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z+N_bsv%j-U{zRLKGW@L00>VgMM7^3pok^5!~Jx=g2D7g3`!<3s2>D2G&{S8aZt8kqOa zv|c!c*p(CiNJ}b^Vz4eDtL8~B9Uk&}TAd_|BJg~t6ry9kUo#!JPynNpm$^|A?Mw0%-2l4ca}PLWnh@5QC%p@a7SNqxVoI5_6N9<<=~Hp8BgtrRWPNmGF*$xn6o95q=_XNx@2|7k`oh^JW2Y*KONu{T&!+qHJUAnn$j!O~{J*B(LqkNtv z|7KApTzb3jiyVbPuG+MTu3pm6M8uaM@cd!wz3zA>^Tc*@&Bw`=%Jd0qT|Bt@O qCNs<<9QF0lc#rd`Y~>x+qm|h$Edd^u1d)%8haG{M`iqJ*!v6t`eV3O2 diff --git a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index 7320c8a..f03c167 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -920,7 +920,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 Antes de poder enunciar el criterio de Saito, necesitamos una definición más. \begin{definicion} - Un polinomio $p \in \Kx$ se dice libre de cuadrados si, en caso de poder expresarlo de la forma $p = a^2b$, se tiene que $a$ es una unidad de \Kx. + Un polinomio $p \in \Kx$ se dice \emph{libre de cuadrados} si, en caso de poder expresarlo de la forma $p = a^2b$, se tiene que $a$ es una unidad de \Kx. \end{definicion} \begin{lema} \label{Lema MCD Parciales} Un polinomio $p \in \Kx$ es libre de cuadrados si y solo si $\mcd{p, \diffp{p}{x_1}, \diffp{p}{x_2}, \dots, \diffp{p}{x_n}}$ es una unidad de \Kx. @@ -933,6 +933,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 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{Producto Exterior} + \todo{Si haces lo de la bilineal lo puedes usar aquí} De cara a la demostración del Criterio de Saito, utilizaremos una operación que nos será de gran ayuda: el producto exterior. Para entender esta, necesitaremos entender primero el producto tensorial. En esencia, el producto tensorial aparece en el siguiente contexto: supongamos que $M$, $N$ y $O$ son \modulo{A}s y tenemos una aplicación $b : M \times N \rightarrow O$ que es bilineal, o sea, que $b\left( \bullet, n \right)$ y $b\left( m, \bullet \right)$ son homorfismos de \modulo{A}s para cualquier $m \in M$ y $n \in N$; 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 que para cada $\left( m, n \right)$ en $M \times N$ exista un elemento $m * n$ en $E$ de forma que $b'\left( m*n \right) = b\left( m, n \right)$. Esta es la motivación de la construcción del producto tensorial. \begin{definicion} Sea $A$ un anillo y $M$ y $N$ dos \modulo{A}s. Tomemos como $E$ al \modulo{A} definido como $\bigoplus_{\substack{m \in M \\ n \in N}} A \cdot \vectorbasecanonica{m, n}$ donde \vectorbasecanonica{m, n} debe entenderse como la tupla $\left( m, n \right)$ sin ningún tipo de operación asociada, es decir, no debemos pensar que $\vectorbasecanonica{m, n} + \vectorbasecanonica{m', n'} = \vectorbasecanonica{m+m', n+n'}$ al igual que no pensamos que $\euno + \edos = \etres$. Ahora, consideremos dentro de $E$ el \submodulo{A} $S$ definido por @@ -944,7 +945,8 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \vectorbasecanonica{m, an} - a\vectorbasecanonica{m, n} \end{gathered} \right\rvert a \in A, m \in M, n \in N}. \end{equation*} - Definimos el producto tensorial de $M$ por $N$ como $M \otimes N := \cociente{E}{S}$ y denotaremos a la clase de equivalencia de cada $\vectorbasecanonica{m, n}$ como $m \otimes n$. + Definimos el p\emph{roducto tensorial} de $M$ por $N$ como $M \otimes N := \cociente{E}{S}$ y denotaremos a la clase de equivalencia de cada $\vectorbasecanonica{m, n}$ como $m \otimes n$ + \todo{Propiedad universal} \end{definicion} \begin{ejemplo} Quizás el ejemplo más natural sea el producto escalar. Así pues, tomemos la aplicación $\productoescalarantilambda{\bullet}{\bullet} : \reales^3 \times \reales^3 \rightarrow \reales$, que es bilineal. En este caso, todo elemento de $\reales^3 \otimes \reales^3$ es de la siguiente forma ---entendiendo $S$ como en la definición de antes---: @@ -972,7 +974,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \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 $m \wedge m'$. + Definiremos el \emph{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 $m \wedge m'$.\todo{Otra propiedad universal} \end{definicion} Nótese que ahora $m \wedge m = 0$. \begin{ejemplo} @@ -1011,7 +1013,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 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: + Sea $A$ un anillo, $M$ y $N$ dos \modulo{A}s y $\phi : M \rightarrow N$ un homomorfismo de \modulo{A}s. Definimos la \emph{aplicación exterior }$\bigwedge^k \phi: \bigwedge^k M \rightarrow \bigwedge^k N$ 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*} From a30299b5f14ddaa0e4bf2063fb130f4414f127ca Mon Sep 17 00:00:00 2001 From: Sixto Date: Tue, 10 Jun 2025 18:28:29 +0200 Subject: [PATCH 2/9] =?UTF-8?q?Cambio=20de=20lugar=20de=20la=20secci=C3=B3?= =?UTF-8?q?n=20del=20Producto=20Exterior.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 461242 -> 463951 bytes TFG_Sixto_José_Martínez_Gan.tex | 212 ++++++++++++++++---------------- 2 files changed, 106 insertions(+), 106 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index d3611f09029d189f79072762e71adee45f59c38d..302bb8cbc038180b2234ae4094e552ff7169e516 100644 GIT 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a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -50,8 +50,8 @@ \usepackage{enumitem} % Bibliografía. -%\usepackage[backend=biber]{biblatex} -%\addbibresource{bibliografia.bib} +\usepackage[backend=biber]{biblatex} +\addbibresource{bibliografia.bib} \usepackage{algpseudocodex} \algrenewcommand\algorithmicwhile{\textbf{mientras}} @@ -542,7 +542,106 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du 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}. - + + \section{Producto Exterior} + \todo{Si haces lo de la bilineal lo puedes usar aquí} + De cara a la demostración del Criterio de Saito, utilizaremos una operación que nos será de gran ayuda: el producto exterior. Para entender esta, necesitaremos entender primero el producto tensorial. En esencia, el producto tensorial aparece en el siguiente contexto: supongamos que $M$, $N$ y $O$ son \modulo{A}s y tenemos una aplicación $b : M \times N \rightarrow O$ que es bilineal, o sea, que $b\left( \bullet, n \right)$ y $b\left( m, \bullet \right)$ son homorfismos de \modulo{A}s para cualquier $m \in M$ y $n \in N$; 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 que para cada $\left( m, n \right)$ en $M \times N$ exista un elemento $m * n$ en $E$ de forma que $b'\left( m*n \right) = b\left( m, n \right)$. Esta es la motivación de la construcción del producto tensorial. + \begin{definicion} + Sea $A$ un anillo y $M$ y $N$ dos \modulo{A}s. Tomemos como $E$ al \modulo{A} definido como $\bigoplus_{\substack{m \in M \\ n \in N}} A \cdot \vectorbasecanonica{m, n}$ donde \vectorbasecanonica{m, n} debe entenderse como la tupla $\left( m, n \right)$ sin ningún tipo de operación asociada, es decir, no debemos pensar que $\vectorbasecanonica{m, n} + \vectorbasecanonica{m', n'} = \vectorbasecanonica{m+m', n+n'}$ al igual que no pensamos que $\euno + \edos = \etres$. Ahora, consideremos dentro de $E$ el \submodulo{A} $S$ definido por + \begin{equation*} + S = \engendrado{ \left.\begin{gathered} + \vectorbasecanonica{m + m', n} - \vectorbasecanonica{m, n} - \vectorbasecanonica{m', n}, \quad + \vectorbasecanonica{m, n + n'} - \vectorbasecanonica{m, n} - \vectorbasecanonica{m, n'}, \\ + \vectorbasecanonica{am, n} - a\vectorbasecanonica{m, n}, \quad + \vectorbasecanonica{m, an} - a\vectorbasecanonica{m, n} + \end{gathered} \right\rvert a \in A, m \in M, n \in N}. + \end{equation*} + Definimos el p\emph{roducto tensorial} de $M$ por $N$ como $M \otimes N := \cociente{E}{S}$ y denotaremos a la clase de equivalencia de cada $\vectorbasecanonica{m, n}$ como $m \otimes n$ + \todo{Propiedad universal} + \end{definicion} + \begin{ejemplo} + Quizás el ejemplo más natural sea el producto escalar. Así pues, tomemos la aplicación $\productoescalarantilambda{\bullet}{\bullet} : \reales^3 \times \reales^3 \rightarrow \reales$, que es bilineal. En este caso, todo elemento de $\reales^3 \otimes \reales^3$ es de la siguiente forma ---entendiendo $S$ como en la definición de antes---: + \begin{equation}\label{Desglose Producto Tensorial} + \begin{aligned} + \vectortres{a}{b}{c} \otimes \vectortres{d}{e}{f} + S &= \left( a\euno + b\edos + c\etres \right) \otimes \left( d\euno + e\edos + f\etres \right) + S \\ + &= ad \euno\otimes\euno + ae\euno\otimes\edos + af\euno\otimes\etres + bd\edos\otimes\euno \\ &\phantom{text} + be\edos\otimes\edos + bf\edos\otimes\etres + cd\etres\otimes\euno + ce\etres\otimes\edos + cf\etres\otimes\etres + S + \end{aligned} + \end{equation} + (De aquí en adelante omitiremos los <<$+S$>> o similares sumas por el \submodulo{A} que estemos haciendo el cociente.) + + 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 \emph{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 $m \wedge m'$.\todo{Otra propiedad universal} + \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. 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, $\dim \bigwedge^k V = \binom{n}{k}$. 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. + \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 \emph{aplicación exterior }$\bigwedge^k \phi: \bigwedge^k M \rightarrow \bigwedge^k N$ 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, $\bigwedge^n V$ 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} \chapter{Bases de Gröbner y Sicigias} A partir de este momento solo trabajaremos con módulos $\Kx^n$ ya que son los involucrados en el módulo de derivaciones. @@ -932,105 +1031,6 @@ 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{Producto Exterior} - \todo{Si haces lo de la bilineal lo puedes usar aquí} - De cara a la demostración del Criterio de Saito, utilizaremos una operación que nos será de gran ayuda: el producto exterior. Para entender esta, necesitaremos entender primero el producto tensorial. En esencia, el producto tensorial aparece en el siguiente contexto: supongamos que $M$, $N$ y $O$ son \modulo{A}s y tenemos una aplicación $b : M \times N \rightarrow O$ que es bilineal, o sea, que $b\left( \bullet, n \right)$ y $b\left( m, \bullet \right)$ son homorfismos de \modulo{A}s para cualquier $m \in M$ y $n \in N$; 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 que para cada $\left( m, n \right)$ en $M \times N$ exista un elemento $m * n$ en $E$ de forma que $b'\left( m*n \right) = b\left( m, n \right)$. Esta es la motivación de la construcción del producto tensorial. - \begin{definicion} - Sea $A$ un anillo y $M$ y $N$ dos \modulo{A}s. Tomemos como $E$ al \modulo{A} definido como $\bigoplus_{\substack{m \in M \\ n \in N}} A \cdot \vectorbasecanonica{m, n}$ donde \vectorbasecanonica{m, n} debe entenderse como la tupla $\left( m, n \right)$ sin ningún tipo de operación asociada, es decir, no debemos pensar que $\vectorbasecanonica{m, n} + \vectorbasecanonica{m', n'} = \vectorbasecanonica{m+m', n+n'}$ al igual que no pensamos que $\euno + \edos = \etres$. Ahora, consideremos dentro de $E$ el \submodulo{A} $S$ definido por - \begin{equation*} - S = \engendrado{ \left.