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6748391..a20e326 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -34,6 +34,7 @@ \usepackage{graphicx} +\usepackage{wrapfig} \graphicspath{{C:/Users/usuario/Dropbox/Carrera_5/Asignaturas/TFG_Sixto/Imagenes_y_Figuras/Identidad_Corporativa_Universidad_de_Zaragoza}} % Si alguno de estos paquetes no los vas a usar, mejor los quitas: \usepackage{amssymb} % para cargar las fuentes de la A.M.S. @@ -66,6 +67,7 @@ %\usepackage{lmodern} % fuente Times \usepackage{tikz} % para los dibujos \usetikzlibrary{babel} % para evitar conflictos entre TikZ y babel +\usepackage{tikz-cd} %\usepackage{longtable} % permite saltos de páginas en tablas largas @@ -381,13 +383,61 @@ urlcolor=black]{hyperref} \begin{definicion} Diremos que una expresión del siguiente estilo: \begin{equation*} - \dots \xrightarrow{\phi_k} M_k \xrightarrow{\phi_{k+1}} M_{k+1} \xrightarrow{\phi_{k+2}} M_{k+2} \xrightarrow{\phi_{k+3}} \dots + \dots \xrightarrow{\phi_{k-1}} M_k \xrightarrow{\phi_{k}} M_{k+1} \xrightarrow{\phi_{k+1}} M_{k+2} \xrightarrow{\phi_{k+2}} \dots \end{equation*} - es una sucesión exacta si se cumple que $\imagen \phi_i = \ker \phi_{i+1}$ para todo $i$. + es una sucesión exacta si se cumple que $\imagen \phi_i = \ker \phi_{i+1}$ para todo $i$, como ilustra este diagrama: + \begin{figure}[h] + \centering + \scriptsize + \begin{tikzpicture}[scale=0.3] + \foreach \veces in {2, 1}{ + \begin{scope}[xshift=\veces*6 cm] + \filldraw[draw=black, fill opacity=0.5, fill=gray] (0, 0) ellipse [x radius = 1, y radius = 2]; + \filldraw[draw=black, fill opacity=0.5, fill=gray] (0, 0) ellipse [x radius = 2, y radius = 4]; + \filldraw[black] (0, 0) ellipse [x radius = 1pt, y radius = 2pt]; + \draw (-5.75, 3.97) -- (0.25, 1.94); + \draw (-5.75, -3.97) -- (0, -2); + \draw (-4.15, 1.4) -- (0, 0); + \draw[dashed] (-4.15, 1.4) -- (-5.75, 1.94); + \draw (-4.15, -1.4) -- (0, 0); + \draw[dashed] (-4.15, -1.4) -- (-5.75, -1.94); + \node at (0.5, 0) {$0$}; + \node at (0, -3) {$M_{k+\veces}$}; + \node at (3, -2) {$\phi_{k+\veces}$}; + \end{scope} + } + \begin{scope}[xshift=0 cm] + \filldraw[draw=black, fill opacity=0.5, fill=gray] (0, 0) ellipse [x radius = 1, y radius = 2]; + \filldraw[draw=black, fill opacity=0.5, fill=gray] (0, 0) ellipse [x radius = 2, y radius = 4]; + \filldraw[black] (0, 0) ellipse [x radius = 1pt, y radius = 2pt]; + \draw (-5.75, 3.97) -- (0.25, 1.94); + \draw (-5.75, -3.97) -- (0, -2); + \draw (-4.15, 1.4) -- (0, 0); + \draw[dashed] (-4.15, 1.4) -- (-5.75, 1.94); + \draw (-4.15, -1.4) -- (0, 0); + \draw[dashed] (-4.15, -1.4) -- (-5.75, -1.94); + \node at (0.5, 0) {$0$}; + \node at (0, -3) { $M_k$}; + \node at (3, -2) {$\phi_{k}$}; + \end{scope} + \begin{scope}[xshift=18 cm] + \draw (-5.75, 3.97) -- (0.25, 1.94); + \draw (-5.75, -3.97) -- (0, -2); + \draw (-4.15, 1.4) -- (0, 