprenderlo, necesitaremos entender primero el producto tensorial. Este aparece en el siguiente contexto: supongamos que $M$, $N$ y $P$ son \modulo{A}s y tenemos una aplicación $b : M \times N \rightarrow O$ que es bilineal; nos gustaría encontrar un \modulo{A} $E$ a partir de $M \times N$ en el cual exista un homomorfismo de \modulo{A}s $b' : E \rightarrow O$ que se comporte como $b$. En resumen, queremos convertir una aplicación bilineal en una lineal. Esta es la motivación de la construcción del producto tensorial. @@ -1501,8 +1520,18 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \begin{equation*} \derivaciones{x^2z + y^3} = \engendrado{\vectorvertical{3x \\ 2y \\ 0}, \vectorvertical{3y^2 \\ -2xz \\ 0 }, \vectorvertical{0 \\ y \\ 3z}, \vectorvertical{0 \\ x^2 \\ -3y^2}, \vectorvertical{0 \\ 0 \\ x^2z + y^3}}. \end{equation*} - - \section{Con ordenador} + + Debido a que no hemos más que mencionado las resoluciones, no ilustraremos su cálculo aquí. De todas formas, como idea intuitiva, obsérvese que para $M := \engendrado{f_1, f_2, \dots, f_k}$, + \begin{equation*} + 0 \longrightarrow \syz{f_1, f_2, \dots, f_k} \hookrightarrow \Kx^k \xrightarrow{\vectorbasecanonica{i} \mapsto f_i} M \longrightarrow 0. + \end{equation*} + Pero, por supuesto, \syz{f_1, f_2, \dots, f_k} no tiene por qué ser libre. Aun así, sí que sabemos que es finitamente generado así que $\syz{f_1, f_2, \dots, f_k} = \engendrado{s_1, s_2, \dots, s_l}$. Pero esto quiere decir que + \begin{equation*} + 0 \longrightarrow \syz{s_1, s_2, \dots, s_l}\hookrightarrow \Kx^l \xrightarrow{\vectorbasecanonica{i} \mapsto s_i} \Kx^k \xrightarrow{\vectorbasecanonica{i} \mapsto f_i} M \longrightarrow 0. + \end{equation*} + Siguiendo de esta forma calcularíamos una resolución, que no tiene por qué ser la mínima. Ilustramos el cálculo por ordenador de una resolución mínima para estos valores en la siguiente sección. + + \section{Con ordenador} \label{Apéndice a 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{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: @@ -1557,6 +1586,25 @@ print(GS);\end{lstlisting} x2z+y3,3z,-3y2,-2z,0, -x2, -3,0, 0, 0 \end{lstlisting} + + Por último, veamos que no es libre calculando una resolución mínima con la instrucción \texttt{mres}: + \begin{lstlisting} +module Siz = [0, 0, x2z+y3], [0, y, 3z], [0,x2,-3y2], [x, 0, -2z], [3y2,-2xz,0]; +mres(Siz,0); + \end{lstlisting} + \newpage + Esto nos devuelve + \begin{lstlisting} + 3 4 1 +R <-- R <-- R + +0 1 2 + \end{lstlisting} + Así pues, una resolución mínima es de la forma + \begin{equation*} + 0 \longrightarrow \Kx^1 \longrightarrow \Kx^4 \longrightarrow \Kx^3 \longrightarrow \derivaciones{x^2z + y^3} \longrightarrow 0, + \end{equation*} + y por el Lema \ref{Lema Longitud Mínima Resolución}, tenemos que \derivaciones{f} no puede ser libre. \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 $0$.