\begin{gathered} - \vectorbasecanonica{m + m', n} - \vectorbasecanonica{m, n} - \vectorbasecanonica{m', n}, \quad - \vectorbasecanonica{m, n + n'} - \vectorbasecanonica{m, n} - \vectorbasecanonica{m, n'}, \\ - \vectorbasecanonica{am, n} - a\vectorbasecanonica{m, n}, \quad - \vectorbasecanonica{m, an} - a\vectorbasecanonica{m, n} - \end{gathered} \right\rvert a \in A, m \in M, n \in N}. - \end{equation*} - Definimos el p\emph{roducto tensorial} de $M$ por $N$ como $M \otimes N := \cociente{E}{S}$ y denotaremos a la clase de equivalencia de cada $\vectorbasecanonica{m, n}$ como $m \otimes n$ - \todo{Propiedad universal} - \end{definicion} - \begin{ejemplo} - Quizás el ejemplo más natural sea el producto escalar. Así pues, tomemos la aplicación $\productoescalarantilambda{\bullet}{\bullet} : \reales^3 \times \reales^3 \rightarrow \reales$, que es bilineal. En este caso, todo elemento de $\reales^3 \otimes \reales^3$ es de la siguiente forma ---entendiendo $S$ como en la definición de antes---: - \begin{equation}\label{Desglose Producto Tensorial} - \begin{aligned} - \vectortres{a}{b}{c} \otimes \vectortres{d}{e}{f} + S &= \left( a\euno + b\edos + c\etres \right) \otimes \left( d\euno + e\edos + f\etres \right) + S \\ - &= ad \euno\otimes\euno + ae\euno\otimes\edos + af\euno\otimes\etres + bd\edos\otimes\euno \\ &\phantom{text} + be\edos\otimes\edos + bf\edos\otimes\etres + cd\etres\otimes\euno + ce\etres\otimes\edos + cf\etres\otimes\etres + S - \end{aligned} - \end{equation} - (De aquí en adelante omitiremos los <<$+S$>> o similares sumas por el \submodulo{A} que estemos haciendo el cociente.) - - 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 \emph{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 $m \wedge m'$.\todo{Otra propiedad universal} - \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. 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, $\dim \bigwedge^k V = \binom{n}{k}$. 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. - \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 \emph{aplicación exterior }$\bigwedge^k \phi: \bigwedge^k M \rightarrow \bigwedge^k N$ 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, $\bigwedge^n V$ 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} @@ -1498,8 +1498,8 @@ x2z+y3,3z,-3y2,-2z,0, a = h + \sum \end{equation*} \todo{Completar la demostración.} - \bibliography{bibliografia} - \bibliographystyle{plain} -% \nocite{*} -% \printbibliography + %\bibliography{bibliografia} + %\bibliographystyle{plain} + 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resolver?}¿Cuáles son estas? ¿Qué estructura siguen? ¿Y cómo podemos calcularlas? - -\chapter{Módulos} + Es decir, con estas derivaciones de \derivaciones{f} estamos obteniendo campos de vectores tangentes a la hipersuperficie $f=0$. Este conjunto será el principal objeto de estudio de este documento. + + A primera vista, es apreciable la relación con los vectores, pero no se tratará de espacios vectoriales, sino de una generalización de estos a coeficientes en un anillo: los módulos. Desarrollaremos una descripción de este tipo de objeto en el capítulo \ref{Capítulo de Módulos}. A continuación, notemos que la condición $\delta\!\left( f \right) = hf$ puede reescribirse como + \begin{equation*} + \sum_{i = 1}^{n}a_i \diffp{f}{x_i} = hf \iff \sum_{i = 1}^{n}a_i \diffp{f}{x_i} - hf = 0 \iff \vectorhorizontal{\diffp{f}{x_1}, \diffp{f}{x_2}, \dots, \diffp{f}{x_n}, f} \vectorvertical{a_1 \\ a_2 \\ \vdots \\ a_n \\ -h} = 0. + \end{equation*} + El vector de coeficientes expresa una <> entre los elementos de \vectorhorizontal{\diffp{f}{x_1}, \diffp{f}{x_2}, \dots, \diffp{f}{x_n}, f} al igual que una solución de un sistema lineal homogéneo expresa una relación entre las columnas de una matriz. El conjunto de todas estas relaciones o soluciones del sistema se conoce como el módulo de sicigias. Para poder calcularlo, necesitaremos las Bases de \Groebner: una herramienta que nos ayuda a la hora de determinar si un elemento pertenece o no a un ideal de \Kx, pero en este caso generalizada también a módulos. Todo esto lo desarrollaremos en el capítulo \ref{Capítulo de Bases de Groebner y Sicigias}. + + Por último, una de las principales dificultades que presentan los módulos son las bases. Así como en cualquier espacio vectorial sabemos que siempre hay una base y que todo subespacio también la tiene, no tiene por qué ser así en el caso de módulos. Para poder discernir en particular si \derivaciones{f} admite una base, la herramienta a utilizar es el Criterio de Saito. Este resultado afirma que \derivaciones{f} admite una base si y solo si se pueden encontrar $\delta_1, \delta_2, \dots, \delta_n \in \derivaciones{f}$ de forma que ---viéndolos como vectores de $\Kx^n$--- se cumpla + \begin{equation*} + \det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf\quad \text{donde $u$ es una unidad de \Kx}. + \end{equation*} + En el capítulo \ref{Capítulo del Criterio de Saito} desarrollaremos la demostración de este criterio basados en \cite{VALLES}. + + De hecho, este concepto de que \derivaciones{f} admita una base no se comprende del todo bien actualmente. Por ejemplo, si tomamos $f := x^2z + y^3 \in \anillodepolinomios{K}{x,y,z}$ nos encontramos con la superficie + \begin{figure}[h] + \centering + \includegraphics[scale=0.3]{primerasuperficie.png}\\ + $x^2z + y^3 = 0$ + \end{figure} + En este caso, \derivaciones{x^2z + y^3} no es libre. Puede encontrarse el cálculo de todo lo comentado hasta ahora para este caso en el apéndice \ref{Apéndice Cálculo}. Sin embargo, añadiendo un plano que cree una especie de <> a esas esquinas que aparecen, que se consigue multiplicando por $x$, nos queda la siguiente superficie: + \begin{figure}[h] + \centering + \includegraphics[scale=0.3]{segundasuperficie.png}\\ + $x\left( x^2z + y^3 \right) = 0$ + \end{figure} + + Y en este caso, \derivaciones{f} sí que admite una base. Para garantizarlo, hacemos uso del Criterio de Saito con + \begin{align*} + \delta_1 &:= \left( x^2z + y^3 \right)\diffp{}{z},\\ + \delta_2 &:= y\diffp{}{y} + 3z\diffp{}{z} \quad \text{y} \\ + \delta_3 &:= x\diffp{}{x} - 2z\diffp{}{z}. + \end{align*} + Viendo estos como vectores de $\Kx^r$, hacemos el determinante: + \begin{equation*} + \det\!\begin{pmatrix} + 0 & 0 & x \\ + 0 & y & 0 \\ + x^2z + y^3 & 3z & -2z + \end{pmatrix} = -y\left[ x\left( x^2z + y^3 \right) \right] = -yf. + \end{equation*} + El criterio de Saito se cumple, y por lo tanto, $\delta_1$, $\delta_2$ y $\delta_3$ forman una base de \derivaciones{f}. + +\chapter{Módulos} \label{Capítulo de Módulos} \section{Conceptos básicos} Los conjuntos con los que trabajaremos son una generalización de los espacios vectoriales donde los coeficientes pertenecen a un anillo en vez de a un cuerpo. A esta generalización la llamamos módulo. En este capítulo veremos solo algunas de las propiedades y definiciones necesarias para nuestros propósitos. Sin embargo, la teoría de módulos se extiende mucho más allá de esta exigua presentación, por lo que recomendamos que se consulte un texto introductorio como \cite{SINGULARLIBRO} o \cite{COCOA}. @@ -642,7 +683,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du \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} -\chapter{Bases de Gröbner y Sicigias} +\chapter{Bases de Gröbner y Sicigias} \label{Capítulo de Bases de Groebner y Sicigias} A partir de este momento solo trabajaremos con módulos $\Kx^n$ ya que son los involucrados en el módulo de derivaciones. @@ -993,7 +1034,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 con sus primeras $r$ componentes nulas y las últimas $k$ coinciden con las de $m_{r+1}$. Por lo tanto, $m_{r+1} \in M_{r+1}$. Concluimos, pues, que $M_{r+1} = \syz{f_1, f_2, \dots, f_k}$. \end{proof} Por lo tanto, para calcular \syz{f_1, f_2, \dots, f_k}, primero calcularemos una base de \Groebner{} $G$ de $$\engendrado{f_1^+ + \vectorbasecanonica{r + 1}, f_2^+ + \vectorbasecanonica{r + 2}, \dots, f_k^+ + \vectorbasecanonica{r + k}}.$$ Por este último teorema, sabemos que $G_{k+1}$ es un conjunto generador de \syz{f_1, f_2, \dots, f_k} y en particular una base de \Groebner{} con respecto al orden que hayamos escogido. Si deseamos una base de \Groebner{} con respecto a otro orden, como decíamos antes de este teorema, basta con aplicar el algoritmo de Buchberger sobre $G_{k+1}$. -\chapter{El Criterio de Saito} +\chapter{El Criterio de Saito} \label{Capítulo del Criterio de Saito} En el capítulo anterior concluíamos con un método para calcular \derivaciones{f} utilizando las sicigias, una herramienta que hemos derivado de las bases de Gröbner. Con estas, podemos obtener un conjunto generador de \derivaciones{f} e incluso una base de Gröbner de este mismo que nos permita determinar si una derivación está o no en el conjunto. Pero otra pregunta de gran importancia aparece cuando consideramos una gran diferencia entre módulos y espacios vectoriales: así como para estos últimos siempre tenemos que los subespacios admiten una base, en el caso de los módulos no tiene por qué ser así. @@ -1254,7 +1295,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \todo[inline]{Para poner un ejemplo espero a la introducción, para que case con el ejemplo que allí pongamos.} \appendix - \chapter{Un ejemplo completo de cálculo} + \chapter{Un ejemplo completo de cálculo}\label{Apéndice Cálculo} El algoritmo para calcular las sicigias que nos permite obtener \derivaciones{f}, en esencia el algoritmo de Buchberger, tienen un alto coste computacional que hace impracticable ejecutarlos <>. En la práctica, se utilizan sistemas de Álgebra Computacional que ya vienen preparados con estos algoritmos en sus versiones más óptimas. Sin embargo, como ocurre con muchas otras operaciones a lo largo de nuestra experiencia matemática, conviene saber cómo hacerlas a mano para tener una mayor intuición. En muchos textos que tratan sobre estos asuntos se suelen omitir los ejemplos a este tipo de operaciones o se dan algunos en casos demasiado sencillos ---ideales de \Kx{} en vez de \modulo{\Kx} de $\Kx^n$ donde da la casualidad que todo se anula a la primera--- por lo que creemos conveniente desarrollar en este apéndice por completo un caso no trivial para que el lector se lleve una imagen realista de qué ocurre detrás de la pantalla. Además, ilustraremos también después cómo se resuelve utilizando un sistema dedicado a este tipo de cálculos. 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-{\color{red} Hago una propuesta alternativa que creo que simplifica y aclara. \begin{definicion} - Dados dos $A$-módulos $M,N$, una forma \emph{bilineal} $\Phi:M\times N\to A$ es una aplicación tal que - \begin{itemize} - \item $\Phi\!\left( a m + a'm', n \right) = a\Phi\!\left( m, n \right) + a' \Phi\!\left( m', n \right)$ y - \item $\Phi\!\left( m, a n + a'n' \right) = a\Phi\!\left( m, n \right) + a' \Phi\!\left( m, n' \right)$ - \end{itemize} -$\forall a,a'\in A$, $\forall m, m'\in M$, $\forall n, n'\in N$. Una forma bilineal induce homomorfismos $\Phi_1:M\to\dual{N}$, $\Phi_2:N\to\dual{M}$, dados por -$\Phi_1(m)(n):=\Phi(m, n)=:\Phi_2(n)(m)$. + Dados dos \modulo{A}s $M$ y $N$, una forma \emph{bilineal} $\Phi:M\times N\to A$ es una aplicación tal que + \begin{equation*} + \begin{cases} + \Phi\!