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z`=wP89(Bg_+rVq>$D(Qf{z~7ku(?Y8vd?D_Hxy0n`5KJc(%XiO27Geulxxvdgt19d z-`MpPWCQ8~?weP#Y%|uMk`xZ1r*%lnW{mFQj3*&$;x=39b?XhaEvWKZ{s@v;vfyV) z5=hwDq~AgN+wlU(u>d5L#|327T9Rm>S&q~D?&~3=DA1CRi9d&oRW@pPgBc1%mNN}2 zQPrRf*B}l3@-0;1>tBw1VU!stk^_^KJr_lcS_r2Hxqqp2e~$rdb_lc=Y4#}{T>e_8?r8TFMjcU$E{X0#?QE z_T5+poxs-%J9)nQ%1^x&(I1Q{H7F}D?WAw30ATf`>> ya los funcionamientos o solo plantear los problemas que queremos resolver?}¿Cuáles son estas? ¿Qué estructura siguen? ¿Y cómo podemos calcularlas? \chapter{Módulos} - \todo[inline]{¿Hace falta introducir las definiciones de anillo y espacio vectorial o se dan por hechas?} - \section{Conceptos básicos} Como hemos podido ver, los conjuntos con los que estamos trabajando son una generalización de los espacios vectoriales donde los coeficientes pertenecen a un anillo en vez de a un cuerpo. A esta generalización la llamamos módulo. \begin{definicion} @@ -228,8 +226,10 @@ urlcolor=black]{hyperref} A lo largo de este trabajo será de gran importancia el anillo de polinomios sobre un cuerpo, $\anillodepolinomios{K}{x_1, x_2, \dots, x_n}$, al que denotaremos \anillodepolinomios{K}{\mathbf{x}}. Sobre todo porque a partir de este construimos el conjunto de vectores de polinomios $\anillodepolinomios{K}{\mathbf{x}}^n$, que es un \modulo{\anillodepolinomios{K}{\mathbf{x}}}. \end{ejemplo} \todo[inline]{¿Haría falta alguna introducción o hilo conductor para esta definición o es mejor yuxtaponerlas?} + + De aquí en adelante, para todos los resultados, daremos por sentado que $A$ sea un anillo y $M$ un \modulo{A}. \begin{definicion} - Sea $A$ un anillo y $M$ un \modulo{A}. Un subconjunto no vacío $N \subseteq M$ se dice que es un \submodulo{A} de $M$ si es un \modulo{A} con las operaciones de $M$ restringidas. + Un subconjunto no vacío $N \subseteq M$ se dice que es un \submodulo{A} de $M$ si es un \modulo{A} con las operaciones de $M$ restringidas. \end{definicion} \begin{ejemplo} Si pensamos en \enteros{} como un \modulo{\enteros}, $2\enteros$ es un \submodulo{\enteros}. Esto debería recordarnos a los ideales de \enteros. De hecho, un ideal de un anillo $A$ es un \submodulo{A} del propio $A$. @@ -240,41 +240,39 @@ urlcolor=black]{hyperref} \end{equation*} es un \submodulo{\anillodepolinomios{\reales}{x, y}}. De hecho, a este tipo de módulos los llamaremos submódulos generados por $\mathbf{u}$ y $\mathbf{v}$. \end{ejemplo} \begin{definicion} - Sea $A$ un anillo, $M$ un \modulo{A} y $N$ un \submodulo{A} de $M$. Diremos que $N$ está generado por $G \subseteq N$ y lo denotaremos como $N = \engendrado{G}$ si + Sea $N$ un \submodulo{A} de $M$. Diremos que $N$ está generado por $G \subseteq N$ y lo denotaremos como $N = \engendrado{G}$ si \begin{equation*} N = \definicionconjunto{a_1g_1 + a_2g_2 + \dots + a_kg_k}{a_i \in A, g_i \in G, k \in \naturales}. \end{equation*} En el caso que exista algún $G = \conjunto{g_1, g_2, \dots, g_n}$ de forma que $N = \engendrado{G}$ diremos que $N$ está finitamente generado y lo escribiremos como $N = \engendrado{g_1, g_2, \dots, g_n}$. \end{definicion} \begin{definicion} - Sea $A$ un anillo, $M$ un \modulo{A}, $N$ un \submodulo{A} de $M$ y $\sim$ la siguiente relación de equivalencia sobre $M$: $m \sim m' \Leftrightarrow m - m' \in N$. Definimos el cociente de un módulo por un submódulo como $\cociente{M}{N} := \cociente{M}{\sim}$. Los elementos de este cociente se denotarán por $m + N$. + Sea $N$ un \submodulo{A} de $M$ y $\sim$ la siguiente relación de equivalencia sobre $M$: $m \sim m' \Leftrightarrow m - m' \in N$. Definimos el cociente de un módulo por un submódulo como $\cociente{M}{N} := \cociente{M}{\sim}$. Los elementos de este cociente se denotarán por $m + N$. \end{definicion} \section{Homomorfismos} \begin{definicion} - Sea $A$ un anillo y $M$ y $N$ dos \modulo{A}s. Una aplicación $\phi: M \longrightarrow N$ se dice que es un homomorfismo de \modulo{A}s si respeta las operaciones de módulo, es decir, si + Sean $M$ y $N$ dos \modulo{A}s. Una aplicación $\phi: M \longrightarrow N$ se dice que es un homomorfismo de \modulo{A}s si respeta las operaciones de módulo, es decir, si \begin{itemize} \item $\phi\!\left( m + m' \right) = \phi\!\left( m \right) + \phi\!\left( m' \right)$ y \item $\phi\!\left( am \right) = a\phi\!\left( m \right)$ \end{itemize} - para todo $m, m' \in M$ y $a \in A$. \todo[inline]{¿Se deberían introducir las nociones de isomorfismo, ser isomorfo, etc?} + para todo $m, m' \in M$ y $a \in A$. \end{definicion} \begin{ejemplo} La inclusión de $\enteros^n$ en $\racionales^n$ es un homomorfismo de \modulo{\enteros}s, sin embargo, no es un homomorfismo de \modulo{\racionales}s ya que $\enteros^n$ no es un \modulo{\racionales}. Por otro lado, la aplicación $\phi: \anillodepolinomios{\complejos}{x, y} \longrightarrow \complejos$ que envía a cada polinomio $p\!\left( x, y \right)$ a $p\!\left(0, 1\right)$ es un homomorfismo de \modulo{\complejos}s. \end{ejemplo} \begin{lema} - Sea un anillo $A$, dos \modulo{A}s $M$ y $N$ y un homomorfismo $\phi$ de \modulo{A}s. $\ker \phi$ es un submódulo de $M$ e $\imagen \phi$ es un submódulo de $N$. + Sean dos \modulo{A}s $M$ y $N$ y un homomorfismo $\phi$ de \modulo{A}s. $\ker \phi$ es un submódulo de $M$ e $\imagen \phi$ es un submódulo de $N$. \end{lema} - \todo[inline]{¿Es apropiado incluir la demostración o es suficientemente obvio?} Con estas definiciones, ya podemos empezar a expresar los objetos que nos encontrábamos al principio en un lenguaje más formal. Para abreviar la notación, escribiremos $\partial_i$ cuando nos refiramos a la aplicación \begin{align*} \partial_i : \anillodepolinomios{K}{\mathbf{x}} &\rightarrow \anillodepolinomios{K}{\mathbf{x}} \\ - p(x, y) &\mapsto \diffp{p}{x_i}(x, y). + p &\mapsto \diffp{p}{x_i}. \end{align*} - \todo[inline]{¿Hasta qué punto interesa o deja de interesar poner los argumentos de los polinomios?