CechCohomologyGroup2 2004-10-10 11:02:30 2007-01-27 10:44:54 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $\mathcal F$ be a sheaf of abelian groups on a topological space $X$ and consider an \PMlinkescapetext{open covering} $\mathcal U=\{U_i\}_{i\in I}$ of $X$. For the sake of simplicity denote $$ U_{i_0i_1\cdots i_q}=U_{i_0}\cap U_{i_1}\cap\dots\cap U_{i_q}. $$ The group $\check C^q(\mathcal U,\mathcal F)$ of \v{C}ech $q$-cochains is the set of families $$ c=(c_{i_0i_1\cdots i_q})\in\prod_{(i_0,\dots,i_q)\in I^{q+1}}\mathcal F(U_{i_0i_1\cdots i_q}). $$ The group \PMlinkescapetext{structure} on $\check C^q(\mathcal U,\mathcal F)$ is the obvious one deduced from the addition law on sections of $\mathcal F$. The \v{C}ech differential $$ \delta^q\colon\check C^q(\mathcal U,\mathcal F)\to\check C^{q+1}(\mathcal U,\mathcal F) $$ is defined by the \PMlinkescapetext{formula} $$ (\delta^q c)_{i_0\cdots i_{q+1}}=\sum_{0\le j\le q+1}(-1)^j c_{i_0\cdots\widehat{i_j}\cdots i_{q+1}}|_{U_{i_0\cdots i_{q+1}}}, $$ and we set $\check C^{q}(\mathcal U,\mathcal F)=0$, $\delta^q=0$ for $q<0$. Easy computations show that $\delta^{q+1}\circ\delta^q=0$. We get therefore a cochain complex $(\check C^\bullet(\mathcal U,\mathcal F),\delta)$, called the complex of \v{C}ech cochains relative to the \PMlinkescapetext{covering} $\mathcal U$. The $q$-th \emph{\v{C}ech cohomology group} of $\mathcal F$ relative to $\mathcal U$ is $$ \check H^q(\mathcal U,\mathcal F)=H^q(\check C^\bullet(\mathcal U,\mathcal F)). $$