\left( a m + a'm', n \right) = a\Phi\!\left( m, n \right) + a' \Phi\!\left( m', n \right) \ \ \text{y} \\ + \Phi\!\left( m, a n + a'n' \right)\; = a\Phi\!\left( m, n \right) + a' \Phi\!\left( m, n' \right) + \end{cases} \quad \forall a,a'\in A, \ \forall m, m'\in M, \ \forall n, n'\in N. + \end{equation*} + Una forma bilineal induce homomorfismos $\Phi_1:M\to\dual{N}$, $\Phi_2:N\to\dual{M}$, dados por + \begin{equation*} + \Phi_1(m)(n):=\Phi(m, n)=:\Phi_2(n)(m). + \end{equation*} \end{definicion} -Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\dual{M}\to A$, definida mediante evaluación, que induce un homomorfismo natural $\Psi:=\Phi^M_1:M\to\dualdual{M}$.} - \begin{definicion} \label{Definición Módulo Reflexivo} - Sea $\psi : M \rightarrow \dualdual{M}$ el homomorfismo de \modulo{A}s que envía un elemento $m$ de $M$ a la función $\psi\!\left( m \right): \dual{M} \rightarrow A$ que a cada $\phi \in \dual{M}$ le asigna el valor $\phi\!\left( m \right)$: +Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\dual{M}\to A$, definida mediante evaluación, que induce un homomorfismo natural $\Psi:=\Phi^M_1:M\to\dualdual{M}$. Esta aplicación envía un elemento $m$ de $M$ a la función $\Psi\!\left( m \right): \dual{M} \rightarrow A$ que a cada $\phi \in \dual{M}$ le asigna el valor $\phi\!\left( m \right)$: \begin{align*} - \psi : M &\longrightarrow \Hom{A}{\Hom{A}{M}{A}}{A} \\ - m &\longmapsto \psi\!\left( m \right) : \Hom{A}{M}{A} \longrightarrow A \\ - &\hspace{3.7cm} \phi \longmapsto \phi\!\left( m \right) + \Psi : M &\longrightarrow \dualdual{M} \\ + m &\longmapsto \Psi\!\left( m \right) : \dual{M} \longrightarrow A \\ + &\hspace{2.1cm} \phi \hspace{0.2cm} \longmapsto \phi\!\left( m \right) \end{align*} - Diremos que $M$ es un \modulo{A} \emph{reflexivo} si $M$ es isomorfo a \dualdual{M} por medio de esta aplicación $\psi$. - \end{definicion} + Diremos que $M$ es un \modulo{A} \emph{reflexivo} si $\Psi$ es un isomorfismo. \begin{ejemplo} \todo{Cambiaría la estructura de este ejemplo usando bases duales y lo simplificaría.} Veamos que todo espacio vectorial de dimensión finita es reflexivo. Sea $V$ un $K$-espacio vectorial de dimensión $n$ y con base $\conjunto{v_1, v_2, \dots, v_n}$. El conjunto \Hom{K}{V}{K} se compone de las aplicaciones lineales de $V$ en $K$. Cada elemento $\phi$ de \Hom{K}{V}{K} puede representarse como la multiplicación por una matriz, puesto que From c108dd743d06dded4f7f529d735906797e3d5b24 Mon Sep 17 00:00:00 2001 From: Sixto Date: Wed, 11 Jun 2025 00:10:25 +0200 Subject: [PATCH 6/9] Cambio en el ejemplo de reflexividad de un espacio vectorial. --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 573909 -> 572702 bytes TFG_Sixto_José_Martínez_Gan.tex | 17 ++++++++++------- 2 files changed, 10 insertions(+), 7 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index 6d16658d19ba3fc9644fa855daf6baf0df4647f4..c2052f7a6bcc8d998b5c0f98642592d23d721f20 100644 GIT binary patch delta 73784 zcmV(e7T{w!&W-(rA&ih&4gemz#RKTA486U7k zbY%xx{25(>RoeD5c7Yn@B2-suBMhr&DrYp9%Ys^m4a{aw(YU| z@!UPE&FUB^xog7R|2ZDu?;ZTZR*vAO1MIV(f$f)@ic(u_xOn!|jVB(<+ z78J|bRpad9%3kHyM!;s66GX}@DQuY9oPtz8tT&Xb>cd+QNJ+aect*v`$un0^VQN_Q zLx1=ArAVt-)K%L)Cqwh%hY3!&qK(A^QjQQ@_fw6YL9 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--- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -50,8 +50,13 @@ \usepackage{enumitem} % Bibliografía. +\newif\ifbibtex +\bibtextrue +\ifbibtex +\else \usepackage[backend=biber]{biblatex} \addbibresource{bibliografia.bib} +\fi \usepackage{algpseudocodex} \algrenewcommand\algorithmicwhile{\textbf{mientras}} @@ -215,8 +220,8 @@ urlcolor=black]{hyperref} \noindent \begin{figure}[h] \centering -\includegraphics[width=100mm]{Identidad_Corporativa_Universidad_de_Zaragoza/ciencias.png}\\% -\includegraphics[width=100mm]{Identidad_Corporativa_Universidad_de_Zaragoza/logoUZ.png} +\includegraphics[width=100mm]{Imagenes_y_Figuras/Identidad_Corporativa_Universidad_de_Zaragoza/ciencias.png}\\% +\includegraphics[width=100mm]{Imagenes_y_Figuras/Identidad_Corporativa_Universidad_de_Zaragoza/logoUZ.png} \end{figure} \vfill @@ -267,13 +272,13 @@ urlcolor=black]{hyperref} \end{equation*} Es decir, con estas derivaciones de \derivaciones{f} estamos obteniendo campos de vectores tangentes a la hipersuperficie $f=0$. Este conjunto será el principal objeto de estudio de este documento. - A primera vista, es apreciable la relación con los vectores, pero no se tratará de espacios vectoriales, sino de una generalización de estos a coeficientes en un anillo: los módulos. Desarrollaremos una descripción de este tipo de objeto en el capítulo \ref{Capítulo de Módulos}. A continuación, notemos que la condición $\delta\!\left( f \right) = hf$ puede reescribirse como + A primera vista, es apreciable la relación con los vectores, pero no se tratará de espacios vectoriales, sino de una generalización de estos a coeficientes en un anillo: los módulos. Desarrollaremos una descripción de este tipo de objeto en el Capítulo \ref{Capítulo de Módulos}. A continuación, notemos que la condición $\delta\!\left( f \right) = hf$ puede reescribirse como \begin{equation*} \sum_{i = 1}^{n}a_i \diffp{f}{x_i} = hf \iff \sum_{i = 1}^{n}a_i \diffp{f}{x_i} - hf = 0 \iff \vectorhorizontal{\diffp{f}{x_1}, \diffp{f}{x_2}, \dots, \diffp{f}{x_n}, f} \vectorvertical{a_1 \\ a_2 \\ \vdots \\ a_n \\ -h} = 0. \end{equation*} - El vector de coeficientes expresa una <> entre los elementos de \vectorhorizontal{\diffp{f}{x_1}, \diffp{f}{x_2}, \dots, \diffp{f}{x_n}, f} al igual que una solución de un sistema lineal homogéneo expresa una relación entre las columnas de una matriz. El conjunto de todas estas relaciones o soluciones del sistema se conoce como el módulo de sicigias. Para poder calcularlo, necesitaremos las Bases de \Groebner: una herramienta que nos ayuda a la hora de determinar si un elemento pertenece o no a un ideal de \Kx, pero en este caso generalizada también a módulos. Todo esto lo desarrollaremos en el capítulo \ref{Capítulo de Bases de Groebner y Sicigias}. + El vector de coeficientes expresa una <> entre los elementos de \vectorhorizontal{\diffp{f}{x_1}, \diffp{f}{x_2}, \dots, \diffp{f}{x_n}, f} al igual que una solución de un sistema lineal homogéneo expresa una relación entre las columnas de una matriz. El conjunto de todas estas relaciones o soluciones del sistema se conoce como el módulo de sicigias. Para poder calcularlo, necesitaremos las Bases de \Groebner: una herramienta que nos ayuda a la hora de determinar si un elemento pertenece o no a un ideal de \Kx, pero en este caso generalizada también a módulos. Todo esto lo desarrollaremos en el Capítulo \ref{Capítulo de Bases de Groebner y Sicigias}. - Por último, una de las principales dificultades que presentan los módulos son las bases. Así como en cualquier espacio vectorial sabemos que siempre hay una base y que todo subespacio también la tiene, no tiene por qué ser así en el caso de módulos. Para poder discernir en particular si \derivaciones{f} admite una base, la herramienta a utilizar es el Criterio de Saito. Este resultado afirma que \derivaciones{f} admite una base si y solo si se pueden encontrar $\delta_1, \delta_2, \dots, \delta_n \in \derivaciones{f}$ de forma que ---viéndolos como vectores de $\Kx^n$--- se cumpla + Por último, una de las principales dificultades que presentan los módulos son las bases\todo{diría saber si son libres y encontrar una base}. Así como en cualquier espacio vectorial sabemos que siempre hay una base y que todo subespacio también la tiene, no tiene por qué ser así en el caso de módulos. Para poder discernir en particular si \derivaciones{f} admite una base, la herramienta a utilizar es el Criterio de Saito. Este resultado afirma que \derivaciones{f} admite una base si y solo si se pueden encontrar $\delta_1, \delta_2, \dots, \delta_n \in \derivaciones{f}$ de forma que ---viéndolos como vectores de $\Kx^n$--- se cumpla \begin{equation*} \det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf\quad \text{donde $u$ es una unidad de \Kx}. \end{equation*} @@ -282,16 +287,16 @@ urlcolor=black]{hyperref} De hecho, este concepto de que \derivaciones{f} admita una base no se comprende del todo bien actualmente. Por ejemplo, si tomamos $f := x^2z + y^3 \in \anillodepolinomios{K}{x,y,z}$ nos encontramos con la superficie \begin{figure}[h] \centering - \includegraphics[scale=0.3]{primerasuperficie.png}\\ + \includegraphics[scale=0.3]{Imagenes_y_Figuras/primerasuperficie.png}\\ $x^2z + y^3 = 0$ \end{figure} En este caso, \derivaciones{x^2z + y^3} no es libre. Puede encontrarse el cálculo de todo lo comentado hasta ahora para este caso en el apéndice \ref{Apéndice Cálculo}. Sin embargo, añadiendo un plano que cree una especie de <> a esas esquinas que aparecen, que se consigue multiplicando por $x$, nos queda la siguiente superficie: \begin{figure}[h] \centering - \includegraphics[scale=0.3]{segundasuperficie.png}\\ + \includegraphics[scale=0.3]{Imagenes_y_Figuras/segundasuperficie.png}\\ $x\left( x^2z + y^3 \right) = 0$ \end{figure} - + \todo[inline]{Si puedes, el plano de otro color y algo traslúcido.} Y en este caso, \derivaciones{f} sí que admite una base. Para garantizarlo, hacemos uso del Criterio de Saito con \begin{align*} \delta_1 &:= \left( x^2z + y^3 \right)\diffp{}{z},\\ @@ -307,6 +312,7 @@ urlcolor=black]{hyperref} \end{pmatrix} = -y\left[ x\left( x^2z + y^3 \right) \right] = -yf. \end{equation*} El criterio de Saito se cumple, y por lo tanto, $\delta_1$, $\delta_2$ y $\delta_3$ forman una base de \derivaciones{f}. + \todo[inline]{Haz referencia al apartado donde está el criterio} \chapter{Módulos} \label{Capítulo de Módulos} \section{Conceptos básicos} @@ -477,7 +483,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du \end{align*} Diremos que $M$ es un \modulo{A} \emph{reflexivo} si $\Psi$ es un isomorfismo. \begin{ejemplo} -\todo{Cambiaría la estructura de este ejemplo usando bases duales y lo simplificaría.} +\todo[inline]{Cambiaría la estructura de este ejemplo usando bases duales y lo simplificaría.} Veamos que todo espacio vectorial de dimensión finita es reflexivo. Sea $V$ un $K$-espacio vectorial de dimensión $n$ y con base $\conjunto{v_1, v_2, \dots, v_n}$. El conjunto \Hom{K}{V}{K} se compone de las aplicaciones lineales de $V$ en $K$. Cada elemento $\phi$ de \Hom{K}{V}{K} puede representarse como la multiplicación por una matriz, puesto que \begin{equation*} \phi\!\left( v \right) = \productoescalar{\vectorhorizontalcuatro{\phi\!\left( v_1 \right)}{\phi\!\left( v_2 \right)}{\hdots}{\phi\!