PicardGroup3 2002-07-27 06:09:39 2007-02-15 10:22:04 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 The {\em Picard group} of a variety, scheme, or more generally locally ringed space $(X,O_X)$ is the group of locally free $O_X$ modules of rank $1$ with tensor product over $O_X$ as the operation, usually denoted by $\operatorname{Pic}(X)$. Alternatively, the Picard group is the group of isomorphism classes of invertible sheaves on $X$, under tensor products. It is not difficult to see that $\operatorname{Pic}(X)$ is isomorphic to ${\rm H}^1(X, O_X^*)$, the first sheaf cohomology group of the multiplicative sheaf $O_X^*$ which consists of the units of $O_X$. Finally, let $\operatorname{CaCl}(X)$ be the group of Cartier divisors on $X$ modulo linear equivalence. If $X$ is an integral scheme then the groups $\operatorname{CaCl}(X)$ and $\operatorname{Pic}(X)$ are isomorphic. Furthermote, if we let $\operatorname{Cl}(X)$ be the class group of Weil divisors (divisors modulo principal divisors) and $X$ is a noetherian, integral and separated locally factorial scheme, then there is a natural isomorphism $\operatorname{Cl}(X)\cong \operatorname{Pic}(X)$. Thus, the Picard group is sometimes called the {\it divisor class group} of $X$.