Grothendieck Group

In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the most universal way. It takes its name from the more general construction in category theory, introduced by Alexander Grothendieck in his fundamental work of the mid-1950s that resulted in the development of K-theory, which led to his proof of the Grothendieck-Riemann-Roch theorem. The Grothendieck group is denoted by K or R.

Read more about Grothendieck GroupUniversal Property, Explicit Construction, Grothendieck Group and Extensions, Grothendieck Groups of Exact Categories, Grothendieck Groups of Triangulated Categories, Examples

Other articles related to "grothendieck group, group":

Direct Sum Of Modules - Grothendieck Group
... can be defined, and every commutative monoid can be extended to an abelian group ... This extension is known as the Grothendieck group ... The construction, detailed in the article on the Grothendieck group, is "universal", in that it has the universal property of being unique, and homomorphic to any other embedding of ...
Grothendieck Group - Examples
... The easiest example of the Grothendieck group construction is the construction of the integers from the natural numbers ... Now when we use the Grothendieck group construction we obtain the formal differences between natural numbers as elements n - m and we have the equivalence relation ... an exact sequence m = l + n, so Thus, the Grothendieck group is isomorphic to Z and is generated by ...

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