\left( v_n \right)}}{v}; @@ -597,7 +603,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du Una introducción a los módulos proyectivos puede encontrarse en \cite[sección 1.2]{LIBROPROYECTIVOS}. \section{Producto Exterior} - \todo{Si haces lo de la bilineal lo puedes usar aquí} + \todo[inline]{Si haces lo de la bilineal lo puedes usar aquí} De cara a la demostración del Criterio de Saito, utilizaremos una operación que nos será de gran ayuda: el producto exterior. Para entender esta, necesitaremos entender primero el producto tensorial. En esencia, el producto tensorial aparece en el siguiente contexto: supongamos que $M$, $N$ y $O$ son \modulo{A}s y tenemos una aplicación $b : M \times N \rightarrow O$ que es bilineal, o sea, que $b\left( \bullet, n \right)$ y $b\left( m, \bullet \right)$ son homorfismos de \modulo{A}s para cualquier $m \in M$ y $n \in N$; 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 que para cada $\left( m, n \right)$ en $M \times N$ exista un elemento $m * n$ en $E$ de forma que $b'\left( m*n \right) = b\left( m, n \right)$. Esta es la motivación de la construcción del producto tensorial. \begin{definicion} Sea $A$ un anillo y $M$ y $N$ dos \modulo{A}s. Tomemos como $E$ al \modulo{A} definido como $\bigoplus_{\substack{m \in M \\ n \in N}} A \cdot \vectorbasecanonica{m, n}$ donde \vectorbasecanonica{m, n} debe entenderse como la tupla $\left( m, n \right)$ sin ningún tipo de operación asociada, es decir, no debemos pensar que $\vectorbasecanonica{m, n} + \vectorbasecanonica{m', n'} = \vectorbasecanonica{m+m', n+n'}$ al igual que no pensamos que $\euno + \edos = \etres$. Ahora, consideremos dentro de $E$ el \submodulo{A} $S$ definido por @@ -610,7 +616,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du \end{gathered} \right\rvert a \in A, m \in M, n \in N}. \end{equation*} Definimos el p\emph{roducto tensorial} de $M$ por $N$ como $M \otimes N := \cociente{E}{S}$ y denotaremos a la clase de equivalencia de cada $\vectorbasecanonica{m, n}$ como $m \otimes n$ - \todo{Propiedad universal} + \todo[inline]{Propiedad universal} \end{definicion} \begin{ejemplo} Quizás el ejemplo más natural sea el producto escalar. Así pues, tomemos la aplicación $\productoescalarantilambda{\bullet}{\bullet} : \reales^3 \times \reales^3 \rightarrow \reales$, que es bilineal. En este caso, todo elemento de $\reales^3 \otimes \reales^3$ es de la siguiente forma ---entendiendo $S$ como en la definición de antes---: @@ -620,7 +626,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du &= ad \euno\otimes\euno + ae\euno\otimes\edos + af\euno\otimes\etres + bd\edos\otimes\euno \\ &\phantom{text} + be\edos\otimes\edos + bf\edos\otimes\etres + cd\etres\otimes\euno + ce\etres\otimes\edos + cf\etres\otimes\etres + S \end{aligned} \end{equation} - (De aquí en adelante omitiremos los <<$+S$>> o similares sumas por el \submodulo{A} que estemos haciendo el cociente.) + De aquí en adelante omitiremos los <<$+S$>> o similares sumas por el \submodulo{A} que estemos haciendo el cociente. 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*} @@ -686,13 +692,14 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du \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. + 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. + \qedhere \end{equation} \end{proof} @@ -712,7 +719,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du El cuerpo de fracciones de un anillo nos permite llevar un \modulo{A} a un \espaciovectorial{\cuerpofracciones{A}}. \begin{definicion} - Llamaremos extensión de escalares del \modulo{A} $M$ al conjunto $M \otimes \cuerpofracciones{A}$. + Llamaremos \emph{extensión de escalares} del \modulo{A} $M$ al \espaciovectorial{\cuerpofracciones{A}} $M \otimes \cuerpofracciones{A}$. \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. @@ -728,13 +735,13 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du &\implies& am \neq 0 & \end{array} \end{equation*} - \todo{La otra implicación.} + \todo{La otra implicación. Creo que quedará más corto definiendo la extensión de escalares como fracciones} \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 $r$. + Sea $M$ un \modulo{A} finitamente generado. Diremos que $M$ tiene \emph{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. + Recordemos que ya en el Lema~\ref{Lema Rango Producto Exterior} había un resultado sobre el rango del producto exterior de los espacios vectoriales. Generalizamos esto a los módulos. \begin{lema} \label{Lema Rango Producto Exterior Módulos} 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} @@ -764,6 +771,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du \text{$M$ es reflexivo} \end{aligned}\right\rbrace \implies \text{$M$ es un módulo libre} \end{gather} + \todo{Esto igual sí que tiene que estar en el último capítulo y aquí alguna cosa de reflexivos.} \chapter{Bases de Gröbner y Sicigias} \label{Capítulo de Bases de Groebner y Sicigias} @@ -1035,7 +1043,7 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición Sea $G$ un subconjunto de $\Kx^r$, este último equipado con un orden monomial de módulos $>$. $G$ es una base de Gröbner de \engendrado{G} si y solo si para toda pareja de elementos $f$ y $g$ en $G$ se tiene que el resto de $\spoly{f}{g} = 0$ entre $G$ es $0$. \end{teorema} \begin{proof} - Véase el apéndice \ref{Apéndice Demostración del Criterio de Buchberger}.\todo{Creo que es mejor gastar energía en otras cosas.} + Véase el apéndice \ref{Apéndice Demostración del Criterio de Buchberger}.\todo[inline]{Creo que es mejor gastar energía en otras cosas.} \end{proof} Un ejemplo de cálculo de base de \Groebner{} puede encontrarse en apéndice \ref{Sección A mano}. @@ -1144,13 +1152,14 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \begin{definicion} Un polinomio $p \in \Kx$ se dice \emph{libre de cuadrados} si, en caso de poder expresarlo de la forma $p = a^2b$, se tiene que $a$ es una unidad de \Kx. \end{definicion} + \todo{Al principio de 2} \begin{lema} \label{Lema MCD Parciales} Un polinomio $p \in \Kx$ es libre de cuadrados si y solo si $\mcd{p, \diffp{p}{x_1}, \diffp{p}{x_2}, \dots, \diffp{p}{x_n}}$ es una unidad de \Kx. \end{lema} El criterio de Saito nos da una forma bastante rápida de determinar si \derivaciones{f} es o no libre. Nos permitimos la licencia de denotar por $\delta_i$ tanto a la derivación como al vector de coeficientes asociado en $\Kx^n$. \begin{teorema}[Criterio de Saito] \label{Criterio de Saito} - Sea $f$ un polinomio libre de cuadrados de \Kx{} y $\delta_1, \delta_2, \dots, \delta_n$ derivaciones de \derivaciones{f}. Entonces, \derivaciones{f} es libre si y solo si $\det\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$ donde $u$ es una unidad de $K$, es decir, un elemento de $K \setminus \left\lbrace 0 \right\rbrace$. + Sea $f$ un polinomio libre de cuadrados de \Kx{} y $\delta_1, \delta_2, \dots, \delta_n$ derivaciones de \derivaciones{f}. Entonces, \derivaciones{f} es libre si y solo si $\det\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$ donde $u$ es una unidad de $\Kx$, es decir, un elemento de $K \setminus \left\lbrace 0 \right\rbrace$. \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. @@ -1170,7 +1179,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 El Lema~\ref{Lema Rango Producto Exterior Módulos} nos garantiza que $\bigwedge^n \derivaciones{f}$ es un \modulo{\Kx} reflexivo de rango $1$. Siguiendo 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 isomorfo a 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 rango $1$. 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 + Ahora bien, recordemos la sucesión exacta \eqref{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 \begin{equation*} \bigwedge^n \derivaciones{f} \xrightarrow{\bigwedge^n \Phi} \bigwedge^n \derivaciones{\Kx}. \end{equation*} @@ -1188,7 +1197,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 Por lo tanto, podemos pensar en que la aplicación $\Phi \circ \iota$ envía a cada $\delta_i$ a $\vectorhorizontalcuatro{\partial_1}{\partial_2}{\dots}{\partial_n} A_i$ donde $A_i$ es la \nolbreaks{$i$-ésima} columna de $A$. Es decir, que las columnas de $A$ se entienden como las coordenadas de los $\delta_i$ en función de la base \conjunto{\partial_1, \partial_2, \dots, \partial_n}\footnote{Es decir, que $\det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = \det A$.}. Pensemos ahora en la aplicación \begin{equation*} - \begin{tikzcd}[sep=small] + \begin{tikzcd}[row sep=0pt] N \arrow[r, "\bigwedge^n \left( \Phi \circ \iota \right)"] & \derivaciones{\Kx} \\ \delta_1 \wedge \delta_2 \wedge \dots \wedge \delta_n \arrow[r, mapsto] & \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n \det A \end{tikzcd} @@ -1324,7 +1333,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{equation*} Lo primero que tendremos que hacer será calcular el submódulo de sicigias \syz{2xz, 3y^2, x^2, x^2z + y^3} siguiendo el método indicado en \ref{Teorema Calculo Sicigias}. Comenzamos ampliando los distintos elementos a vectores para que sus componentes extra <> de las anulaciones que hacemos. Así, tomamos \begin{equation*} - g_1 := \vectorvertical{2xz \\ 1 \\ 0 \\ 0 \\ 0}\!\!, \ g_2 := \vectorvertical{3y^2 \\ 0 \\ 1 \\ 0 \\ 0}\!\!, \ g_3 := \vectorvertical{x^2 \\ 0 \\ 0 \\ 1 \\ 0}\ \text{y} \ \ g_4' := \vectorvertical{x^2z + y^3 \\ 0 \\ 0 \\ 0 \\ 1}. + g_1 := \vectorvertical{2xz \\ 1 \\ 0 \\ 0 \\ 0} , \ g_2 := \vectorvertical{3y^2 \\ 0 \\ 1 \\ 0 \\ 0}, \ g_3 := \vectorvertical{x^2 \\ 0 \\ 0 \\ 1 \\ 0}\ \text{y} \ \ g_4' := \vectorvertical{x^2z + y^3 \\ 0 \\ 0 \\ 0 \\ 1}. \end{equation*} Calculamos ahora una base de Gr\"obner a partir de estos elementos. Para empezar con el mejor pie posible, lo que haremos será convertir este conjunto en un conjunto reducido, es decir, que buscaremos garantizar que ninguno de los términos de $g_i$ sea divisible por algún \LM{g_j} donde $j \neq i$. En este caso, vemos que $g_1$, $g_2$ y $g_3$ no tienen ningún termino divisible, sin embargo $g_4'$ sí, así que lo reducimos: \begin{equation} \label{Ejemplo División Compuesta} @@ -1345,14 +1354,14 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{blockarray} \end{equation} Por ser esta la primera división con más de un divisor que hacemos, la desglosaremos algo más siguiendo la notación del algoritmo en \ref{Algoritmo de la División Compuesta}. - \begin{description} + \begin{description}[wide, leftmargin=0pt,itemindent=6pt] \item[1\textsuperscript{a} iteración: ] $\LT{D} = x^2z\euno$ es divisible por $\LM{g_1}$ y $\LM{g_3}$. Escogemos, por pura preferencia personal, $g_1$ y reducimos ---la primera resta---. Después de reducir, $\LT{D} = y^3\euno$ es divisible por $\LM{g_2}$ así que reducimos por este. Tras esta reducción, $\LT{D} = -\frac{1}{2}x\edos$, que no es divisible por ninguno de los monomios líderes de $g_1$, $g_2$ y $g_3$. Se lo sustraemos a $D$ y lo añadimos al resto. Esto deja $D = -\frac{1}{3}y\etres + \ecinco$, $c_1 = \frac{1}{2}x$, $c_2 = \frac{1}{3}y$, $c_3 = 0$ y $r = -\frac{1}{2}x\edos$. \item[2\textsuperscript{a} iteración: ] $\LT{D} = -\frac{1}{3}y\etres$ no es divisible por ninguno de los términos líderes de los divisores. Así pues, no reducimos y sustraemos este término de $D$ para pasarlo a $r$. Esto deja $D = \ecinco$, $r = -\frac{1}{2}x\edos - \frac{1}{3}y\etres$ y los cocientes como antes. \item[3\textsuperscript{a} iteración: ] $\LT{D} = \ecinco$ y nos encontramos con la misma situación que antes. Al sustraer este de $D$, $D$ pasa a ser $0$, por lo que el algoritmo termina. Nos queda $r = -\frac{1}{2}x\edos - \frac{1}{3}y\etres + \ecinco$. \end{description} De aquí en adelante, no escribiremos todas las <> desde el principio, sino que las iremos añadiendo conforme las necesitemos. Tampoco seguiremos tachando ni indicando hasta dónde alcanza cada iteración para descargar las divisiones. - El resto de la anterior división es irreducible por los demás elementos, así que lo usaremos como elemento $g_4$. Multiplicar por un escalar un elemento no afecta al resultado, por lo tanto, intentamos obtener la expresión más sencilla para este y así + El resto de la anterior división es irreducible por los demás elementos, así que lo usaremos como elemento~$g_4$. Multiplicar por un escalar un elemento no afecta al resultado, por lo tanto, intentamos obtener la expresión más sencilla para este y así \begin{equation*} g_4 := -6\left( -\frac{1}{2}x\edos - \frac{1}{3}y\etres +\ecinco\right) = \vectorvertical{0 \\ 3x \\ 2y \\ 0 \\ -6}. \end{equation*} @@ -1553,8 +1562,11 @@ x2z+y3,3z,-3y2,-2z,0, a = h + \sum \end{equation*} \todo{Completar la demostración.