---
tag: 5FDMY7PDYY
kind: concept
name: Grothendieck group
msc: [19A49, 18F30, "16E20"]
sources:
  - url: https://en.wikipedia.org/wiki/Grothendieck_group
same_as:
  wikidata: Q1128678
  wikipedia: Grothendieck group
---

## Summary

The Grothendieck group of a commutative [[monoid|0N6R902FG1]] is the [[abelian group|082GHNPJ53]] obtained by adding formal differences, in the universal way.
It turns the natural numbers into the integers.
Applied to isomorphism classes of finitely generated [[projective modules|BFFA03GE3V]] under direct sum, it gives the group $K_0$ of a ring, the start of [[algebraic K-theory|5GVFZWMWC8]].
The same idea applies to vector bundles and to abelian categories.

## Explanations

### L1 human

Some structures can be added but not subtracted, such as natural numbers, or vector spaces under direct sum.
The Grothendieck group adds negatives in the most economical way.
Its elements are formal differences $a - b$, with $a - b = c - d$ when $a + d$ and $c + b$ become equal after adding a common term.
From the natural numbers one gets the integers.
From finite-dimensional vector spaces over a field one also gets the integers, through the dimension.