K-group, zeroth (Ex)
From Manifold Atlas
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(Created page with "<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 ...") |
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− | * 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 | + | * 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. |
− | $R$-modules under direct sum. | + | |
* Compute $K_0(\Cc[S_3])$, where $S_3$ is the [[symmetric group]] on three elements. | * Compute $K_0(\Cc[S_3])$, where $S_3$ is the [[symmetric group]] on three elements. | ||
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{{#RefList:}} | {{#RefList:}} | ||
[[Category:Exercises]] | [[Category:Exercises]] | ||
− | [[Category:Exercises | + | [[Category:Exercises with solution]] |
Latest revision as of 18:16, 29 August 2013
- Show that the group is isomorphic to the group obtain by applying the Grothendieck construction to the abelian monoid of isomorphism classes of finitely generated projective -modules under direct sum.
- Compute , where is the symmetric group on three elements.
- Show that the -module of sections of the tangent bundle is finitely generated projective and even stably finitely generated free, but not finitely generated free.