} - %\bibliography{bibliografia} - %\bibliographystyle{plain} +\ifbibtex + \bibliography{bibliografia} + \bibliographystyle{plain} +\else \nocite{*} \printbibliography +\fi \end{document} From 37ad5118ce51309d1c2e75a5d3f69523fd4aaf9b Mon Sep 17 00:00:00 2001 From: Sixto Date: Wed, 11 Jun 2025 02:19:49 +0200 Subject: [PATCH 8/9] Resto de correcciones incluidas y resumen. --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 572702 -> 574199 bytes TFG_Sixto_José_Martínez_Gan.tex | 164 +++++++++++++++++++------------- 2 files changed, 96 insertions(+), 68 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf 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z0w5c(u@Wfzz2@6XlbDz5fVxal<0vkl%nW3oSnXLx+dQ|lMX`AN?4f?xLrDBn!Hrim zTW21*`YPaPe9zHT{_D8s`{kVU(+xjA(UxHE!XMfDXxvdHViFo<_b7|Qu3%n$?Dm#B z_`Sz8P~U7uUDqG4;5?>ChK6_*Ef?B&{!Iowcf3(G9G#NbF&s|-`PR7>vI}Vchy*-Q z5&FK`Vax>j_mp6i|9(Ja@NgHk&f?UdKUlK&f)#p@L(@WstnmspCabDzVM7vsd7?sc zlrnKtTZi_aEN?paRo_Mt(*oKn9e3#mVwB|UUF+=5XS?ofy}2KLr+?C%{uNZnpVb%d zS4AJv=}&rmL4)~$O=ixx-EFu>gsVuE_dp1p*XKW5p`PRk`QR=_m!($^Xx9f4@gBuX zqV4T=B6o*4=z+`z`AmZrT7}l`(k|Ztwz;@rM#Am693Fyk`TX39+6}KBALn7uFHc2I ziN*^yk%tieoEnS$8TA`}MceHBX1Bdc^$S&Pjv^b4@0IT&D~(s)?S5nBH$zhgD?QcR zmnQd0fPPD-BHn6lOQkTvfPL7UW#?yv*^Bjk3Asjb5$^=8zZ;(uT>t&XB?$&fxY|@ic=Iq(Qdly&>k5C|2tW`U!BZHYzs$L%KjDEXxH(} V80caaW(9&&!2l^KqnpNn{{T3WhuHuC diff --git a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index 7d31719..86d077d 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -238,8 +238,9 @@ urlcolor=black]{hyperref} %--- RESUMEN \chapter{Resumen} - - + En este trabajo respondemos a varias preguntas acerca del módulo de derivaciones polinómicas de un polinomio: qué clase de objeto es, para lo cual necesitaremos utilizar la teoría de módulos; cómo calcularlo, donde explicaremos herramientas como las bases de \Groebner{} y las sicigias; y por último, una demostración del Criterio de Saito, que nos permite saber si el dicho módulo de derivaciones admite una base o no. + + In this paper we answer several questions about the module of polynomial derivations of a polynomial: which kind of object it is, where we will need to use module theory; how to compute it, with an explanation of tools like \Groebner{} bases and syzygies; and lastly, a proof of Saito's Criterion, which allows us to know whether the derivations module admits a basis or not. %--- FIN DEL RESUMEN \clearemptydoublepage @@ -285,6 +286,7 @@ urlcolor=black]{hyperref} \includegraphics[scale=0.3]{primerasuperficie.png}\\ $x^2z + y^3 = 0$ \end{figure} + En este caso, \derivaciones{x^2z + y^3} no es libre. Puede encontrarse el cálculo de todo lo comentado hasta ahora para este caso en el apéndice \ref{Apéndice Cálculo}. Sin embargo, añadiendo un plano que cree una especie de <> a esas esquinas que aparecen, que se consigue multiplicando por $x$, nos queda la siguiente superficie: \begin{figure}[h] \centering @@ -329,7 +331,7 @@ urlcolor=black]{hyperref} Un módulo sobre un cuerpo es un espacio vectorial pero no todas las propiedades de espacios vectoriales se trasladan a módulos. - %La definición de espacio vectorial es más restrictiva que la de módulo, por lo que todo \nolbreaks{$C$-espacio vectorial} donde $C$ es un cuerpo es también un \modulo{C}. Por ejemplo, $\reales^n$ es un \modulo{\reales}. + %La definición de espacio vectorial es más restrictiva que la de módulo, por lo que todo \nolbreaks{$C$-espacio vectorial} donde $C$ es un cuerpo es también un \modulo{C}. Por ejemplo, $\racionales^n$ es un \modulo{\racionales}. A lo largo de este trabajo será de gran importancia el anillo de polinomios $\anillodepolinomios{K}{x_1, x_2, \dots, x_n}$ sobre un cuerpo $K$, al que denotaremos \anillodepolinomios{K}{\mathbf{x}}. Sobre todo porque a partir de este construimos el conjunto de vectores de polinomios $\anillodepolinomios{K}{\mathbf{x}}^n$, que es un \modulo{\anillodepolinomios{K}{\mathbf{x}}}. \end{ejemplo} @@ -341,10 +343,10 @@ urlcolor=black]{hyperref} \begin{ejemplo} Si pensamos en \enteros{} como un \modulo{\enteros}, $2\enteros$ es un \submodulo{\enteros}. Esto debería recordarnos a los ideales de \enteros. De hecho, un ideal de un anillo $A$ es un \submodulo{A} del propio $A$. - Consideremos el \modulo{\anillodepolinomios{\reales}{x, y}} $\anillodepolinomios{\reales}{x, y}^2$ y tomemos un par de elementos, como pueden ser $\formatovector{u} := \vectordos{x^2y + xy^2}{x + y}$ y $\formatovector{v} := \vectordos{y^3}{x^3}$. El conjunto de combinaciones lineales de estos vectores, + Consideremos el \modulo{\anillodepolinomios{\racionales}{x, y}} $\anillodepolinomios{\racionales}{x, y}^2$ y tomemos un par de elementos, como pueden ser $\formatovector{u} := \vectordos{x^2y + xy^2}{x + y}$ y $\formatovector{v} := \vectordos{y^3}{x^3}$. El conjunto de combinaciones lineales de estos vectores, \begin{equation*} - \engendrado{\formatovector{u}, \formatovector{v}}_{\anillodepolinomios{\reales}{x, y}} := \definicionconjunto{a \formatovector{u} + b\formatovector{v}}{a, b \in \anillodepolinomios{\reales}{x, y}} = \definicionconjunto{\vectordos{a\left( x^2y + xy^2 \right) + by^3}{a\left( x + y \right) + bx^3}}{a, b \in \anillodepolinomios{\reales}{x, y}}. - \end{equation*} es un \submodulo{\anillodepolinomios{\reales}{x, y}}. De hecho, a este tipo de módulos los llamaremos submódulos generados por $\mathbf{u}$ y $\mathbf{v}$. + \engendrado{\formatovector{u}, \formatovector{v}}_{\anillodepolinomios{\racionales}{x, y}} := \definicionconjunto{a \formatovector{u} + b\formatovector{v}}{a, b \in \anillodepolinomios{\racionales}{x, y}} = \definicionconjunto{\vectordos{a\left( x^2y + xy^2 \right) + by^3}{a\left( x + y \right) + bx^3}}{a, b \in \anillodepolinomios{\racionales}{x, y}}. + \end{equation*} es un \submodulo{\anillodepolinomios{\racionales}{x, y}}. De hecho, a este tipo de módulos los llamaremos submódulos generados por $\mathbf{u}$ y $\mathbf{v}$. \end{ejemplo} \begin{definicion} Sea $N$ un \submodulo{A} de $M$ y $G$ un subconjunto cualquiera de $N$. Diremos que $N$ \emph{está generado por} $G$ y lo denotaremos como $N = \engendrado{G}_A$ si @@ -441,7 +443,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. Sea $M$ un \modulo{A} noetheriano. Los submódulos y módulos cocientes de $M$ son también noetherianos. \end{lema} \begin{lema} - Que $M$ sea noetheriano es equivalente a que toda cadena ascendente de submódulos $N_1 \subseteq N_2 \subseteq \dots$ sea eventualmente estacionaria. + Que $M$ sea noetheriano es equivalente a que toda cadena ascendente de submódulos $N_1 \subseteq N_2 \subseteq \dots$ llega a ser estacionaria en algún momento. \end{lema} \begin{definicion} @@ -478,7 +480,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \begin{definicion} Diremos que $M$ es un \modulo{A} \emph{reflexivo} si $\Psi$ es un isomorfismo. \end{definicion} - \begin{ejemplo} + \begin{ejemplo} \label{Ejemplo Espacio Vectorial es Reflexivo} Veamos que todo espacio vectorial de dimensión finita es reflexivo. Sea $V$ un $K$-espacio vectorial de dimensión $n$ y con base $\conjunto{v_1, v_2, \dots, v_n}$. El conjunto \dual{V} admite como base a la base dual compuesta por los $\mathbf{e}^*_{i}$ definidos como \begin{equation*} \mathbf{e}^{*}_{i}\!\left( \vectorbasecanonica{j} \right) := \begin{cases} @@ -486,7 +488,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. 0 & \text{si no.} \end{cases} \end{equation*} - Como una base de \dual{V} tiene tantos elementos como una de $V$, $V$ y \dual{V} deben ser isomorfos y, por ende, \dual{V} y \dualdual{V} también. Así pues, $V$ y \dualdual{V} son isomorfos. Pero para verificar que $V$ sea reflexivo debemos cerciorarnos de que la aplicación $\Psi$ sea un isomorfismo. Para ello, supongamos que $\Psi\!\left( v \right) = \Psi\!\left( v' \right)$; entonces, como $\mathrm{id} \in \dual{V}$, $\mathrm{id}\!\left( v \right) = \mathrm{id}\!\left( v' \right)$ por lo que $\Psi$ es inyectiva, lo cual ya nos garantiza que sea un isomorfismo. + Como una base de \dual{V} tiene tantos elementos como una de $V$, $V$ y \dual{V} deben ser isomorfos y, por ende, \dual{V} y \dualdual{V} también. Así pues, $V$ y \dualdual{V} son isomorfos. Pero para verificar que $V$ sea reflexivo debemos cerciorarnos de que la aplicación $\Psi$ sea un isomorfismo. Para ello, supongamos que $\Psi\!\left( v \right) = \Psi\!\left( v' \right)$; entonces, como $\mathrm{id} \in \dual{V}$, $\mathrm{id}\!\left( v \right) = \mathrm{id}\!\left( v' \right)$ por lo que $\Psi$ es inyectiva, lo cual ya nos garantiza que sea un isomorfismo. Este ejemplo aplica también a módulos libres de rango finito. \end{ejemplo} \begin{ejemplo} Consideremos ahora el \modulo{\enteros} \cociente{\enteros}{p\enteros} donde $p$ es un número primo. Sabemos que $\cociente{\enteros}{p\enteros} = \engendrado{1}$; por lo tanto, cualquier $\phi \in \dual{\left( \cociente{\enteros}{p\enteros} \right)}$ está unívocamente determinado por la imagen de $1$. Supongamos que $\phi\!\left( 1 \right) = x \in \enteros$, se debe cumplir que @@ -495,6 +497,39 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \end{equation*} Es decir, que $\dual{\left( \cociente{\enteros}{p\enteros} \right)} = \conjunto{\dual{\mathbf{0}}}$ donde $\dual{\mathbf{0}}$ envía cualquier elemento de \cociente{\enteros}{p\enteros} al $0$ de \enteros. Al estar \dual{\left( \cociente{\enteros}{p\enteros} \right)} solo compuesto por la aplicación que todo lo lleva al cero, su dual, $\dualdual{\left( \cociente{\enteros}{p\enteros} \right)}$, solo estará compuesto por la aplicación \dualdual{\mathbf{0}} que envía todo al $0$ de \enteros{} también. Al ser todos estos conjuntos finitos, podemos ver que la cardinalidad de \dualdual{\left( \cociente{\enteros}{p\enteros} \right)} y \cociente{\enteros}{p\enteros} no coinciden y por lo tanto no pueden ser isomorfos, impidiendo esto a su vez que \cociente{\enteros}{p\enteros} sea un \modulo{\enteros} reflexivo. \end{ejemplo} + + + \begin{wrapfigure}[7]{r}{0.3\textwidth} + \vspace{-1.2cm} + \begin{equation*} + \begin{tikzcd} + M_\mathrm{Q} \times \left( \dual{M} \right)_\mathrm{Q} \arrow[r, "\Phi^{M_\mathrm{Q}}"] & \cuerpofracciones{\Kx} & \\ + M \hspace{0.05cm}\times \hspace{0.4cm}\dual{M} \arrow[u, shift left=6mm] \arrow[u, shift right=6mm] \arrow[r, "\Phi^{M}"] & \Kx \arrow[u] & \\ + 0 \hspace{17pt} \phantom{. \times } 0\arrow[u, shift left=6mm] \arrow[u, shift right=6mm] & 0 \arrow[u] & + \end{tikzcd} + \end{equation*} + \caption{Diagrama de aplicaciones duales.} \label{Diagrama Dual Reflexivo} + \end{wrapfigure} + + \begin{lema} + Si $M$ es un \modulo{\Kx} finitamente generado, entonces \dual{M} es reflexivo. + \end{lema} + \begin{proof} + Para cualquier \modulo{A} $B$, denotaremos $B_\mathrm{Q} := B \otimes \cuerpofracciones{A}$ (véase la Definición \ref{Definición Extensión de Escalares}). Consideremos el diagrama de homomorfismos de \modulo{\Kx} de la figura \ref{Diagrama Dual Reflexivo}. Las inyecciones que aquí mostramos son las naturales: en el caso de las de la izquierda, mandan los elementos $m$ bien de $M$ o \dual{M} a los correspondientes $m \otimes 1$; y en el de la derecha se manda $p$ a $\frac{p}{1}$. Estas son inyecciones por el Lema \ref{Lema Extensión es Inyección}. + + Podemos ver los conjuntos de la parte baja del diagrama como subconjuntos de los de arriba. Así, pensaremos en $\dual{M}$ como en + \begin{equation*} + \definicionconjunto{h \in \left( \dual{M} \right)_\mathrm{Q}}{h\!\left( m \right) \in A \ \forall m \in M}, + \end{equation*} + o en otras palabras, como todos aquellos homomorfismos en el conjunto ampliado a las fracciones que <> a $M$ funcionan como un elemento de \dual{M}. + Ahora recordemos por el \ref{Ejemplo Espacio Vectorial es Reflexivo} que $M_\mathrm{Q}$ es reflexivo así que $\dualdual{M_Q} \simeq M_Q$. Esto nos permite ver \dualdual{M} como + \begin{equation*} + \definicionconjunto{m \in M_\mathrm{Q}}{ \phi(m) \in A \ \forall \phi \in \dual{M}}. + \end{equation*} + + \todo[inline]{Esto terminarlo.} + \end{proof} + La reflexividad jugará un papel importante en la demostración del Criterio de Saito que ilustramos en el esquema de \eqref{Esquema Demostración}. Más propiedades acerca de los módulos reflexivos pueden encontrarse en \cite[\href{https://stacks.math.columbia.edu/tag/0AUY}{Tag 0AUY}]{STACKS}. En ocasiones, encadenaremos homomorfismos y podemos necesitar expresar las relaciones entre las imágenes y los núcleos de las distintas aplicaciones. Para expresar esto, utilizaremos las sucesiones exactas. @@ -600,8 +635,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. Una introducción a los módulos proyectivos puede encontrarse en \cite[sección 1.2]{LIBROPROYECTIVOS}. \section{Producto Exterior} - \todo{Si haces lo de la bilineal lo puedes usar aquí} - De cara a la demostración del Criterio de Saito, utilizaremos una operación que nos será de gran ayuda: el producto exterior. Para entender esta, necesitaremos entender primero el producto tensorial. En esencia, el producto tensorial aparece en el siguiente contexto: supongamos que $M$, $N$ y $O$ son \modulo{A}s y tenemos una aplicación $b : M \times N \rightarrow O$ que es bilineal, o sea, que $b\left( \bullet, n \right)$ y $b\left( m, \bullet \right)$ son homorfismos de \modulo{A}s para cualquier $m \in M$ y $n \in N$; 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 que para cada $\left( m, n \right)$ en $M \times N$ exista un elemento $m * n$ en $E$ de forma que $b'\left( m*n \right) = b\left( m, n \right)$. Esta es la motivación de la construcción del producto tensorial. + 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. \begin{definicion} Sea $A$ un anillo y $M$ y $N$ dos \modulo{A}s. Tomemos como $E$ al \modulo{A} definido como $\bigoplus_{\substack{m \in M \\ n \in N}} A \cdot \vectorbasecanonica{m, n}$ donde \vectorbasecanonica{m, n} debe entenderse como la tupla $\left( m, n \right)$ sin ningún tipo de operación asociada, es decir, no debemos pensar que $\vectorbasecanonica{m, n} + \vectorbasecanonica{m', n'} = \vectorbasecanonica{m+m', n+n'}$ al igual que no pensamos que $\euno + \edos = \etres$. Ahora, consideremos dentro de $E$ el \submodulo{A} $S$ definido por \begin{equation*} @@ -612,11 +646,20 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \vectorbasecanonica{m, an} - a\vectorbasecanonica{m, n} \end{gathered} \right\rvert a \in A, m \in M, n \in N}. \end{equation*} - Definimos el p\emph{roducto tensorial} de $M$ por $N$ como $M \otimes N := \cociente{E}{S}$ y denotaremos a la clase de equivalencia de cada $\vectorbasecanonica{m, n}$ como $m \otimes n$ - \todo{Propiedad universal} + Definimos el p\emph{roducto tensorial} de $M$ por $N$ como $M \otimes N := \cociente{E}{S}$ y denotaremos a la clase de equivalencia de cada $\vectorbasecanonica{m, n}$ como $m \otimes n$. \end{definicion} + Ahora sí, tenemos la siguiente propiedad universal que buscábamos al principio: para todo mapa bilineal $b$ podemos encontrar otro mapa lineal $l: M \otimes N \to P$ de forma que $b\!\left( m, n \right) = l\!\left( m\otimes n \right)$. + \begin{figure}[h] + \begin{equation*} + \begin{tikzcd} + M \otimes N \arrow[rd, "l"]& & \\ + M \times N \arrow[r, "b"] \arrow[u]& P + \end{tikzcd} + \end{equation*} + \end{figure} + \vspace{-0.7cm} \begin{ejemplo} - Quizás el ejemplo más natural sea el producto escalar. Así pues, tomemos la aplicación $\productoescalarantilambda{\bullet}{\bullet} : \reales^3 \times \reales^3 \rightarrow \reales$, que es bilineal. En este caso, todo elemento de $\reales^3 \otimes \reales^3$ es de la siguiente forma ---entendiendo $S$ como en la definición de antes---: + Quizás el ejemplo más natural sea el producto escalar. Así pues, tomemos la aplicación $\productoescalarantilambda{\bullet}{\bullet} : \racionales^3 \times \racionales^3 \rightarrow \racionales$, que es bilineal. En este caso, todo elemento de $\racionales^3 \otimes \racionales^3$ es de la siguiente forma ---entendiendo $S$ como en la definición de antes---: \begin{equation}\label{Desglose Producto Tensorial} \begin{aligned} \vectortres{a}{b}{c} \otimes \vectortres{d}{e}{f} + S &= \left( a\euno + b\edos + c\etres \right) \otimes \left( d\euno + e\edos + f\etres \right) + S \\ @@ -625,7 +668,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \end{equation} (De aquí en adelante omitiremos los <<$+S$>> o similares sumas por el \submodulo{A} que estemos haciendo el cociente.) - 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, + Vemos pues que \definicionconjunto{\vectorbasecanonica{i} \otimes \vectorbasecanonica{j}}{ i, j \in \conjunto{1, 2, 3}} es una base de $\racionales^3 \otimes \racionales^3$. Por lo tanto, para encontrar una aplicación lineal $\phi : \racionales^3 \otimes \racionales^3 \rightarrow \racionales$ 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 $i = j$;}\\ @@ -641,11 +684,11 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \begin{equation*} T = \engendrado{m \otimes m' - m' \otimes m \vert m, m' \in M}. \end{equation*} - Definiremos el \emph{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 $m \wedge m'$.\todo{Otra propiedad universal} + Definiremos el \emph{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 $m \wedge m = 0$. + Nótese que ahora $m \wedge m = 0$ y que en este caso tenemos una propiedad universal similar a la del producto tensorial pero con aplicaciones bilineales alternadas. \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}. + El ejemplo natural en el caso de aplicaciones bilineales alternadas es el producto vectorial $\bullet \times \bullet : \racionales^3 \times \racionales \rightarrow \racionales^3$. 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; \\ @@ -653,7 +696,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. &= 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 + Vemos por lo tanto que $\racionales^3 \wedge \racionales^3$ 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*} @@ -664,14 +707,14 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. 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 + Consideremos $\racionales^3 \wedge \racionales^3 \wedge \racionales^3$. 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. + Vemos, por lo tanto, que $\racionales^3\wedge\racionales^3\wedge\racionales^3$ es en realidad $\racionales\left( \euno\wedge\edos\wedge\etres \right)$, un espacio isomorfo a \racionales. \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} @@ -714,10 +757,22 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \end{definicion} El cuerpo de fracciones de un anillo nos permite llevar un \modulo{A} a un \espaciovectorial{\cuerpofracciones{A}}. - \begin{definicion} + \begin{definicion} \label{Definición Extensión de Escalares} Llamaremos extensión de escalares del \modulo{A} $M$ al conjunto $M \otimes \cuerpofracciones{A}$. \end{definicion} - \begin{lema} + \begin{definicion} + Sea $S$ un subconjunto de $A$ multiplicativamente cerrado. Definimos la \emph{localización de $M$ con respecto a $S$} como el \modulo{A} + \begin{equation*} + S^{-1}M := \definicionconjunto{\frac{m}{s}}{m \in M, s \in S}, + \end{equation*} + donde $\frac{m}{s}$ denota la clase de equivalencia de $\left( m, s \right) \in M \times S$ con respecto a la relación de equivalencia\footnote{La definición de localización es, en realidad, más general: \begin{equation*} + \left( m, s \right) \sim \left( m', s' \right) \iff \exists\sigma \in S \ \text{tal que} \ \sigma\left( ms' - m's \right) = 0. + \end{equation*} Pero en para nosotros, como asumimos que no hay divisores del $0$, nuestra definición es suficiente.} + \begin{equation*} + \left( m, s \right) \sim \left( m', s' \right) \iff ms' = m's. + \end{equation*} + \end{definicion} + \begin{lema} \label{Lema Extensión es Inyección} 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} @@ -731,7 +786,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. &\implies& am \neq 0 & \end{array} \end{equation*} - \todo{La otra implicación.} + Vemos también gracias a esto que todo elemento de $M \otimes \cuerpofracciones{A}$ puede escribirse como $\mu \otimes \frac{1}{q}$. Consideremos el homomorfismo de $M \otimes \cuerpofracciones{A}$ en $\left( A \setminus \left\lbrace 0 \right\rbrace \right)$ dado por $\mu \otimes \frac{1}{q} \mapsto \frac{\mu}{q}$. Vemos por ella que $\frac{\mu}{q} = 0$ si y solo si $\mu = 0$; es decir, que $\mu \otimes \frac{1}{q} = 0$ si y solo si $\mu = 0$. Esto demuestra que la última implicación de la cadena anterior funciona también en el sentido contrario, completando la demostración. \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 $r$. @@ -776,8 +831,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. Veamos cómo se afronta este problema a distintos niveles. \begin{ejemplo} \label{Ejemplo Pertenencia Ideal Una Variable} -\todo{Es una tontería, pondría los racionales en lugar de los reales, también para otros ejemplos.