From 64589681fcb89b87436a836321c7a25efe8efaa4 Mon Sep 17 00:00:00 2001 From: Sixto Date: Mon, 2 Jun 2025 14:16:02 +0200 Subject: [PATCH] =?UTF-8?q?A=C3=B1adida=20la=20primera=20de=20las=20tres?= =?UTF-8?q?=20partes=20de=20la=20demostraci=C3=B3n=20que=20comentamos=20el?= =?UTF-8?q?=20profesor=20Artal=20y=20yo:=20Der(f)=20=3D=20.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 292826 -> 299370 bytes TFG_Sixto_José_Martínez_Gan.tex | 51 ++++++++++++++++++++++++++------ 2 files changed, 42 insertions(+), 9 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index adcb821d1baad79545a6fd5a320778a43b17eabe..f8d068a5dae7af1dd58ce095398866d582ddcdc8 100644 GIT binary patch delta 41165 zcmV(_K-9n5?-A;v60iqP0Wp&i0x5s(S50r5KoGtASL_iV*v@`2%QdO%rc$M36tnxa)S?t#jSJ#I5tM2Y>NyTU2L#igEmW7{6eK1w~wDIG270argoa z;M^B^l>Djmx)#zq;N_q<5o|HmsMUYJ5}l!Ja3#;kM8`sX4u 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LISM5uMNdWw84W~U diff --git a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index 1d9c1de..d55ad09 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -390,9 +390,9 @@ 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{\iota} \derivaciones{\Kx} \xrightarrow{\kappa} \Kx^n \xrightarrow{\Phi} J \xrightarrow{\pi} \frac{J}{J \cap \engendrado{f}} \rightarrow 0. + 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. \end{equation} - La función $\iota$ es una inclusión, la función $\kappa$ es el homomorfismo que envía $\partial_i$ a \vectorbasecanonica{i}, $\Phi$ 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}, $\iota$ 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} @@ -816,8 +816,8 @@ urlcolor=black]{hyperref} \end{align} Vemos, por lo tanto, que $\reales^3\wedge\reales^3\wedge\reales^3$ es en realidad $\reales\left( \euno\wedge\edos\wedge\etres \right)$, un espacio isomorfo a \reales. \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. - \begin{lema} + 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, $\dim \bigwedge^k V = \binom{n}{k}$. \end{lema} @@ -829,9 +829,9 @@ urlcolor=black]{hyperref} \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, $\wedge^n V$ y $\wedge^n W$ 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; por ende, - \begin{equation*} + \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*} + \end{equation} \section{Demostración del Criterio de Saito} @@ -861,7 +861,7 @@ urlcolor=black]{hyperref} \end{definicion} En general, la cadena de implicaciones que nos serán útiles a lo largo de la demostración del Criterio de Saito serán las siguientes: - \begin{gather} + \begin{gather} \label{Esquema Demostración} \left.\begin{aligned} \left.\begin{aligned} \text{$M$ módulo reflexivo} \xRightarrow{\text{\cite[\href{https://stacks.math.columbia.edu/tag/0AV0}{Tag 0AV0}]{STACKS}}} \text{$M$ libre de torsión}& \\ @@ -871,9 +871,42 @@ urlcolor=black]{hyperref} \end{aligned}\right\rbrace \implies \text{$M$ es un módulo libre} \end{gather} + Lo primero que haremos es ver que \derivaciones{f} es siempre un módulo reflexivo de rango $n$ tal y como demuestra en \cite{SAITO}. + \todo[inline]{En el artículo de Vallés pone que tiene que ser \textit{square-free}.} + \begin{lema} + \derivaciones{f} es un \modulo{\Kx} reflexivo. + \end{lema} + \begin{proof} + \todo[inline]{No entiendo la demostración de Saito.} + \end{proof} + + Si vemos 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} + \begin{proof} + Recordemos que ya en \ref{Lema Rango Producto Exterior} este resultado aparecía tras observar lo que ocurría con los elementos de las bases. Ahora consideremos $\left( \bigwedge^s M \right) \otimes \cuerpofracciones{A}$. Hay un isomorfismo de \modulo{\cuerpofracciones{A}}s entre $\bigwedge^s \otimes \cuerpofracciones{A}$ y $\bigwedge^s\left( M \otimes \cuerpofracciones{A} \right)$: + \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) \\ + \left( m_1 \wedge m_2 \wedge \dots \wedge m_s\right)\otimes \frac{a_1a_2\dots