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a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index 9df446b..e8b4f6f 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -1267,7 +1267,6 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \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{El anillo $\Kx$ es un dominio de integridad por lo que podemos simplificar,} p_1'a_1 + p_2a_2' + \dots + p_k'a_k &= 1. @@ -1281,19 +1280,16 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{teorema} En este capítulo explicaremos la demostración de este resultado que aparecen en \cite{VALLES}. Pero primero, deberemos ilustrar algunas construcciones que se utilizan en dicha demostración. - - \section{Demostración del Criterio de Saito} - - A lo largo de esta demostración, utilizaremos fuertemente el hecho de que \derivaciones{f} es un \modulo{\Kx} reflexivo. Esto se debe a que podemos encontrar ciertas relaciones entre los módulos reflexivos y los módulos libres. Por un lado, tenemos la siguiente relación: - \begin{gather*} - \text{$M$ módulo libre} \xRightarrow{\text{\referenciamodomatematico{lema}{Lema Libre implica Proyectivo}}} \text{$M$ módulo proyectivo} \xRightarrow{\phantom{fd}} \text{$M$ módulo reflexivo} \\ - \text{$M$ módulo libre} \ \xLeftarrow[\substack{\text{Teorema de Quillen-Suslin} \\ \text{\cite{ALGEBRA} }}]{\text{si $M$ es un \modulo{\Kx}}} \text{$M$ módulo proyectivo} \ \xLeftarrow{\text{si $M$ es un \modulo{\anillodepolinomios{K}{x, y}}}} \text{$M$ módulo reflexivo} - \end{gather*} - Gracias a estas implicaciones podemos asegurar que todo módulo libre es reflexivo. Sin embargo, solo podemos asegurar que los módulos reflexivos sean libres cuando son \modulo{\anillodepolinomios{K}{x, y}}, es decir, cuando admitan la estructura de módulo asociada a un anillo de polinomios en dos variables. Esto, por lo tanto, nos garantiza que \derivaciones{f} es siempre libre cuando $f \in \anillodepolinomios{K}{x,y}$. Más que solucionar nuestro problema, estos resultados nos sirven para ilustrar la gran dificultad que supone garantizar que un módulo reflexivo sea libre. + % A lo largo de esta demostración, utilizaremos fuertemente el hecho de que \derivaciones{f} es un \modulo{\Kx} reflexivo. Esto se debe a que podemos encontrar ciertas relaciones entre los módulos reflexivos y los módulos libres. Por un lado, tenemos la siguiente relación: + % \begin{gather*} + % \text{$M$ módulo libre} \xRightarrow{\text{\referenciamodomatematico{lema}{Lema Libre implica Proyectivo}}} \text{$M$ módulo proyectivo} \xRightarrow{\phantom{fd}} \text{$M$ módulo reflexivo} \\ + % \text{$M$ módulo libre} \ \xLeftarrow[\substack{\text{Teorema de Quillen-Suslin} \\ \text{\cite{ALGEBRA} }}]{\text{si $M$ es un \modulo{\Kx}}} \text{$M$ módulo proyectivo} \ \xLeftarrow{\text{si $M$ es un \modulo{\anillodepolinomios{K}{x, y}}}} \text{$M$ módulo reflexivo} + % \end{gather*} + % Gracias a estas implicaciones podemos asegurar que todo módulo libre es reflexivo. Sin embargo, solo podemos asegurar que los módulos reflexivos sean libres cuando son \modulo{\anillodepolinomios{K}{x, y}}, es decir, cuando admitan la estructura de módulo asociada a un anillo de polinomios en dos variables. Esto, por lo tanto, nos garantiza que \derivaciones{f} es siempre libre cuando $f \in \anillodepolinomios{K}{x,y}$. Más que solucionar nuestro problema, estos resultados nos sirven para ilustrar la gran dificultad que supone garantizar que un módulo reflexivo sea libre. - \subsection{Comienzo de la demostración} - 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. + \section{Comienzo de la demostración} + Puede encontrarse en \cite[corolario 1.7]{SAITO} una demostración de que \derivaciones{f} es un módulo reflexivo de rango $n$. Siguiendo el esquema siguiente, 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. Esta cadena de implicaciones que recogemos aquí nos será muy útil a lo largo de esta demostración: \begin{gather} \label{Esquema Demostración} @@ -1313,7 +1309,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \vspace{-0.15cm}\end{equation*} Recordemos lo que ocurría en \ref{Aparición Determinante Función Exterior}. De forma similar, aquí tendremos que $\bigwedge^n \Phi$ se corresponde con multiplicar por el escalar $p$ ---o $up$ donde $u$ es una unidad de \Kx---. Sin embargo, el hecho de que no tengamos base de \derivaciones{f} impide que se pueda ver $\Phi$ como una matriz. Aun así, nos permitimos el abuso de notación de llamar $\det\Phi := p$, pero no es más que una convención, no se trata