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UNIVERSAL PROPERTY


In its simplest form, the Grothendieck group of a commutative monoid is the universal way of making that monoid into an abelian group. For example, let ''M'' be a commutative monoid. Its Grothendieck group ''N'' should have the following universal property: There exists a map

i


such that for any map

f


from a commutative monoid ''M'' to an abelian group ''A'', there is a unique map

g


such that

f


In other words, in the conventional language of Category Theory , the Forgetful Functor from the Category of abelian groups to the category of commutative monoids has a Left Adjoint .


EXPLICIT CONSTRUCTION


To construct the Grothendieck group of a commutative monoid ''M'', one forms the Cartesian product

M


The two coordinates are meant to represent a positive part and a negative part:

:(''m'', ''n'')

is meant to correspond to

m


Addition is defined coordinate-wise:

:(''m1'', ''m2'') + (''n1'', ''n2'') = (''m1'' + ''n1'', ''m2'' + ''n2''). Next we define an equivalence relation on ''M''×''M''. We say that (''m1'', ''m2'') is equivalent to (''n1'', ''n2'') if, for some element ''k'' of ''M'', ''m1'' + ''n2'' + ''k'' = ''m2'' + ''n1'' + ''k''. It is easy to check that the addition operation is compatible with the equivalence relation. The identity element is now any element of the form (''m'', ''m''), and the inverse of (''m1'', ''m2'') is (''m2'', ''m1'').

In this form, the Grothendieck group is the fundamental construction of K-theory . The group ''K0(M)'' of a Manifold ''M'' is defined to be the Grothendieck group of the commutative monoid of all Vector Bundle s on ''M'' with the group operation given by direct sum.


GENERALIZATION


To apply the Grothendieck group to purely algebraic settings, it is useful to generalize it to the case of an Essentially Small Abelian Category . To do this, let \mathcal A be an essentially small abelian category. Let ''F'' be the Free Abelian Group generated by isomorphism classes of objects of the category. (This is where the hypothesis of essential smallness is necessary; without it, ''F'' would not be a set.) We will impose some relations on ''F''. Call ''R'' the subgroup of ''F'' generated as follows: For each exact sequence 0→''A''→''B''→''C''→0 in \mathcal A, the element

: + [''C'' - [''B'']

is in ''R''. Then the Grothendieck group K_0({\mathcal A}) is ''F''/''R''.

''K0'' of an abelian category has a similar universal property to ''K0'' of a commutative monoid. We make a preliminary definition: A function χ from isomorphism classes of objects of an abelian category \mathcal A to an abelian group ''A'' is called ''additive'' if, for each exact sequence 0→''A''→''B''→''C''→0, we have χ(''A'') + χ(''C'') - χ(''B'') = 0. Then, for any additive function χ:\mathcal A→''A'', there is a unique abelian group homomorphism ''f'':K_0{\mathcal A}→''A'' such that χ factors through ''f'' and the map that takes each object of \mathcal A to the element representing its isomorphism class in K_0({\mathcal A}).

This universal property makes K_0({\mathcal A}) the 'universal receiver' of generalized Euler Characteristic s. In particular, for every Bounded Complex of objects in {\mathcal A}
: \cdots o 0 o 0 o A^n o A^{n+1} o \cdots o A^{m-1} o A^m o 0 o 0 o \cdots
we have a canonical element
  • ]" class="copylinks" target="_blank">= \sum_i (-1)^i [A^i = \sum_i (-1)^i (A^---) \in K_0.

    • In fact the Grothendieck group was originally introduced for the study of Euler characteristics.



SPLITTING PRINCIPLE


The relationship between ''K0'' of a commutative monoid and ''K0'' of an abelian category comes from the Splitting Principle . According to the splitting principle, we can always take an exact sequence 0→''A''→''B''→''C''→0 and find a closely related exact sequence in which the middle term splits, that is, it is the direct sum of the other two terms. Because of this, the Grothendieck group of the commutative monoid of vector bundles on a smooth manifold is the same as the Grothendieck group of the abelian category of vector bundles on that same smooth manifold.

''K0'' is often defined for a s over the ring. For a ringed space (''X'',''OX''), one lets the abelian category \mathcal A be the category of all Coherent Sheaves on ''X''. This makes ''K0'' into a functor.

There is another Grothendieck group of a ring or a ringed space which is sometimes useful. The Grothendieck group ''G0'' of a ring is the Grothendieck group associated to the category of all finitely generated modules over a ring. Similarly, the Grothendieck group ''G0'' of a ringed space is the Grothendieck group associated to the category of all quasicoherent sheaves on the ringed space. ''G0'' is ''not'' a functor, but nevertheless it carries important information.


EXAMPLE


In the abelian category of finite dimensional Vector Space s over a Field k , two vector spaces are isomorphic if and only if they have the same dimension. Thus, for a vector space ''V'' the class = [k^{\mbox{dim}(V)} in K_0(Vect_{fin}). Moreover for an exact sequence
: 0 o k^l o k^m o k^n o 0
''m = l + n'', so
: = [k^l + = (l+n)[k .
  • ,

  • ]" class="copylinks" target="_blank">= \chi(V^---)[k

    • where \chi is the standard Euler characteristic defined by

  • )= \sum_i (-1)^i \mbox{dim}(V) = \sum_i (-1)^i \mbox{dim}(H^i(V^---))