a_s}{b_1 b_2 \dots b_s} &\longleftarrow \left( m_1 \otimes \frac{a_1}{b_1} \right) \wedge \left( m_2 \otimes \frac{a_2}{b_2} \right) \wedge \dots \wedge \left( m_s \otimes \frac{a_s}{b_s} \right) + \end{align*} + Aunque pudiera parecer que no es un isomorfismo puesto que los lados derechos no lucen iguales, recordemos que por las propiedades del producto tensorial y exterior y por ser ambos conjuntos \modulo{\cuerpofracciones{A}} tenemos que + \begin{align*} + \left( m_1 \otimes \frac{a_1}{b_1} \right) \wedge \left( m_2 \otimes \frac{a_2}{b_2} \right) \wedge \dots \wedge \left( m_s \otimes \frac{a_s}{b_s} \right) &= \left[\frac{a_1}{b_1} \left( m_1 \otimes 1 \right) \right] \wedge \left[\frac{a_2}{b_2} \left( m_2 \otimes 1 \right) \right] \wedge \dots \wedge \left[\frac{a_s}{b_s} \left( m_s \otimes 1 \right) \right] \\ + &= \frac{a_1a_2\dots a_s}{b_1 b_2 \dots b_s} \left( m_1 \otimes 1 \right) \wedge \left( m_2 \otimes 1 \right) \wedge \dots \wedge \left( m_s \otimes 1 \right) \\ + &= \left( m_1 \otimes \frac{a_1a_2\dots a_s}{b_1 b_2 \dots b_s} \right) \wedge \left( m_2 \otimes 1 \right) \wedge \dots \wedge \left( m_s \otimes 1 \right). + \end{align*} + Ahora bien, por hipótesis $M \otimes \cuerpofracciones{A}$ es un \espaciovectorial{\cuerpofracciones{A}} de rango $n$ así que $\bigwedge^n \left( M \otimes \cuerpofracciones{A} \right)$ será también un \espaciovectorial{\cuerpofracciones{A}}, en este caso de rango $\binom{n}{s}$. Gracias al isomorfismo, $\left( \bigwedge^s M \right) \otimes \cuerpofracciones{A}$ es igualmente otro \espaciovectorial{\cuerpofracciones{A}} de rango $\binom{n}{s}$ lo cual quiere decir que $M$ es de rango $\binom{n}{s}$. + \end{proof} + + Este lema nos garantiza que $\bigwedge^n \derivaciones{f}$ es un \modulo{\Kx} reflexivo de rango $1$. Por el esquema en \ref{Esquema Demostración}, tenemos que $\bigwedge^n \derivaciones{f}$ es también libre de torsión y, por lo tanto, $\bigwedge^n \derivaciones{f}$ es un ideal de \Kx. Por último, aplicando la última de las implicaciones, tenemos que $\bigwedge^n \derivaciones{f}$ es un \modulo{\Kx} libre y además de dimensión $1$. Sumado a esto tenemos que $\bigwedge^n \derivaciones{f}$ es un \submodulo{\Kx} de $\bigwedge^n \derivaciones{\Kx} = \engendrado{\partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n}$, por lo tanto, $\derivaciones{f} = \engendrado{p \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n}$ para algún $p$. Nótese que este $p$ está bien definido salvo por alguna unidad de \Kx, es decir, tanto nos daría poner $p$ que $kp$ siempre y cuando exista $k^{-1} \in \Kx$. + + Ahora bien, recordemos la sucesión exacta de \ref{Secuencia Exacta Definición Módulo Derivaciones}. Podemos aplicar productos exteriores a los elementos, pero esto hace que la sucesión pueda dejar de ser exacta. Sin embargo, habrá un homomorfismo $\bigwedge^n \derivaciones{f} \xrightarrow{\bigwedge^n \Phi} \bigwedge^n \derivaciones{\Kx}$. Recordemos lo que ocurría en \ref{Aparición Determinante Función Exterior}. De forma similar, \todo[inline]{Esto de antes lo teníamos ya que el módulo era libre. Habría que justificar que para módulos finitamente generados de rango $n$ también ocurre.} aquí tendremos que $\bigwedge^n \Phi$ se corresponde con multiplicar por el determinante de $\Phi$. Como además $\Phi$ jugaba el papel de inclusión, llegamos a que es este determinante el valor de $p$, es decir, $\bigwedge^n\derivaciones{f} = \engendrado{\det\Phi \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n}$. + + \todo[inline]{Aquí vendría la parte en la que demostramos que $\det \Phi = f$.} + - \todo[inline]{Demostrar que Der R es reflexivo. Propiedades de los modulos reflexivos (libre implica proyectivo implica reflexivo y los conversos condicionados), reflexivo implica libre de torsión, etc como esquema.} - \nocite{*} \printbibliography \end{document} \ No newline at end of file