de un determinante de una matriz como ocurre en el caso de las aplicaciones entre espacios vectoriales. Tenemos pues que $\bigwedge^n\derivaciones{f} = \engendrado{\det\Phi \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n}$. - \subsection{\derivaciones{f} libre implica $\det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$} + \section{\derivaciones{f} libre implica $\det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$} Ahora, por un lado, definamos $N := \engendrado{\delta_1, \delta_2, \dots, \delta_n}$ como un \submodulo{\Kx} de \derivaciones{f} donde los $\delta_i$ forman una base de \derivaciones{f} y consideremos la inclusión $N \xrightarrow{\iota} \derivaciones{f}$. Para aclararnos algo más, escribimos el siguiente esquema de los homomorfismos ---¡ojo!, esto no es una sucesión exacta---: \begin{equation}\label{Secuencia Demostración Saito} N = \engendrado{\delta_1, \delta_2, \dots, \delta_n} \xrightarrow{\phantom{s}\iota\phantom{s}} \derivaciones{f} \xrightarrow{\Phi\phantom{s}} \derivaciones{\Kx}. @@ -1332,9 +1328,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{equation*} Vemos\footnote{De nuevo, recordemos que $\det\Phi$ no es el determinante de una matriz, sino una convención que hemos explicado antes. Por otro lado, $\det\left( \Phi \circ \iota \right)$ sí que es el determinante de una matriz ya que tanto $N$ como \derivaciones{\Kx} son módulos libres.} por ella que $\det\left( \Phi \circ \iota \right) = \det\Phi \det\iota = \det A$. Como $\iota$ es un monomorfismo, puede representarse como una matriz que tiene inversa en $\mathrm{Mat}\!\left( n \times n, \Kx\right)$ y por la regla de Cramer\footnote{Por si no la recordamos, esta dice que toda matriz $C$ inversible cumple que $C^{-1} = \frac{1}{\det C} \mathrm{adj}\!\left( C \right)$, con $\mathrm{adj}\!\left( C \right)$ la matriz de adjuntos.} para la matriz inversa, $\det i$ debe ser unidad de \Kx. - \subsubsection{El valor del determinante} - - Por un lado, fijamos $f\partial_i \in \derivaciones{f}$; y por otro, definimos $\dertheta{i}{j} := \diffp{f}{x_i}\partial_j - \diffp{f}{x_j}\partial_i \in \derivaciones{f}$. Así, + Nos dedicaremos ahora a obtener el valor del determinante. Por un lado, fijamos $f\partial_i \in \derivaciones{f}$; y por otro, definimos $\dertheta{i}{j} := \diffp{f}{x_i}\partial_j - \diffp{f}{x_j}\partial_i \in \derivaciones{f}$. Así, \begin{equation*} \textstyle \dertheta{i}{1} \wedge \dertheta{i}{2} \wedge \dots \wedge \dertheta{i}{i-1} \wedge f\partial_i \wedge \dertheta{i}{i+1} \wedge \dots \wedge \dertheta{i}{n-1} \wedge \dertheta{i}{n} \in \bigwedge^n \derivaciones{f}. \end{equation*} @@ -1342,7 +1336,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \begin{equation*} \textstyle f\left( \diffp{f}{x_i} \right)^{n-1} \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n \in \bigwedge^n \derivaciones{f} = \engendrado{\det\Phi \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n}. \end{equation*} - Esto quiere decir que $\det\Phi \vert f\left( \diffp{f}{x_i} \right)^{n-1}$ para todo $i$. Asimismo, el máximo común divisor de todos ellos también será divisible por $\det\Phi$: + Esto quiere decir que $\det\Phi \vert f\left( \diffp{f}{x_i} \right)^{n-1}$ para todo $i$. Esto también ocurre con su máximo común divisor: \begin{equation*} \textstyle \det\Phi \left\lvert f\mcd{f^{n-1}, \left( \diffp{f}{x_1} \right)^{n-1}, \left(\diffp{f}{x_2} \right)^{n-1}, \dots, \left( \diffp{f}{x_n} \right)^{n-1}}\right.. \end{equation*} @@ -1361,7 +1355,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \vspace{-0.3cm} \begin{proof} Sea $\delta \in \derivaciones{f}$ y llamemos $f_{-i} := f_1 f_2 \cdots f_{i-1} f_{i+1} \cdots f_{r-1} f_r$. Por la regla del producto tenemos que - \begin{align*} + \vspace{-0.4cm}\begin{align*} \delta\!\left( f \right) &= \delta\!\left( f_1 f_2 \cdots f_r \right) \\ pf_1 f_2 \cdots f_r &= \delta\!\left( f_i \right) f_{-i} + f_i \delta\!\left( f_{-i} \right) \\ pf_1 f_2 \cdots f_r - f_i \delta\!\left( f_{-i} \right) &= \delta\!\left( f_i \right) f_{-i} \\ @@ -1386,7 +1380,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = \det A = \det \left( \Phi \circ \iota \right) = \det\Phi \det\iota = u'f \det\iota = uf, \quad \text{con $u$ una unidad de \Kx.} \end{equation*} - \subsection{$\det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$ implica que \derivaciones{f} es libre} + \section{$\det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$ implica que \derivaciones{f} es libre} 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*}