0); + \draw[dashed] (-4.15, 1.4) -- (-5.75, 1.94); + \draw (-4.15, -1.4) -- (0, 0); + \draw[dashed] (-4.15, -1.4) -- (-5.75, -1.94); + \end{scope} + \draw (-4.15, -1.4) -- (-5.75, -1.94); + \draw (-4.15, 1.4) -- (-5.75, 1.94); + \node at (-3, -2) {$\phi_{k-1}$}; + \end{tikzpicture} + \end{figure} Para indicar que una aplicación $\phi : M \rightarrow N$ es inyectiva, escribiremos $0 \rightarrow M \xrightarrow{\phi} N$; y para indicar que es sobreyectiva, $M \xrightarrow{\phi} N \rightarrow 0$. \end{definicion} + Con las secuencias exactas, podemos resumir nuestro problema en una sola expresión. Recordemos que estamos estudiando el \modulo{\Kx} \derivaciones{f}, que es un \submodulo{\Kx} de \derivaciones{\Kx}. Por otro lado, cada derivación $\delta$, cuando se aplica a $f$ nos da lugar a un elemento en el \submodulo{\Kx} $J := \engendrado{\partial_1 f, \dots, \partial_n f}$. Finalmente, a nosotros los elementos que nos interesan son aquellos que sean múltiplos de $f$, es decir, que pertenezcan al \submodulo{\Kx} \engendrado{f}; para poder identificarlo, haremos el cociente de $J$ por \engendrado{f} para que así todas las expresiones de $J$ que sean múltiplos de $f$ vayan al cero. Todo esto lo podemos expresar como sigue: \begin{equation} \label{Secuencia Exacta Definición Módulo Derivaciones} 0 \rightarrow \derivaciones{f} \xrightarrow{\Phi} \derivaciones{\Kx} \xrightarrow{\kappa} \Kx^n \xrightarrow{\iota} J \xrightarrow{\pi} \frac{J}{J \cap \engendrado{f}} \rightarrow 0. @@ -398,7 +448,13 @@ urlcolor=black]{hyperref} \begin{definicion} \label{Definición Módulo Proyectivo} Un \modulo{A} $P$ es proyectivo si para cada epimorfismo de \modulo{A}s $e : M \rightarrow N$ y cada homomorfismo de \modulo{A} $h: P \rightarrow N$ existe un homomorfismo de \modulo{A}s $\phi : P \rightarrow M$ de forma que $h = e \circ \phi$. \end{definicion} - \todo[inline]{Aquí habrá que poner el diagrama de las secuencias exactas que expresan esto como en el Lectures.} + Suele ser común referirse a esto diciendo que todo homomorfismo $h$ puede ser <> a través de cualquier epimorfismo $e$ a un homomorfismo $\phi$ y queda ilustrado por el siguiente diagrama de secuencias exactas: + \begin{equation*} + \begin{tikzcd} + & P \arrow[d, "h"] \arrow[ld, "\phi" above] & \\ + M \arrow[r, "e"] & N \arrow[r] & 0 + \end{tikzcd} + \end{equation*} \begin{ejemplo} Sean $M = \cociente{\enteros}{4\enteros}$, $N = \cociente{\enteros}{2\enteros}$ y $P = \cociente{\enteros}{2\enteros}$ vistos como \modulo{\enteros}s. Como $e$ es un epimorfismo de \modulo{\enteros}s, necesariamente tendremos que $e\!\left( 0 \right) = e\!\left( 2 \right) = 0$ y $e\!\left( 1 \right) = e\!\left( 3 \right) = 1$. Como $h$ simplemente tomaremos la identidad. Sea ahora $\phi$ un homomorfismo de \modulo{\enteros}s cualquiera; existen dos posibilidades para $\phi\!\left( 1 \right)$: