From 5e4d31c9662f84bb272b52421df700f7398c157b Mon Sep 17 00:00:00 2001 From: Sixto Date: Sun, 1 Jun 2025 17:30:56 +0200 Subject: [PATCH] Varios. --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 271721 -> 292826 bytes TFG_Sixto_José_Martínez_Gan.tex | 170 ++++++++++++++++++++++++++++++-- bibliografia.bib | 21 ++++ 3 files changed, 184 insertions(+), 7 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index c09d067b668b02bf90457f4d2386c9fed83ab686..adcb821d1baad79545a6fd5a320778a43b17eabe 100644 GIT binary patch delta 109253 zcmZs>Lv)}G)3qC`V>{{Cwr$(C(J}7WM#r{o+qSKaI!VX+pYIIL=pEErHK>}^U|)OJ z#U%3PElNErRPvnyT9X9S1}GR8Yw}MR7cg$trc@Yr*koIIgk(Axz2sXK#HJurINY=( zCOghu5CCxr0 zBy@TjZ+AM28Nwb#xC*l2QosJ?FD#4+STI;rP#z47MsK|Dz7|bv5H>$Io`N}Hnew5x z9I9p3u{6%jHqk)Zfa+x!YIUT;4)QVRh}hli&~x#orL+L2GB$wycT@x>4A$79s>RcZ zKM&#ARms5dB4ZADaa6A1G7$FZXPs;sO)M8Hc9 zc5GeB^dT3k;1Lis$oMNl{*sIj-(@z*(^L&Gu(+n;scNZnoaumXw+oE4a7{kSwJncr 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Vemos pues que \definicionconjunto{\vectorbasecanonica{i} \otimes \vectorbasecanonica{j}}{ i, j \in \conjunto{1, 2, 3}} es una base de $\reales^3 \otimes \reales^3$. Por lo tanto, para encontrar una aplicación lineal $\phi : \reales^3 \otimes \reales^3 \rightarrow \reales$ que coincida con el producto escalar debemos dar las imágenes de la base. En este caso, + \begin{equation*} + \phi\!\left(\vectorbasecanonica{i} \otimes \vectorbasecanonica{j} \right) = \productoescalarantilambda{\vectorbasecanonica{i}}{\vectorbasecanonica{j}} = \begin{cases} + 1 &\text{si $i = j$;}\\ + 0 &\text{si $i \neq j$.} + \end{cases} + \end{equation*} + Si evaluamos esta aplicación lineal $f$ en el producto anterior veremos que, en efecto, nos devuelve el valor que queremos. Por lo tanto, gracias al producto tensorial hemos convertido una aplicación bilineal en una aplicación lineal a costa de tener un dominio algo más complicado. + \end{ejemplo} + + Ahora supongamos que aquello en lo que queremos trabajar son aplicaciones bilineales alternadas, es decir, $b : M \times M \rightarrow N$ donde $b\!\left(m, m'\right) = -b\!\left( m', m \right)$. En esta situación tendremos que imponer que $m \otimes n = - n\otimes m$. + \begin{definicion} + Sea $A$ un anillo y $M$ un \modulo{A}. Consideremos el \submodulo{A} $T$ de $M \otimes M$ definido como + \begin{equation*} + T = \engendrado{m \otimes m' - m' \otimes m \vert m, m' \in M}. + \end{equation*} + Definiremos el producto exterior de $M$ por $M$ como $M \wedge M := \cociente{\left( M \otimes M \right)}{T}$ y denotaremos la clase de equivalencia de cada $\left( m, m' \right)$ como $m \wedge m'$. + \end{definicion} + Nótese que ahora $m \wedge m = 0$. + \begin{ejemplo} + El ejemplo natural en el caso de aplicaciones bilineales alternadas es el producto vectorial $\bullet \times \bullet : \reales^3 \times \reales \rightarrow \reales^3$. Podemos aplicar el mismo proceso que en \eqref{Desglose Producto Tensorial}. + \begin{align*} + \vectortres{a}{b}{c} \wedge \vectortres{d}{e}{f} &= \left( a\euno + b\edos + c\etres \right) \wedge \left( d\euno + e\edos + f\etres \right) \\ + &= ad \euno\wedge\euno + ae\euno\wedge\edos + af\euno\wedge\etres + bd\edos\wedge\euno \\ &\phantom{text} + be\edos\wedge\edos + bf\edos\wedge\etres + cd\etres\wedge\euno + ce\etres\wedge\edos + cf\etres\wedge\etres; \\ + \intertext{suprimimos los elementos de la forma $\vectorbasecanonica{i} \wedge \vectorbasecanonica{i}$ y ponemos todas las parejas $\vectorbasecanonica{i} \wedge \vectorbasecanonica{j}$ de forma que $i < j$;} + &= ae\euno\wedge\edos + af\euno\wedge\etres - bd\euno\wedge\edos + bf\edos\wedge\etres - cd\euno\wedge\etres - ce\edos\wedge\etres \\ + &= \left( ae - bd \right)\euno\wedge\edos + \left( af - cd \right)\euno\wedge\etres + \left( bf - ce \right)\edos\wedge\etres. + \end{align*} + Vemos por lo tanto que $\reales^3 \wedge \reales^3$ tiene como base $\left\lbrace \euno\wedge\edos, \euno\wedge\etres, \edos\wedge\etres \right\rbrace$ y las imágenes de estos son + \begin{equation*} + \phi\!\left( \euno\wedge\edos \right) = \euno \times \edos = \etres, \quad \phi\!\left( \euno\wedge\etres \right) = \euno \times \etres = -\edos \ \text{y} \ \phi\!