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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*}