Picard group

The Picard group of a variety, scheme, or more generally locallyringed 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 ${\\rm H}^{1}(X,O_{X}^{*})$ , thefirst sheaf cohomology group of the multiplicative sheaf $O_{X}^{*}$ which consists of theunits 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 divisor class group of $X$ .

单词	PicardGroup
释义	Picard group The Picard group of a variety, scheme, or more generally locallyringed 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 ${\\rm H}^{1}(X,O_{X}^{})$ , thefirst sheaf cohomology group of the multiplicative sheaf $O_{X}^{}$ which consists of theunits 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 divisor class group of $X$ .
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