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Grothendieck group
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Definition/Summary
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A Grothendieck group is a group constructed from a commutative monoid by a generalization of subtraction.
A simple example is the construction of the group of integers over addition from the monoid of nonnegative integers over addition. |
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Recent forum threads on Grothendieck group
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Breakdown
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Mathematics
> Algebra
>> Group Theory
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Extended explanation
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For a commutative monoid M with operation +, we define the elements of its Grothendieck group as equivalence classes of pairs (a1,a2) where a1 and a2 are in M.
The equivalence classes are defined with this equivalence relation on the pairs:
(a0,a1) ~ (b0,b1) if a0+b1 = a1+b0
The group operation on equivalence classes A and B is defined as follows: take (a0,a1) from A and (b0,b1) from B, and find the equivalence class A+B for
(a0+b0, a1+b1)
It is easy to prove associativity: (A+B)+C = A+(B+C) and also commutativity: A+B = B+A for equivalence classes A, B, C.
The identity is the equivalence class that contains all pairs (a,a), and the inverse of an equivalence class A is the equivalence class for (a1,a0) where (a0,a1) is in A.
One may even identify monoid elements a with the equivalence classes for (a,0), where 0 is the monoid's identity. |
Commentary
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