4
$\begingroup$

On the wiki page for Grothendieck group, the first sentence says the Grothendieck group is the "best possible" way to construct an abelian group from a commutative monoid.

What actually does this mean formally?

  • 4
    It means it satisfies a universal property. In this case, the Grothendieck group of $M$ is the unique (up to isomorphism) abelian group $G$ and a homomorphism $M \to G$ such that every monoid homomorphism $M \to H$ with $H$ a group factors uniquely through $M \to G$.2012-06-25
  • 0
    Paolo Aluffi has a nice (elementary) discussion of the Grothendieck group in Algebra Chapter 0. I recommend that you check it out. The exercises deals with some elementary instances of it too.2012-06-25
  • 0
    @Dedalus Can you please share the link of the note of Paolo Aluffi?2016-03-06

1 Answers 1