From fa9e605508dc461f3ca371e4cfb15ced9f9cf131 Mon Sep 17 00:00:00 2001 From: Sixto Date: Wed, 11 Jun 2025 15:50:05 +0200 Subject: [PATCH] Cambio de los productos tensoriales por fracciones. --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 580788 -> 580280 bytes TFG_Sixto_José_Martínez_Gan.tex | 36 +++++++++++++++----------------- 2 files changed, 17 insertions(+), 19 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index 9d88b5f4cafbd53d0fe343d98392bd48e083f3a8..3c05ffe8a5e96f3de7f9598e404c8b0ed5228380 100644 GIT binary patch delta 42445 zcmV(^K-Isrxg)r>BY=bfv;wx30x~d{aZ3X!m*JHH7Ju4FGAnh~ifg~wX@@>yJW6$O ztfMr^ukW`3kOW9j(mCU4GhMz&q&5NUEA|a!i}wq*`0B~Oyngc8m&zizlSO4Q5d zZhJg*5PzFCeEwJsu^3Z!^f-b&-bf(c?_UmA5U+k=HcR-hMp43 zd)(58_SbOttLCu!?)BHef~Qz0RadNbK++@N2<8ckSOZurxT>Twf`Hc@^jfmY>N-3l zDp_GdhE@GLeJ-2B`(uvE`4B$$Z>5o&^X^z$3V&MwrecwlU(h2aK5EiO{pF6xDwgfj zc-hbdZh1J~Z3!?wuynE?(}ycW##NB+QSAsERf^S-nFOgeGsTX 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100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -718,13 +718,13 @@ es la \emph{localización} de $A$ respecto de $S$. Si $S=A \setminus \left\lbrac \begin{equation*} \left( m, s \right) \sim \left( m', s' \right) \iff ms' = m's. \end{equation*} -Si $S=A \setminus \left\lbrace 0 \right\rbrace$, denotaremos la localización como $M_Q$ y la llamaremos $Q(A)$-\emph{extensión de escalares} de~$M$. +Si $S=A \setminus \left\lbrace 0 \right\rbrace$, denotaremos la localización como $M_\mathrm{Q}$ y la llamaremos $Q(A)$-\emph{extensión de escalares} de~$M$. \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 $\varphi: M \rightarrow M_Q$ dada por $m \mapsto \frac{m}{1}$ es inyectiva si y solo si $M$ es libre de torsión. + La extensión de escalares de un \modulo{A} $M$ es un \espaciovectorial{\cuerpofracciones{A}}. Además, la aplicación $\varphi: M \rightarrow M_\mathrm{Q}$ dada por $m \mapsto \frac{m}{1}$ es inyectiva si y solo si $M$ es libre de torsión. \end{lema} \begin{proof} - Para la primera afirmación, vemos que $M_Q$ es un \modulo{\cuerpofracciones{A}} y, como \cuerpofracciones{A} es un cuerpo, tenemos un espacio vectorial. + Para la primera afirmación, vemos que $M_\mathrm{Q}$ es un \modulo{\cuerpofracciones{A}} y, como \cuerpofracciones{A} es un cuerpo, tenemos un espacio vectorial. Para la segunda, véase que \begin{align*} \ker\varphi = 0 &\iff \frac{m}{1} \neq 0 \quad\forall m \in M \setminus \left\lbrace 0 \right\rbrace \\ @@ -732,7 +732,7 @@ Si $S=A \setminus \left\lbrace 0 \right\rbrace$, denotaremos la localización co &\iff am \neq 0 \\ &\iff \mathrm{Tors}\!\left( M \right) = \left\lbrace 0 \right\rbrace.