\left( \edos\wedge\etres \right) = \edos \times \etres = \euno. + \end{equation*} + Vemos de nuevo que la aplicación $\phi$ es lineal y nos devuelve el producto vectorial. + \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 $\bigwedge^n M := M \wedge M \wedge \dots \wedge M$. + + Aunque pueda parecer que los productos exteriores hacen un espacio <>, esto no tiene por qué ser así. + \begin{ejemplo} + Consideremos $\reales^3 \wedge \reales^3 \wedge \reales^3$. Los elementos son de la forma + \begin{align} + \vectortres{a}{b}{c} \wedge \vectortres{d}{e}{f} \wedge \vectortres{g}{h}{i} &= aei\euno\wedge\edos\wedge\etres + afh\euno\wedge\etres\wedge\edos + bdi\edos\wedge\euno\wedge\etres \nonumber \\ + &\phantom{text} + bfg\edos\wedge\etres\wedge\euno + cdh\etres\wedge\euno\wedge\edos + ceg\etres\wedge\edos\wedge\euno; \nonumber \\ + \intertext{donde podemos ver que, por ejemplo, $\edos\wedge\etres\wedge\euno = -\edos\wedge\euno\wedge\etres = \euno\wedge\edos\wedge\etres$ y así expresamos todos estos productos en términos de $\euno\wedge\edos\wedge\etres$;} + &= \left( aei + bfg + cdh - \left( ceg + afh + bdi \right) \right) \euno\wedge\edos\wedge\etres. \label{Producto Exterior Como Determinante} + \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} + Sea $V$ un espacio vectorial de dimensión $n$. Entonces, $\dim \bigwedge^k V = \binom{n}{k}$. + \end{lema} + + Por otro lado, podemos extender los homomorfismos de \modulo{A}s que ya conocemos a productos exteriores de una forma natural. + \begin{definicion} + Sea $A$ un anillo, $M$ y $N$ dos \modulo{A}s y $\phi : M \rightarrow N$ un homomorfismo de \modulo{A}s. Definimos la aplicación $\bigwedge^k \phi: \bigwedge^k M \rightarrow \bigwedge^k N$ como sigue: + \begin{equation*} + 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, $\wedge^n V$ 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*} + \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*} + + \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[\text{Teorema de Quillen-Suslin}]{\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*} + \todo[inline]{¿Lo de que reflexivo implica proyectivo es si es un módulo de con como mucho dos variables o que es de rango 2? ¿Y la referencia dónde está?} + 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 $f \in \anillodepolinomios{K}{x,y}$. + + En un módulo libre, la noción de rango es simplemente la cantidad de elementos que tiene la base. Pero como ya hemos visto a lo largo de este trabajo, hay módulos que no tienen por qué ser libres. Por ende, necesitamos generalizar la noción de rango para poder tratar con estos. + + \begin{definicion} + Sea $A$ un anillo y $\sim$ la relación de equivalencia sobre $A\times \left( A \setminus \left\lbrace 0 \right\rbrace \right)$ dada por $\left( a, b \right) \sim \left( \alpha, \beta \right) \iff a\beta = \alpha b$. A la clase de equivalencia de $\left( a, b \right)$ la denotaremos $\frac{a}{b}$. El conjunto + \begin{equation*} + \cuerpofracciones{A} := \cociente{A\times \left( A \setminus \left\lbrace 0 \right\rbrace \right)}{\sim} + \end{equation*} + junto con las operaciones + \begin{equation*} + \frac{a}{b} + \frac{\alpha}{\beta} := \frac{a\beta + \alpha b}{b\beta} \quad \text{y} \quad \frac{a}{b} \cdot \frac{\alpha}{\beta} := \frac{a\alpha}{b\beta} + \end{equation*} + forma un cuerpo al cual llamaremos cuerpo de fracciones de $A$. + \end{definicion} + \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 $r$. + \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} + \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}& \\ + \text{$M$ de rango 1}& + \end{aligned}\right\rbrace \implies \text{$M$ es un ideal de $\Kx$}& \\ + \text{$M$ es reflexivo} + \end{aligned}\right\rbrace \implies \text{$M$ es un módulo libre} + \end{gather} + \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{*} diff --git a/bibliografia.bib b/bibliografia.bib index e00f279..6400d2f 100644 --- a/bibliografia.bib +++ b/bibliografia.bib @@ -12,6 +12,7 @@ shorthand = "CoCoA1", author= "Martin Kreuzer and Lorenzo Robbiano", title= "Computational Commutative Algebra 1", + edition= {1}, publisher= "Springer Berlin, Heidelberg", year= "2000", isbn= "978-3-540-67733-8", @@ -28,6 +29,26 @@ doi = {https://doi.org/10.1007/978-3-319-16721-3}, series = {Undergraduate Texts in Mathematics} } +@Book{ALGEBRA, + shorthand = {Lang}, + author = {Serge Lang}, + title = {Algebra}, + edition = {3}, + publisher = {Springer New York}, + year = {2002}, + isbn = {978-0-387-95385-4}, + doi = {https://doi.org/10.1007/978-1-4613-0041-0}, + series = {Graduate Texts in Mathematics}} +@Book{LIBROPROYECTIVOS, + shorthand = {Lectures}, + author = {T. Y. Lam}, + title = {Lectures on Modules and Rings}, + edition = {1}, + publisher = {Springer New York}, + year = {1998}, + isbn = {978-0-387-98428-5}, + doi = {https://doi.org/10.1007/978-1-4612-0525-8}, + series = {Graduate Texts in Mathematics}} % El lugar donde se puede encontrar publicado el artículo de Saito es aquí: https://repository.dl.itc.u-tokyo.ac.jp/records/39646. @article{SAITO, shorthand = {Saito},