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Esta biyección es un isomorfismo de \modulo{K}s, por lo que tenemos que $V$ es isomorfo a \dual{V} y, por consiguiente, a \dualdual{V} también. Con esto hemos demostrado que \dualdual{V} es un \espaciovectorial{K} de la misma dimensión de $V$, pero para verificar que este sea reflexivo debemos cerciorarnos de que la aplicación $\psi$ sea un isomorfismo. Para ello, basta darse cuenta que el kernel de esta es necesariamente el vector nulo, que se corresponde con evaluar todas las aplicaciones de \dual{V} en el $\mathbf{0}$, y por lo tanto $\psi$ es inyectiva; como $V$ y \dualdual{V} son de la misma dimensión, concluimos que $\psi$ es un isomorfismo. \end{ejemplo} \begin{ejemplo} - \todo[inline]{Poner ejemplo de un módulo no reflexivo.} + 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 $\phi \in \dual{\cociente{\enteros}{p\enteros}}$ está unívocamente determinado por la imagen de $1$. Supongamos que $\phi\!\left( 1 \right) = x \in \enteros$, se debe cumplir que + \begin{equation*} + px = p\phi\!\left( 1 \right) = \phi\!\left( p\cdot 1 \right) = \phi\!\left( 0 \right) = 0 \ \Leftrightarrow \ px = 0 \ \Leftrightarrow \ x = 0. + \end{equation*} + Es decir, que $\dual{\left( \cociente{\enteros}{p\enteros} \right)} = \conjunto{\dual{\mathbf{0}}}$ donde $\mathbf{0}$ 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. 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} Una discusión más profunda acerca de los módulos reflexivos y sus propiedades puede encontrarse en \cite[\href{https://stacks.math.columbia.edu/tag/0AUY}{Tag 0AUY}]{STACKS}. @@ -940,16 +944,7 @@ urlcolor=black]{hyperref} \end{aligned}\right\rbrace \implies \text{$M$ es un módulo libre} \end{gather} - Lo primero que haremos es ver que \derivaciones{f} es siempre un módulo reflexivo de rango $n$ tal y como demuestra en \cite{SAITO}. - \todo[inline]{En el artículo de Vallés pone que tiene que ser \textit{square-free}.} - \begin{lema} - \derivaciones{f} es un \modulo{\Kx} reflexivo. - \end{lema} - \begin{proof} - \todo[inline]{No entiendo la demostración de Saito.} - \end{proof} - - Si vemos el esquema, tenemos que \derivaciones{f} es libre de torsión, pero necesitaríamos que tuviese rango $1$ para continuar hacia delante. Lo que haremos será estudiar el módulo $\bigwedge^n \derivaciones{f}$ para reducir este rango y poder continuar. + Puede encontrarse en \cite[corolario 1.7]{SAITO} una demostración de que \derivaciones{f} es un módulo reflexivo. Siguiendo el esquema, tenemos que \derivaciones{f} es libre de torsión, pero necesitaríamos que tuviese rango $1$ para continuar hacia delante. Lo que haremos será estudiar el módulo $\bigwedge^n \derivaciones{f}$ para reducir este rango y poder continuar. \begin{lema} Sea $M$ un \modulo{A} de rango $n$ y $s \in \naturales$. Entonces, $\bigwedge^s M$ es un \modulo{A} de rango $\binom{n}{s}$. \end{lema}