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%       pLaTeX2e + amsmath      (Takuya Ikuta)
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%
%
\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb,amsthm,amscd,cases}
\usepackage[notref]{showkeys}

\sloppy\raggedbottom
%
%
%%%
%
%       environments
%
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}
\newenvironment{defn**}[1][]{\noindent\textbf{#1.}\enspace\ignorespaces}{}
\renewcommand{\labelenumi}{(\theenumi)}         % for enumerate

\newcommand{\kl}{\text{[}}
\newcommand{\km}{\text{][}}
\newcommand{\kr}{\text{]}}

\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}

\newcommand{\matX}{\textsl{Mat}_X(\CC^*)}         % Matrix
\newcommand{\matY}{\textsl{Mat}_Y(\CC^*)}         % Matrix
\newcommand{\matZ}{\textsl{Mat}_Z(\CC^*)}         % Matrix

% 2010-07-05 に宗政先生からご指摘
\DeclareMathOperator{\LCM}{LCM}

\newcommand{\nexteq}{\displaybreak[0]\\ &=}

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\begin{document}


\end{document}