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b/TFG_Sixto_José_Martínez_Gan.tex @@ -792,21 +792,13 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. 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} - Para la primera afirmación, vemos que $M \otimes \cuerpofracciones{A}$ es un \modulo{\cuerpofracciones{A}} y, como \cuerpofracciones{A} es un cuerpo, tenemos un espacio vectorial. - - Para la segunda, véase que - \begin{equation*} - \begin{array}{rcll} - \ker\varphi = 0 &\iff& m \otimes 1 \neq 0 &\forall m \in M\setminus\conjunto{0} \\ - &\iff & am \otimes \displaystyle\frac{1}{a} \neq 0 &\forall a \in A \setminus \conjunto{0} \\ - &\implies& am \neq 0 & - \end{array} - \end{equation*} - - 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. - - \todo{La otra implicación. Creo que quedará más corto definiendo la extensión de escalares como fracciones} - + La aplicación $m \otimes \frac{1}{q} \mapsto \frac{m}{q}$ de es un isomorfismo así podemos ver la extensión de escalares como $M \otimes \cuerpofracciones{A}$ o $\left( A \setminus \left\lbrace 0 \right\rbrace \right)^{-1}M$ según nos convenga. En este caso, utilizaremos la segunda para ver que + \begin{align*} + \ker\varphi = 0 &\iff \frac{m}{1} \neq 0 \quad\forall m \in M \setminus \left\lbrace 0 \right\rbrace \\ + &\iff \frac{am}{a} \neq 0 \quad \forall a \in A \setminus \left\lbrace 0 \right\rbrace \\ + &\iff am \neq 0 \\ + &\iff \mathrm{Tors}\!\left( M \right) = \left\lbrace 0 \right\rbrace. + \end{align*} \end{proof} \begin{definicion} 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 $r$.