16D10-Support.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{Support}
\pmcreated{2013-03-22 12:10:22}
\pmmodified{2013-03-22 12:10:22}
\pmowner{djao}{24}
\pmmodifier{djao}{24}
\pmtitle{support}
\pmrecord{6}{31406}
\pmprivacy{1}
\pmauthor{djao}{24}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{16D10}
\pmclassification{msc}{13C99}

\endmetadata

\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
%%%\usepackage{xypic}
\newcommand{\C}{\mathbb{C}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\barQ}{\overline{\Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\F}{\mathbb{F}}
\renewcommand{\S}{\mathcal{S}}
\renewcommand{\a}{{\mathfrak{a}}}
\renewcommand{\b}{{\mathfrak{b}}}
\newcommand{\m}{{\mathfrak{m}}}
\newcommand{\p}{{\mathfrak{p}}}
\renewcommand{\P}{{\mathfrak{P}}}
\newcommand{\q}{{\mathfrak{q}}}
\renewcommand{\c}{{\bf{c}}}
\renewcommand{\d}{{\bf{d}}}
\newcommand{\x}{{\bf{x}}}
\renewcommand{\Re}{\operatorname{Re}}
\newcommand{\0}{\bf{0}}
\newcommand{\category}[1]{\mbox{\boldmath $\mathsf{{#1}}$}}
\newcommand{\qr}[2]{{\mbox{$\left(\frac{{#1}}{{#2}}\right)$}}}
\newcommand{\s}[1]{\EuScript{{#1}}}
\newcommand{\conj}[1]{{\overline{{#1}}}}

\newlength{\algcwidth} \newlength{\algcheight}
\newcommand{\algc}[1]{%
\settowidth{\algcwidth}{#1} \settoheight{\algcheight}{#1}%
\raisebox{1.2\algcheight}[0pt]{%
\makebox[0pt][l]{\hspace{.4\algcwidth}%
\rule{.6\algcwidth}{0.05\algcheight}}}%
{{#1}}}

\newcommand{\jacobi}[2]{{\left(\frac{#1}{#2}\right)}}
\renewcommand{\H}{\category{H}}
\newcommand{\A}{\category{A}}
\newcommand{\B}{\category{B}}
\renewcommand{\O}{\mathcal{O}}
\newcommand{\M}{\mathfrak{M}}
\newcommand{\<}{\langle}
\renewcommand{\>}{\rangle}
\newcommand{\lra}{\longrightarrow}
\newcommand{\ra}{\rightarrow}
\newcommand{\iso}{\cong}
\newcommand{\dsum}{\oplus}
\newcommand{\bigdsum}{\bigoplus}
\newcommand{\conv}{\circ}
\newcommand{\comp}{\circ}
%\newcommand{\st}{\text{\ s.t.\ }}
\newcommand{\st}{\mid}
\renewcommand{\div}{\mid}
\newcommand{\ndiv}{\nmid}
\newcommand{\intersect}{\cap}
\newcommand{\union}{\cup}
\newcommand{\bigintersect}{\bigcap}
\newcommand{\bigunion}{\bigcup}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\cross}{\times}
\newcommand{\tensor}{\otimes}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Sym}{\operatorname{Sym}^2}
\newcommand{\Alt}{\operatorname{Alt}^2}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\Res}{\operatorname{Res}}
\newcommand{\Spec}{\operatorname{Spec}}
\newcommand{\Nil}{\operatorname{Nil}}
\newcommand{\Ann}{\operatorname{Ann}}
\newcommand{\Ass}{\operatorname{Ass}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Co}{\operatorname{Co}}
\newcommand{\Supp}{\operatorname{Supp}}
\newcommand{\cl}{\operatorname{cl}}
\newcommand{\coker}{\operatorname{coker}}
\newcommand{\disc}{\operatorname{disc}}
\newcommand{\Gal}{\operatorname{Gal}}
\newcommand{\lcm}{\operatorname{lcm}}
\newcommand{\sign}{\operatorname{sign}}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\GL}{\operatorname{GL}}
\newcommand{\semidirect}{\times}
\newcommand{\bd}{\partial}
\newcommand{\dual}{\vee}
\newcommand{\pth}{{\mbox{$p^{\text{th}}$}}}
\newcommand{\qth}{{\mbox{$q^{\text{th}}$}}}
\begin{document}
The {\em support} $\Supp(M)$ of a module $M$ over a ring $R$ is the set of all prime ideals $\p \subset R$ such that the localization $M_\p$ is nonzero.

The {\em maximal support} $\Supp_m(M)$ of a module $M$ over a ring $R$ is the set of all maximal ideals $\m \subset R$ such that $M_\m$ is nonzero.
%%%%%
%%%%%
%%%%%
\end{document}