} Con estas aplicaciones, pasamos a definir el módulo de derivaciones, el conjunto donde residen todas las posibles combinaciones lineales de derivadas parciales: \begin{equation*} \derivaciones{\anillodepolinomios{K}{\mathbf{x}}} := \definicionconjunto{a_1\partial_1 + a_2\partial_2 + \dots + a_n\partial_n}{a_i \in \anillodepolinomios{K}{\mathbf{x}} \, \forall i}. @@ -286,37 +284,37 @@ urlcolor=black]{hyperref} Puede verse que si identificamos $\partial_i$ con \vectorbasecanonica{i}, tenemos que \derivaciones{f} es isomorfo a un submódulo de $\Kx^n$. \begin{definicion} - Sea $A$ un anillo. Diremos que $A$ es noetheriano si todos sus ideales son finitamente generados. + Diremos que $A$ es noetheriano si todos sus ideales son finitamente generados. Diremos también que un \modulo{A} $M$ es noetheriano si todo submódulo $N \subseteq M$ está finitamente generado. Es decir, si existen $n_1, n_2, \dots, n_k$ en $N$ de forma que $N = \engendrado{n_1, n_2, \dots, n_k}_A$. \end{definicion} \begin{lema} - Sea $A$ un anillo y $M$ un \modulo{A} noetheriano.\todo[inline]{¿A partir de qué momento se pueden suprimir las premisas genéricas?} Los submódulos y módulos cocientes de $M$ son también noetherianos. + Sea $M$ un \modulo{A} noetheriano.\todo[inline]{¿A partir de qué momento se pueden suprimir las premisas genéricas?} Los submódulos y módulos cocientes de $M$ son también noetherianos. \end{lema} \begin{proof} [\textsc{Demostración}] \end{proof} \begin{lema} - Sea $A$ un anillo y $M$ un \modulo{A}. Que $M$ sea noetheriano es equivalente a que toda cadena ascendente de submódulos $N_1 \subseteq N_2 \subseteq \dots$ sea eventualmente estacionaria. + Que $M$ sea noetheriano es equivalente a que toda cadena ascendente de submódulos $N_1 \subseteq N_2 \subseteq \dots$ sea eventualmente estacionaria. \end{lema} \begin{proof} [\textsc{Demostración}] \end{proof} \begin{definicion} - Sea $A$ un anillo y $M$ y $N$ dos \modulo{A}s. El conjunto de homomorfismos de \modulo{A}s de $M$ a $N$ se denota por + Sean $M$ y $N$ dos \modulo{A}s. El conjunto de homomorfismos de \modulo{A}s de $M$ a $N$ se denota por \begin{equation*} \Hom{A}{M}{N} := \definicionconjunto{\phi : M \longrightarrow N}{\text{$\phi$ es un homomorfismo de \modulo{A}s}}. \end{equation*} \end{definicion} \begin{definicion} - Sea $A$ un anillo y $M$ un \modulo{A}. Se define el módulo dual \dual{M} como el conjunto de homomorfismos de \modulo{A}s de $M$ en $A$: + Se define el módulo dual \dual{M} como el conjunto de homomorfismos de \modulo{A}s de $M$ en $A$: \begin{equation*} \dual{M} := \Hom{A}{M}{A}. \end{equation*} \end{definicion} \begin{definicion} - Sea $A$ un anillo y $M$ un \modulo{A}. Diremos que $M$ es un \modulo{A} reflexivo si $M$ es isomorfo a \dual{\dual{M}}. + Diremos que $M$ es un \modulo{A} reflexivo si $M$ es isomorfo a \dual{\dual{M}}. \end{definicion} \begin{ejemplo} \todo[inline]{Poner ejemplo de un módulo reflexivo.} @@ -364,7 +362,7 @@ urlcolor=black]{hyperref} \begin{align*} x^3 + 2x^2 - x + 1 &= 0q + x^3 + 2x^2 - x + 1 \\ x^3 + 2x^2 - x + 1 &= xq + - \end{align*} \todo{Completar.} + \end{align*} \todo[inline]{Completar.} Como puede apreciarse, el funcionamiento de este algoritmo depende muy fuertemente de la noción de grado de un polinomio. Sin embargo, en el caso de módulos el grado no siempre está claro. Por lo tanto, introducimos el concepto de que un anillo o un módulo esté <>. \begin{definicion} Sea $A$ un anillo. Se dice que es un anillo graduado si existe una descomposición $\bigoplus_{g=0}^{\infty} A_g$ donde $A_g$ es un grupo abeliano para todo $g$ y se cumple que $A_g A_h \subseteq A_{g+h}$ para todo $g$ y $h$.