} - Consideremos el \modulo{\reales} \anillodepolinomios{\reales}{x} ---solo una variable--- y a \nolbreaks{$f := x^{3} - 6x^{2} + 11x - 6$} y \nolbreaks{$g := x^{3} - 9x^{2} + 26x - 24$} dos elementos de este. Queremos determinar si un polinomio cualquiera, por ejemplo \nolbreaks{$p = 3x^7 + 4x^5 - 3x^2 + x + 1$}, pertenece al submódulo ---o en este caso, ideal--- $\engendrado{f, g}$. Como \anillodepolinomios{\reales}{x} es un dominio euclídeo, es dominio de ideales principales y, por lo tanto, + Consideremos el \modulo{\racionales} \anillodepolinomios{\racionales}{x} ---solo una variable--- y a \nolbreaks{$f := x^{3} - 6x^{2} + 11x - 6$} y \nolbreaks{$g := x^{3} - 9x^{2} + 26x - 24$} dos elementos de este. Queremos determinar si un polinomio cualquiera, por ejemplo \nolbreaks{$p = 3x^7 + 4x^5 - 3x^2 + x + 1$}, pertenece al submódulo ---o en este caso, ideal--- $\engendrado{f, g}$. Como \anillodepolinomios{\racionales}{x} es un dominio euclídeo, es dominio de ideales principales y, por lo tanto, \begin{equation*} \engendrado{f, g} = \engendrado{\mcd{f, g}} = \engendrado{x^2 - 5x + 6}. \end{equation*} @@ -785,16 +839,16 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \begin{equation*} p = \underbrace{\left( 3x^5 + 15x^4 + 61x^3 + 215x^2 + 709x + 2252 \right)}_{\text{cociente $q$}} \underbrace{\left( x^2 - 5x + 6 \right)}_{\text{divisor $d$}} + \underbrace{7007x - 13511}_{\text{resto $r$}}. \end{equation*} - Por el Teorema de la División, sabemos que este resto es el único polinomio de \anillodepolinomios{\reales}{x} que satisface $p = q d + r$ con grado menor que $d$. Por lo tanto, al no ser $0$, podemos afirmar que $p$ no está en \engendrado{f,g}. + Por el Teorema de la División, sabemos que este resto es el único polinomio de \anillodepolinomios{\racionales}{x} que satisface $p = q d + r$ con grado menor que $d$. Por lo tanto, al no ser $0$, podemos afirmar que $p$ no está en \engendrado{f,g}. \end{ejemplo} - Este ejemplo también ilustra un cambio de perspectiva que será muy importante a lo largo de toda esta sección: la división deja de ser importante por los valores exactos que arroja y su relevancia es más bien cualitativa. Con esto nos referimos a que, como tal, los valores concretos del cociente y el resto no tienen ninguna relevancia a la hora de determinar si $p$ pertenece o no a \engendrado{f, g}. Conocidas las propiedades de este algoritmo de la división en \anillodepolinomios{\reales}{x}, solo nos interesa si el resto es $0$ o no, puesto que es lo que determina si $p$ pertenece o no a \engendrado{f, g}. En base a esto, nuestra motivación para la división debe cambiar de ser <> a un más abstracto <>. \todo{En realidad con este lenguaje, $r$ es importante ya que determina la clase de $p$ en el cociente.} + Este ejemplo también ilustra un cambio de perspectiva que será muy importante a lo largo de toda esta sección: la división deja de ser importante por los valores exactos que arroja y su relevancia es más bien cualitativa. Con esto nos referimos a que, como tal, el valor concreto del cociente no tienen ninguna relevancia a la hora de determinar si $p$ pertenece o no a \engendrado{f, g} y el del resto nos importa ya que nos da la clase de equivalencia, pero si hubiese más de un posible representante, daría igual cuál nos fuese devuelto. En concreto, en \anillodepolinomios{\racionales}{x}, solo nos interesa si el resto es $0$ o no, puesto que es lo que determina si $p$ pertenece o no a \engendrado{f, g}. En base a esto, nuestra motivación para la división debe cambiar de ser <> a un más abstracto <>. Visto así, el Ejemplo \ref{Ejemplo Pertenencia Ideal Una Variable} supone una situación idílica: por ser un dominio de ideales principales, sabemos que vamos a poder representar a cualquier conjunto generador con un solo elemento; por lo tanto, sobre cualquier conjunto generador siempre habrá definida una división, que consistirá en dividir como ya conocemos por el elemento generador. Además, por el Teorema de la División, conocemos que el representante $r$ que satisfaga $\grado r < \grado d$ es único, lo cual nos permite discernir si $p$ está en \engendrado{f, g} viendo si $r = 0$ o no. En espacios más complicados, muchas de estas buenas propiedades desaparecen e impiden extender el proceso anterior. \begin{description}[wide, leftmargin=0pt,itemindent=6pt] - \item[Más variables:] Consideremos ahora el anillo de polinomios $\anillodepolinomios{\reales}{x, y, z}$, un elemento $f: = 4x^2y+ 12xy^2 + \frac{5}{2}xz^3 + x^2yz$ y el ideal $I := \engendrado{f}$. Si bien este espacio no tiene por qué ser un dominio de ideales principales, al solo haber un generador podemos obviar de momento esa parte. Siguiendo la línea del Ejemplo~\ref{Ejemplo Pertenencia Ideal Una Variable}, para ver si $p \in \anillodepolinomios{\reales}{x, y, z}$ deberíamos dividir por $f$ y ver si el resto es $0$ o no. Pero llegados a este punto, ¿cómo deberíamos dividir? Podemos intentar imitar el algoritmo usual de la división, pero este necesita siempre una noción de orden entre los términos del divisor y también del dividendo. En este caso, ¿deberíamos tomar $4x^2y$ como <> de $f$ o igual sería mejor tomar a uno de mayor grado como $\frac{5}{2}xz^3$? Pero $\frac{5}{2}xz^3$ y $x^2yz$ tienen el mismo grado, ¿cuál escoger, entonces? La elección de este <> en dividendo y divisor cambiará el funcionamiento del algoritmo, así que debemos cerciorarnos de cuáles de estas opciones son válidas y también revisar si el algoritmo tradicional adaptado de esta manera sirve su propósito. - \item[Más generadores:] Sin cambiar de espacio, consideremos ahora el ideal $I := \engendrado{x + y, xyz, x^2 + y^2 + z^2}$. En este caso, el que \anillodepolinomios{\reales}{x, y, z} no sea un dominio de ideales principales es un problema. ¿Cómo podemos dividir siquiera? Otra dificultad a la que nos enfrentamos aquí una vez hayamos entendido cómo dividir será saber si nuestro sistema de generadores es compatible con la división ya que, por ejemplo, a $x^2 + y^2 + z^2$ se le puede restar $\left( x+y \right)^2$ e $I$ podría expresarse como $\engendrado{x+y, xyz, -2xy + z^2}$. ¿Sirven ambas representaciones? + \item[Más variables:] Consideremos ahora el anillo de polinomios $\anillodepolinomios{\racionales}{x, y, z}$, un elemento $f: = 4x^2y+ 12xy^2 + \frac{5}{2}xz^3 + x^2yz$ y el ideal $I := \engendrado{f}$. Si bien este espacio no tiene por qué ser un dominio de ideales principales, al solo haber un generador podemos obviar de momento esa parte. Siguiendo la línea del Ejemplo~\ref{Ejemplo Pertenencia Ideal Una Variable}, para ver si $p \in \anillodepolinomios{\racionales}{x, y, z}$ deberíamos dividir por $f$ y ver si el resto es $0$ o no. Pero llegados a este punto, ¿cómo deberíamos dividir? Podemos intentar imitar el algoritmo usual de la división, pero este necesita siempre una noción de orden entre los términos del divisor y también del dividendo. En este caso, ¿deberíamos tomar $4x^2y$ como <> de $f$ o igual sería mejor tomar a uno de mayor grado como $\frac{5}{2}xz^3$? Pero $\frac{5}{2}xz^3$ y $x^2yz$ tienen el mismo grado, ¿cuál escoger, entonces? La elección de este <> en dividendo y divisor cambiará el funcionamiento del algoritmo, así que debemos cerciorarnos de cuáles de estas opciones son válidas y también revisar si el algoritmo tradicional adaptado de esta manera sirve su propósito. + \item[Más generadores:] Sin cambiar de espacio, consideremos ahora el ideal $I := \engendrado{x + y, xyz, x^2 + y^2 + z^2}$. En este caso, el que \anillodepolinomios{\racionales}{x, y, z} no sea un dominio de ideales principales es un problema. ¿Cómo podemos dividir siquiera? Otra dificultad a la que nos enfrentamos aquí una vez hayamos entendido cómo dividir será saber si nuestro sistema de generadores es compatible con la división ya que, por ejemplo, a $x^2 + y^2 + z^2$ se le puede restar $\left( x+y \right)^2$ e $I$ podría expresarse como $\engendrado{x+y, xyz, -2xy + z^2}$. ¿Sirven ambas representaciones? \item[Más <>:] Por último, también debemos tener en cuenta que por lo general trabajaremos en $\Kx^r$, lo cual hará que los elementos tengan más de una componente. ¿Cómo integrar las componentes en las divisiones? \end{description} @@ -839,7 +893,7 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \end{equation*} Este orden monomial es el \nolbreaks{graduado-lexicográfico} y se denota por \mayorquedeglex. \end{ejemplo} - Para poder generalizar estos órdenes a un módulo como $\Kx^r$ primero tenemos que aclarar qué es un monomio en este anillo. La elección es bastante natural cuando consideramos que un elemento de, por ejemplo, $\anillodepolinomios{\reales}{x, y}^3$ como puede ser $\begin{psmallmatrix} + Para poder generalizar estos órdenes a un módulo como $\Kx^r$ primero tenemos que aclarar qué es un monomio en este anillo. La elección es bastante natural cuando consideramos que un elemento de, por ejemplo, $\anillodepolinomios{\racionales}{x, y}^3$ como puede ser $\begin{psmallmatrix} 2xy + y^2 \\ x^3 \\ y \end{psmallmatrix}$ puede escribirse como $2xy\euno + y^2\euno + x^3\edos + y\etres$. \begin{definicion} @@ -897,7 +951,7 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición \end{definicion} La única novedad de este algoritmo con respecto al que ya conocemos de polinomios en una sola variable es la instrucción de la línea 6. En ella, le sustraemos a $D$ su término líder, lo añadimos al resto y repetimos todo el proceso otra vez con el resto de términos de $D$. Esto lo hacemos para reducir todo lo posible el resto, garantizando que ninguno de sus términos sea divisible por \LM{g}. De no hacer esto, solo podríamos garantizar que, al final de la ejecución, $\LT{D}$ no es divisible por $\LT{g}$. \begin{ejemplo} - Sean $f := 3x^4\euno + 2x^3\euno + 2x^2y\edos + x^2\euno$ y $g := x^2\euno + 2y\etres$ dos elementos de $\anillodepolinomios{\reales}{x, y}^3$ con el orden monomial \mayorquedeglexpos. Aunque hayamos expresado en la definición todo de una forma muy <>, en esencia, lo que hacemos es bastante similar a una división usual: + Sean $f := 3x^4\euno + 2x^3\euno + 2x^2y\edos + x^2\euno$ y $g := x^2\euno + 2y\etres$ dos elementos de $\anillodepolinomios{\racionales}{x, y}^3$ con el orden monomial \mayorquedeglexpos. Aunque hayamos expresado en la definición todo de una forma muy <>, en esencia, lo que hacemos es bastante similar a una división usual: \begin{center} \begin{blockarray}{rrrrrrrcl} \begin{block}{rrrrrrrc|l} @@ -943,9 +997,9 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición \Until{$D = 0$} \end{algorithmic} \end{definicion} - Un ejemplo como el de \eqref{Ejemplo División Compuesta} muestra cómo funciona este algoritmo. Ahora, haría falta generalizar todavía más para ser capaces también de dividir por un número infinito de divisores, pero los siguientes resultado nos permitirán ver que esto no es necesario. + Un ejemplo como el de \eqref{Ejemplo División Compuesta} muestra cómo funciona este algoritmo. Ahora, haría falta generalizar todavía más para ser capaces también de dividir por un número infinito de divisores, pero los siguientes resultado nos permitirán ver que esto no es necesario. El primero de estos, el Teorema de la base de Hilbert, nos garantiza el buen comportamiento de los anillos de polinomios en el sentido de ser noetherianos, lo cual jugará un papel fundamental en el funcionamiento de las bases de \Groebner. \begin{teorema}[de la base de Hilbert] - Sea $A$ un anillo noetheriano. Entonces el anillo de polinomios \anillodepolinomios{A}{x_1, \dots, x_n} es también noetheriano. \todo{Di algo más.