From 4c74f25e7f448a15ab4e85e3a2714d86f1c060f3 Mon Sep 17 00:00:00 2001 From: Sixto Date: Mon, 9 Jun 2025 00:50:06 +0200 Subject: [PATCH] =?UTF-8?q?Eliminaci=C3=B3n=20de=20la=20secci=C3=B3n=20sob?= =?UTF-8?q?re=20graduaci=C3=B3n=20de=20anillos.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 472929 -> 470808 bytes TFG_Sixto_José_Martínez_Gan.tex | 29 ----------------------------- 2 files changed, 29 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index 7274490825ef461a2cfaeceb01f3cf1a00ace870..c71117e169aa88cec6ff792783c6343eab1c9ca6 100644 GIT binary patch delta 86816 zcmXtfV{o8N6K!nU=Ek;dJK1Ps+j(MRW9`PaZQFLTv2EP8lwX$XYzg#G`%c+w6lQiB5;JRT_9|Ic7; zxSNR+b6 zHfC$m_=}tf7Xt)HQaiqEe=^Ma+Kp)^8{$7Q*xZmHxJvn(N#JscW+M@MKf8Y|3G;V94z^R@(mMn_Gk^QKWF?EHKYoK8J5@*^NTda)q-` zt)PhQ!11r-RyFgaJ!=)1w`tm2lkTN_A3WG0NwnoG!HT_0?D+3S1i_KT-N}B@? 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el cociente, ver cómo dividir por más de un generador e integrar a las distintas componentes. - \section{Graduación de Módulos} - Pensemos con detalle en el algoritmo de la división en un anillo de polinomios familiar como es \anillodepolinomios{\reales}{x}. Al dividir $p = x^3 + 2x^2 - x + 1$ entre $q = x^2 - 1$, buscamos anular el término de mayor grado de $p$ ---$x^3$--- haciendo al mayor término de $q$ ---$x^2$--- alcanzar a este. Esto es básicamente en lo que pensamos en todo momento mientras realizamos el algoritmo de la división <>: - \begin{center} - \begin{blockarray}{rrrrcl} - \begin{block}{rrrrc|l} - $x^3$ & $+2x^2$ & $-x\phantom{)}$ & $+1\phantom{)}$ & & \cocientecaja{x^2 - 1} \\ - $-(x^3$ & & $-x)$ & & & $x + 2$ \\ - \end{block} - \begin{block}{rrrrcl} - \cmidrule{1-4} - \tiny 1\textsuperscript{a} iteración: &$+2x^2$ & & $+1\phantom{)}$ & & \\ - & $-(2x^2$ & & $-2)$ & & \\ - \cmidrule{2-4} - & & \tiny 2\textsuperscript{a} iteración: & $+3\phantom{)}$ & & \\ - \end{block} - \end{blockarray} - \end{center} - Como puede apreciarse, el funcionamiento de este algoritmo depende muy fuertemente de la noción de grado de un polinomio. Al estar generalizando todo, es oportuno que generalicemos también la noción de grado a anillos y módulos. - \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 $h$. - - Sea ahora $M$ un \modulo{A}. Se dice que es un \modulo{A} graduado si existe una descomposición $\bigoplus_{g\in \enteros} M_g$ donde $M_g$ es un grupo abeliano para todo $g$ y se cumple que $A_{h} M_{g} \subseteq M_{g+h}$. - - Los elementos de $A_g$ y de $M_g$ se denominan homogeneos de grado $g$. - \end{definicion} - - - El problema de estos grados es que no nos permiten ordenar los distintos monomios que nos aparecen de forma unívoca. Por ejemplo, en $xyz + x^2y + xy^2$ los tres monomios que aparecen son distintos el uno del otro aunque son del mismo grado. Para solucionar esto, debemos introducir el concepto de orden monomial. - \section{Órdenes monomiales} El concepto de orden monomial nos permite construir un orden total sobre los monomios de \Kx. Para poder definirlo bien, debemos identificar un monomio en $n$ variables con un vector en $\naturales^n$ que represente sus exponentes, es decir, \begin{equation*}