Editing Ring theory (section)

===Finitely generated projective module over a ring and Picard group===
Let ''R'' be a commutative ring and <math>\mathbf{P}(R)</math> the set of isomorphism classes of finitely generated [[projective module]]s over ''R''; let also <math>\mathbf{P}_n(R)</math> subsets consisting of those with constant rank ''n''. (The rank of a module ''M'' is the continuous function <math>\operatorname{Spec}R \to \mathbb{Z}, \, \mathfrak{p} \mapsto \dim M \otimes_R k(\mathfrak{p})</math>.<ref>{{harvnb|Weibel|2013|loc=Ch I, Definition 2.2.3}}</ref>) <math>\mathbf{P}_1(R)</math> is usually denoted by Pic(''R''). It is an abelian group called the [[Picard group]] of ''R''.<ref>{{harvnb|Weibel|2013|loc=Definition preceding Proposition 3.2 in Ch I}}</ref> If ''R'' is an integral domain with the field of fractions ''F'' of ''R'', then there is an exact sequence of groups:<ref>{{harvnb|Weibel|2013|loc=Ch I, Proposition 3.5}}</ref>
:<math>1 \to R^* \to F^* \overset{f \mapsto fR}\to \operatorname{Cart}(R) \to \operatorname{Pic}(R) \to 1</math>
where <math>\operatorname{Cart}(R)</math> is the set of [[fractional ideal]]s of ''R''. If ''R'' is a [[Regular ring|regular]] domain (i.e., regular at any prime ideal), then Pic(R) is precisely the [[divisor class group]] of ''R''.<ref>{{harvnb|Weibel|2013|loc=Ch I, Corollary 3.8.1}}</ref>

For example, if ''R'' is a principal ideal domain, then Pic(''R'') vanishes. In algebraic number theory, ''R'' will be taken to be the [[ring of integers]], which is Dedekind and thus regular. It follows that Pic(''R'') is a finite group ([[finiteness of class number]]) that measures the deviation of the ring of integers from being a PID.<!-- discuss coordinate ring -->

One can also consider the [[group completion]] of <math>\mathbf{P}(R)</math>; this results in a commutative ring K<sub>0</sub>(R). Note that K<sub>0</sub>(R) = K<sub>0</sub>(S) if two commutative rings ''R'', ''S'' are Morita equivalent.

{{See also|Algebraic K-theory}}