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单词 GrothendieckGroup
释义

Grothendieck group


Let S be an abelian semigroup.The Grothendieck group of S is K(S)=S×S/,where is the equivalence relationMathworldPlanetmath:(s,t)(u,v) if there exists rS such that s+v+r=t+u+r.This is indeed an abelian groupMathworldPlanetmath with zero elementMathworldPlanetmath (s,s) (any sS), inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath -(s,t)=(t,s) and addition given by(s,t)+(u,v)=(s+u,t+v).It is common to use the suggestive notation t-s for (t,s).

The Grothendieck group construction is a functorMathworldPlanetmath from the categoryMathworldPlanetmath of abelian semigroups to the category of abelian groups.A morphismMathworldPlanetmath f:ST induces a morphism K(f):K(S)K(T)which sends an element (s+,s-)K(S) to (f(s+),f(s-))K(T).

Example 1

Let (N,+) be the semigroup of natural numbersMathworldPlanetmath with composition given by addition.Then, K(N,+)=Z.

Example 2

Let (Z-{0},×) be the semigroup of non-zero integers with composition given by multiplication.Then, K(Z-{0},×)=(Q-{0},×).

Example 3

Let G be an abelian group, then K(G)G via the identification (g,h)g-h(or (g,h)gh-1 if G is multiplicative).

Let C be a (essentially small) symmetric monoidal category.Its Grothendieck group is K([C]),i.e. the Grothendieck group of the isomorphism classes of objects of C.

TitleGrothendieck group
Canonical nameGrothendieckGroup
Date of creation2013-03-22 13:38:24
Last modified on2013-03-22 13:38:24
Ownermhale (572)
Last modified bymhale (572)
Numerical id11
Authormhale (572)
Entry typeDefinition
Classificationmsc 16E20
Classificationmsc 13D15
Classificationmsc 18F30
Synonymgroup completion
Related topicAlgebraicKTheory
Related topicKTheory
Related topicAlgebraicTopology
Related topicGrothendieckCategory
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更新时间:2025/5/4 21:11:00