From aa0387125b5d7f616df3b37917fb4e998fa0a384 Mon Sep 17 00:00:00 2001 From: Sixto Date: Sat, 24 May 2025 18:18:50 +0200 Subject: [PATCH] =?UTF-8?q?Inclusi=C3=B3n=20de=20la=20definici=C3=B3n=20de?= =?UTF-8?q?=20secuencia=20exacta=20y=20los=20convenios=20para=20expresar?= =?UTF-8?q?=20dentro=20de=20estas=20que=20una=20funci=C3=B3n=20es=20inyect?= =?UTF-8?q?iva=20o=20sobreyectiva.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 196939 -> 197627 bytes TFG_Sixto_José_Martínez_Gan.tex | 14 +++++++++++--- 2 files changed, 11 insertions(+), 3 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index 2fc4329d27cdf963fdd0ae67d1fe2e0a0c3fed7f..2e2f7c544c1f0072ddbaa4a12ed99fb8367729d4 100644 GIT binary patch delta 14800 zcmaibQ*b3f*KN*;ZQJ&VJ+W4thWOC>G>egSk>VLRTz1QyD z-7k&WYwey2MtYAxYT$%PCWAw27J=CZ0D1V5$>CgqynM~6aGvo0+JDeG`IZBvIT!S+ 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zg$0-@uX$;V7%EczriyX#Y7M8*8Rd}h=5EJrb+8X zqmG2rhWjhB`8&i``x}Qcya(=Q2L>Ter{-V4xqG+3|5s>qb2o8y_ja|ggk$01;O60g Mqo9ybl7#zz01vOo)c^nh diff --git a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index 70e8dd7..8f2de0b 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -339,11 +339,19 @@ urlcolor=black]{hyperref} y viceversa, es decir, para cualquier vector $w$ de $V$, $v \mapsto \productoescalar{w}{v}$ es un elemento de \Hom{K}{V}{K}. 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.}hgfh + \todo[inline]{Poner ejemplo de un módulo no reflexivo.} \end{ejemplo} - \section{Secuencias Exactas} - + En ocasiones, encadenaremos homomorfismos y podemos necesitar expresar las relaciones entre las imágenes y los núcleos de las distintas aplicaciones. Para expresar esto, utilizaremos las sucesiones exactas. + \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 + \end{equation*} + es una sucesión exacta si se cumple que $\imagen \phi_i = \ker \phi_{i+1}$ para todo $i$. + + 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} \chapter{Bases de Gröbner y Syzygias} A partir de este momento nos fijaremos en $\Kx^n$ ya que sus elementos son los involucrados en el módulo de derivaciones. En particular nos fijaremos en sus submódulos.