\qedhere \end{align*} -% Vemos también gracias a esto que todo elemento de $M_Q$ puede escribirse como $ \frac{m}{q}$ para algún $m\in M$ y algún $q\in A \setminus \left\lbrace 0 \right\rbrace$. Consideremos el homomorfismo de $M_Q$ 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. +% Vemos también gracias a esto que todo elemento de $M_\mathrm{Q}$ puede escribirse como $ \frac{m}{q}$ para algún $m\in M$ y algún $q\in A \setminus \left\lbrace 0 \right\rbrace$. Consideremos el homomorfismo de $M_Q$ 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. % % \todo{La otra implicación. Creo que quedará más corto definiendo la extensión de escalares como fracciones} % @@ -760,13 +760,13 @@ Consideremos el diagrama de homomorfismos de \modulo{\Kx}s de la Figura \ref{Dia Podemos ver los conjuntos de la parte baja del diagrama como subconjuntos de los de arriba. Así, identificamos \begin{equation*} - \dual{M}\equiv\definicionconjunto{h \in \left( \dual{M} \right)_Q}{h\!\left( m \right) \in A \ \forall m \in M}, + \dual{M}\equiv\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 Ejemplo~\ref{Ejemplo Espacio Vectorial es Reflexivo} que $M_\mathrm{Q}$ es reflexivo así que $\dualdual{M_Q} \simeq M_Q$. Esto nos identificar + Ahora recordemos por el Ejemplo~\ref{Ejemplo Espacio Vectorial es Reflexivo} que $M_\mathrm{Q}$ es reflexivo así que $\dualdual{M_\mathrm{Q}} \simeq M_\mathrm{Q}$. Esto nos identificar \begin{equation*} - \dualdual{M}\equiv\definicionconjunto{m \in M_Q}{ \phi(m) \in A \ \forall \phi \in \dual{M}},\quad -M^{***}\equiv\definicionconjunto{m \in \left( \dual{M} \right)_Q}{h\!\left( m \right) \in A \ \forall m \in M}.\qedhere + \dualdual{M}\equiv\definicionconjunto{m \in M_\mathrm{Q}}{ \phi(m) \in A \ \forall \phi \in \dual{M}},\quad +M^{***}\equiv\definicionconjunto{m \in \left( \dual{M} \right)_\mathrm{Q}}{h\!\left( m \right) \in A \ \forall m \in M}.\qedhere \end{equation*} % % \todo{Esto terminarlo.} @@ -774,7 +774,7 @@ M^{***}\equiv\definicionconjunto{m \in \left( \dual{M} \right)_Q}{h\!\left( m \r \begin{definicion} - Sea $M$ un \modulo{A} finitamente generado. Diremos que $M$ tiene \emph{rango} $r$ si $M_Q$ es un \espaciovectorial{\cuerpofracciones{A}} de dimensión $r$. + Sea $M$ un \modulo{A} finitamente generado. Diremos que $M$ tiene \emph{rango} $r$ si $M_\mathrm{Q}$ es un \espaciovectorial{\cuerpofracciones{A}} de dimensión $r$. \end{definicion} Un caso particular importante es el de cualquier ideal $I$, que independientemente del número de generadores que tenga siempre es de rango $1$. @@ -898,19 +898,17 @@ Ahora sí, tenemos la siguiente propiedad universal que buscábamos al principio Sea $M$ un \modulo{A} de rango $n$ y $s \in \naturales$. Entonces, $\bigwedge^s M$ es un \modulo{A} de rango $\binom{n}{s}$. \end{lema} \begin{proof} - Consideremos $\left( \bigwedge^s M \right) \otimes \cuerpofracciones{A}$. Hay un isomorfismo de \modulo{\cuerpofracciones{A}}s entre $\bigwedge^s M_Q$ y $\bigwedge^s\left( M_Q \right)$: + Consideremos $\left( \bigwedge^s M \right)_\mathrm{Q}$. Hay un isomorfismo de \modulo{\cuerpofracciones{A}}s entre $\left( \bigwedge^s M \right)_\mathrm{Q}$ y $\bigwedge^s\left( M_\mathrm{Q} \right)$: \begin{align*} - \bigwedge^sM_Q &\longleftrightarrow \bigwedge^s\left( M_Q \right) \\ - \left( m_1 \wedge m_2 \wedge \dots \wedge m_s \right) \otimes \frac{a}{b} &\longrightarrow \left( m_1 \otimes \frac{a}{b} \right) \wedge \left( m_2 \otimes 1 \right) \wedge \dots \wedge \left( m_s \otimes 1 \right) \\ - \left( m_1 \wedge m_2 \wedge \dots \wedge m_s\right)\otimes \frac{a_1a_2\dots a_s}{b_1 b_2 \dots b_s} &\longleftarrow \left( m_1 \otimes \frac{a_1}{b_1} \right) \wedge \left( m_2 \otimes \frac{a_2}{b_2} \right) \wedge \dots \wedge \left( m_s \otimes \frac{a_s}{b_s} \right) + \left( \bigwedge^sM \right)_\mathrm{Q} &\longleftrightarrow \bigwedge^s\left( M_\mathrm{Q} \right) \\ + \frac{a}{b}\left( m_1 \wedge m_2 \wedge \dots \wedge m_s \right) &\longrightarrow \left( \frac{a}{b} m_1\right) \wedge \left( m_2 \right) \wedge \dots \wedge \left( m_s \right) \\ + \frac{a_1a_2\dots a_s}{b_1 b_2 \dots b_s}\left( m_1 \wedge m_2 \wedge \dots \wedge m_s\right) &\longleftarrow \left( \frac{a_1}{b_1}m_1 \right) \wedge \left( \frac{a_2}{b_2}m_2 \right) \wedge \dots \wedge \left( \frac{a_s}{b_s} m_s \right) \end{align*} Aunque pudiera parecer que no es un isomorfismo puesto que los lados derechos no lucen iguales, recordemos que por las propiedades del producto tensorial y exterior y por ser ambos conjuntos \modulo{\cuerpofracciones{A}} tenemos que - \begin{align*} - \left( m_1 \otimes \frac{a_1}{b_1} \right) \wedge \left( m_2 \otimes \frac{a_2}{b_2} \right) \wedge \dots \wedge \left( m_s \otimes \frac{a_s}{b_s} \right) &= \left[\frac{a_1}{b_1} \left( m_1 \otimes 1 \right) \right] \wedge \left[\frac{a_2}{b_2} \left( m_2 \otimes 1 \right) \right] \wedge \dots \wedge \left[\frac{a_s}{b_s} \left( m_s \otimes 1 \right) \right] \\ - &= \frac{a_1a_2\dots a_s}{b_1 b_2 \dots b_s} \left( m_1 \otimes 1 \right) \wedge \left( m_2 \otimes 1 \right) \wedge \dots \wedge \left( m_s \otimes 1 \right) \\ - &= \left( m_1 \otimes \frac{a_1a_2\dots a_s}{b_1 b_2 \dots b_s} \right) \wedge \left( m_2 \otimes 1 \right) \wedge \dots \wedge \left( m_s \otimes 1 \right). - \end{align*} - Ahora bien, por hipótesis $M_Q$ es un \espaciovectorial{\cuerpofracciones{A}} de rango $n$ así que $\bigwedge^n \left( M_Q \right)$ será también un \espaciovectorial{\cuerpofracciones{A}}, en este caso de rango $\binom{n}{s}$. Gracias al isomorfismo, $\left( \bigwedge^s M \right) \otimes \cuerpofracciones{A}$ es igualmente otro \espaciovectorial{\cuerpofracciones{A}} de rango $\binom{n}{s}$ lo cual quiere decir que $M$ es de rango $\binom{n}{s}$. + \begin{equation*} + \left( \frac{a_1}{b_1}m_1 \right) \wedge \left( \frac{a_2}{b_2}m_2 \right) \wedge \dots \wedge \left( \frac{a_s}{b_s} m_s \right) = \left( \frac{a_1a_2\dots a_s}{b_1 b_2 \dots b_s}m_1 \right) \wedge \left( m_2 \right) \wedge \dots \wedge \left( m_s \right). + \end{equation*} + Ahora bien, por hipótesis $M_\mathrm{Q}$ es un \espaciovectorial{\cuerpofracciones{A}} de rango $n$ así que $\bigwedge^n \left( M_\mathrm{Q} \right)$ será también un \espaciovectorial{\cuerpofracciones{A}}, en este caso de rango $\binom{n}{s}$. Gracias al isomorfismo, $\left( \bigwedge^s M \right)_\mathrm{Q}$ es igualmente otro \espaciovectorial{\cuerpofracciones{A}} de rango $\binom{n}{s}$ lo cual quiere decir que $M$ es de rango $\binom{n}{s}$. \end{proof}