} + Sea $A$ un anillo noetheriano. Entonces el anillo de polinomios \anillodepolinomios{A}{x_1, \dots, x_n} es también noetheriano. \end{teorema} \begin{proposicion} \label{Proposición Kx^r noetheriano}%Proposición 2.1.29 de Singular. Sea $A$ un anillo noetheriano y $M$ un \modulo{A} finitamente generado. Entonces, $M$ es un \modulo{A} noetheriano. @@ -954,7 +1008,7 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición Con respecto a los problemas que se presentaban al principio, ya hemos resuelto varios: tenemos un orden para los monomios que nos permite tener un convenio de cuáles son los términos principales de los polinomios, hemos integrado las distintas componentes de un vector de $\Kx^r$ al concepto de monomio y sabemos cómo dividir por ellas, y también sabemos cómo afrontar la división con más de un divisor. Sin embargo, todavía podemos encontrarnos con casos como el siguiente. \begin{ejemplo} \label{Ejemplo Fallo División} - Supongamos que $N := \engendrado{x\euno + y\edos, y\euno + x\edos}$ es un \submodulo{\anillodepolinomios{\reales}{x,y}} de $\anillodepolinomios{\reales}{x, y}^2$ y queremos saber si $f := x^2\edos - y^2\edos$ pertenece a $N$. Al aplicar el algoritmo de la división siguiendo el orden \mayorqueposlex{} de \ref{Algoritmo de la División Compuesta} nos encontramos que ninguno de los dos generadores de $N$ puede reducir a $f$, por lo tanto el resto es el propio $f$. Como el resto es distinto de $0$, estaríamos tentados a decir que $f$ no pertenece a $N$, pero démonos cuenta de que $f = -y\left( x\euno + y\edos \right) + x\left( y\euno + x\edos \right)$ y, por ende, sí que pertenece. + Supongamos que $N := \engendrado{x\euno + y\edos, y\euno + x\edos}$ es un \submodulo{\anillodepolinomios{\racionales}{x,y}} de $\anillodepolinomios{\racionales}{x, y}^2$ y queremos saber si $f := x^2\edos - y^2\edos$ pertenece a $N$. Al aplicar el algoritmo de la división siguiendo el orden \mayorqueposlex{} de \ref{Algoritmo de la División Compuesta} nos encontramos que ninguno de los dos generadores de $N$ puede reducir a $f$, por lo tanto el resto es el propio $f$. Como el resto es distinto de $0$, estaríamos tentados a decir que $f$ no pertenece a $N$, pero démonos cuenta de que $f = -y\left( x\euno + y\edos \right) + x\left( y\euno + x\edos \right)$ y, por ende, sí que pertenece. \end{ejemplo} Para entender bien lo que ocurre en este ejemplo, necesitamos el siguiente objeto. \begin{definicion} @@ -1004,7 +1058,7 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición Con todo esto, ya tenemos una solución al problema de la pertenencia de un elemento a un submódulo: obtener la base de Gröbner del submódulo y dividir por ella para ver si el resto es nulo o no. Sin embargo, no tenemos garantías de que existan bases de Gröbner para todos los submódulos y, en caso de que sí las haya, tampoco sabemos cómo calcularlas. - Recordemos el problema que nos habíamos encontrado en el Ejemplo~\ref{Ejemplo Fallo División}. En este ejemplo, había monomios líderes ocultos en los términos de los generadores. Para desenmascararlos, lo que haremos será lo que vimos en ese ejemplo: anular los términos libres para forzar a que estos aparezcan. Para eso, dados $f = x^\mathbf{\alpha}\vectorbasecanonica{i}$ y $g = x^\mathbf{\alpha}\vectorbasecanonica{j}$ elementos de $\Kx^r$ y $a$ y $b$ elementos de \Kx{} de forma que $a\LM{f} = b\LM{g} = \mathrm{mcm}\left(\LM{f}, \LM{g}\right)$\footnote{Aquí podría haber un pequeño malentendido con la noción de mínimo común múltiplo: por ejemplo, ¿cuál es el mínimo común múltiplo de $xy^2$ y $x^2y$ en \anillodepolinomios{\reales}{x, y}? Lo natural sería decir $x^2y^2$, que es nuestra convención, pero también sería válido decir $3x^2y^2$ ya que ambos tienen el mismo orden monomial. Por simplicidad, siempre nos referiremos a aquel que no está multiplicado por constante.}, definimos la siguiente expresión: + Recordemos el problema que nos habíamos encontrado en el Ejemplo~\ref{Ejemplo Fallo División}. En este ejemplo, había monomios líderes ocultos en los términos de los generadores. Para desenmascararlos, lo que haremos será lo que vimos en ese ejemplo: anular los términos libres para forzar a que estos aparezcan. Para eso, dados $f = x^\mathbf{\alpha}\vectorbasecanonica{i}$ y $g = x^\mathbf{\alpha}\vectorbasecanonica{j}$ elementos de $\Kx^r$ y $a$ y $b$ elementos de \Kx{} de forma que $a\LM{f} = b\LM{g} = \mathrm{mcm}\left(\LM{f}, \LM{g}\right)$\footnote{Aquí podría haber un pequeño malentendido con la noción de mínimo común múltiplo: por ejemplo, ¿cuál es el mínimo común múltiplo de $xy^2$ y $x^2y$ en \anillodepolinomios{\racionales}{x, y}? Lo natural sería decir $x^2y^2$, que es nuestra convención, pero también sería válido decir $3x^2y^2$ ya que ambos tienen el mismo orden monomial. Por simplicidad, siempre nos referiremos a aquel que no está multiplicado por constante.}, definimos la siguiente expresión: \begin{equation*} \spoly{f}{g} = \begin{cases} \frac{a}{\LC{f}}f - \frac{b}{\LC{g}}g &\text{si $i = j$;} \\ @@ -1012,7 +1066,7 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición \end{cases} \end{equation*} \begin{ejemplo} - Sean $f = 3x^2y\euno + 2x\edos$ y $g = 13x\euno + y^5\edos$ elementos de $\anillodepolinomios{\reales}{x, y}^2$ con el orden \mayorquelexpos. Como $\LM{f} = x^2y\euno$ y $\LM{g} = x\euno$ están en la misma componente ---\euno---, y $\mathrm{mcm}\left(\LM{f}, \LM{g}\right) = x^2y = \LM{f} = xy\LM{g}$, tenemos que + Sean $f = 3x^2y\euno + 2x\edos$ y $g = 13x\euno + y^5\edos$ elementos de $\anillodepolinomios{\racionales}{x, y}^2$ con el orden \mayorquelexpos. Como $\LM{f} = x^2y\euno$ y $\LM{g} = x\euno$ están en la misma componente ---\euno---, y $\mathrm{mcm}\left(\LM{f}, \LM{g}\right) = x^2y = \LM{f} = xy\LM{g}$, tenemos que \begin{equation*} \spoly{f}{g} = \frac{1}{3}\left( 3x^2y\euno + 2x\edos \right) + \frac{xy}{13} \left( 13x\euno + y^5\edos \right) = \frac{2}{3}x\edos + \frac{1}{13}xy^6\edos. \end{equation*} @@ -1057,13 +1111,12 @@ A^k\rar["\sigma"]&M\\ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2f_2 + \dots + a_kf_k \end{tikzcd} \end{equation} - + + Con la definición, ya podemos dar por respondida la primera pregunta que nos hacíamos en la introducción: ¿qué es \derivaciones{f}? \begin{definicion} Sean $A$ un anillo, $M$ un \modulo{A} y $f_1, f_2, \dots, f_k$ elementos de $M$. Denotaremos por $\syz{f_1, f_2, \dots, f_k}$ \emph{al \submodulo{A} de $A^k$ formado por todas las sicigias de} $f_1, f_2, \dots, f_k$. \end{definicion} - -\todo{Dale un poco más de bombo a este resultado} - Véase que \derivaciones{f}, tal y como está definido en \eqref{Definicion Modulo Derivaciones de f} puede expresarse en términos de sicigias: + Vemos que \derivaciones{f}, tal y como está definido en \eqref{Definicion Modulo Derivaciones de f} puede expresarse prácticamente como un módulo de sicigias: \begin{align*} \derivaciones{f} &= \definicionconjunto{a_1\partial_1 + a_2\partial_2 + \dots + a_n\partial_n}{a_1\partial_1f + a_2\partial_2f + \dots + a_n\partial_nf = pf, p \in \Kx}; \\ \intertext{recordemos que podemos ver $\partial_i$ como \vectorbasecanonica{i};} @@ -1253,31 +1306,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{equation*} \subsection{$\det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$ implica que \derivaciones{f} es libre} - - \begin{wrapfigure}[7]{r}{0.3\textwidth} - \begin{equation*} - \begin{tikzcd} - \left( \dual{M} \right)_\mathrm{Q} \times M_\mathrm{Q} \arrow[r, "\mathrm{eval}"] & \cuerpofracciones{\Kx} & \\ - \phantom{(}\dual{M}\phantom{)_\mathrm{Q}} \times M\hspace{5pt} \arrow[u, shift left=6mm] \arrow[u, shift right=6mm] \arrow[r, "\mathrm{eval}"] & \Kx \arrow[u] & \\ - 0 \hspace{17pt} \phantom{. \times } 0\arrow[u, shift left=6mm] \arrow[u, shift right=6mm] & 0 \arrow[u] & - \end{tikzcd} - \end{equation*} - \end{wrapfigure} - - Antes de embarcarnos en la otra implicación, necesitaremos un resultado auxiliar. - \begin{lema} - Si $M$ es un \modulo{\Kx} finitamente generado, entonces \dual{M} es reflexivo. - \end{lema} - \begin{proof} - Para cualquier \modulo{A} $B$, denotaremos $B_\mathrm{Q} := B \otimes \cuerpofracciones{A}$. Consideremos el diagrama de homomorfismos de \modulo{\Kx} que aparece a la derecha. Las inyecciones que aquí mostramos son las naturales: en el caso de las de la izquierda, mandan los elementos $m$ bien de $M$ o \dual{M} a los correspondientes $m \otimes 1$; y en el de la derecha se manda $p$ a $\frac{p}{1}$. Estas son inyecciones si y solo si $M$ es libre de torsión ya que en caso de que no, un elemento como $m \otimes 1 = fm\otimes \frac{1}{f}$ podría ser $0$ si $fm = 0$. Con el mapa $\mathrm{eval}$ cometemos un abuso de notación y lo reutilizamos en ambos casos aunque no sea la misma aplicación. Esta envía un par $\left( f, m \right)$ a $f\!\left( m \right)$ en ambos casos, es decir, evalúa la aplicación en el punto que se le dé. - - Podemos ver los conjuntos de la parte baja del diagrama como subconjuntos de los de arriba. Así, pensaremos en $\dual{M}$ como en - \begin{equation*} - \definicionconjunto{h \in \left( \dual{M} \right)_\mathrm{Q}}{\mathrm{eval}\!\left( h, m \right) \in \Kx \ \forall m \in M}, - \end{equation*} - o en otras palabras, como todos aquellos homomorfismos en el conjunto ampliado a las fracciones que <> a $M$ funcionan como un elemento de \dual{M}. - \todo[inline]{Esto terminarlo.} - \end{proof} + 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*} \begin{array}{rcccl} @@ -1309,8 +1338,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \delta_1 \wedge \delta_2 \wedge \dots \wedge \delta_{i-1} \wedge \delta \wedge \delta_{i+1} \wedge \dots \wedge \delta_{n-1} \wedge \delta_n \in \bigwedge^n N. \end{equation*} Percatémonos de que esta expresión, una vez eliminamos todos los productos exteriores nulos, queda como $\overline{b}_i \delta_1 \wedge \delta_2 \wedge \dots \wedge \delta_n \in \bigwedge^n N$, lo cual es solo posible si $\overline{b}_i \in \Kx$. Con esto concluimos que $\delta \in N$ y, por tanto, que $N = \derivaciones{f}$. - - \todo[inline]{Para poner un ejemplo espero a la introducción, para que case con el ejemplo que allí pongamos.} + \appendix \chapter{Un ejemplo completo de cálculo}\label{Apéndice Cálculo} From e160be7380d97f5d7cb59d9c0eb407fe513896e0 Mon Sep 17 00:00:00 2001 From: Enrique Artal Date: Wed, 11 Jun 2025 08:01:56 +0200 Subject: [PATCH 9/9] if --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 578343 -> 574944 bytes TFG_Sixto_José_Martínez_Gan.tex | 21 +++++++++++---------- 2 files changed, 11 insertions(+), 10 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index 5f08e07e91c994b888f4eefdb2c091fe4677dd33..d7e40d4c1ea3f6f503c53315dc58a973f05845fa 100644 GIT binary patch delta 496393 zcmYg%1yoeu7cQZoqzFiZh_v(o(jwgg(p^J?baM$sB&A~riJ^PwMmmPBAtfB7r5oP( z`@i+po3-YyIs5GMop0~`opWcMxl>rc@GFv`N*)X2txy6H1}gy7P*Gdre%7|7GcqAQ zKff_%XfV`|LqNLrJ|z^9&#vQuHy9FR5}Uwc70PxN#Lvk6 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