K-group, zeroth (Ex)

• Show that the group $K_0(R)$$<wikitex>; * Show that the group $K_0(R)$ is isomorphic to the group obtain by applying the Grothendieck construction to the abelian monoid of isomorphism classes of finitely generated projective $R$-modules under direct sum. * Compute $K_0(\Cc[S_3])$, where $S_3$ is the [[symmetric group]] on three elements. * Show that the $C(S^2)$-module of sections of the tangent bundle $TS^2$ is finitely generated projective and even stably finitely generated free, but not finitely generated free. </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises with solution]]K_0(R)$ is isomorphic to the group obtain by applying the Grothendieck construction to the abelian monoid of isomorphism classes of finitely generated projective $R$$R$-modules under direct sum.

• Compute $K_0(\Cc[S_3])$$K_0(\Cc[S_3])$, where $S_3$$S_3$ is the symmetric group on three elements.

• Show that the $C(S^2)$$C(S^2)$-module of sections of the tangent bundle $TS^2$$TS^2$ is finitely generated projective and even stably finitely generated free, but not finitely generated free.