Abelian group extending a commutative monoid
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid "M" is a certain abelian group. This abelian group is constructed from "M" in the most universal way, in the sense that any abelian group containing a homomorphic image of "M" will also contain a homomorphic image of the Grothendieck group of "M". The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.
Grothendieck group of a commutative monoid.
Motivation.
Given a commutative monoid M, "the most general" abelian group K that arises from M is to be constructed by introducing inverse elements to all elements of M. Such an abelian group K always exists; it is called the Grothendieck group of M. It is characterized by a certain universal property and can also be concretely constructed from M.
If M does not have the cancellation property (that is, there exists "a", "b" and c in M such that formula_1 and formula_2), then the Grothendieck group K cannot contain M. In particular, in the case of a monoid operation denoted multiplicatively that has a zero element satisfying formula_3 for every formula_4 the Grothendieck group must be the trivial group (group with only one element), since one must have 
formula_5
for every x.
Universal property.
Let "M" be a commutative monoid. Its Grothendieck group is an abelian group "K" with a monoid homomorphism formula_6 satisfying the following universal property: for any monoid homomorphism formula_7 from "M" to an abelian group "A", there is a unique group homomorphism formula_8 such that formula_9
This expresses the fact that any abelian group "A" that contains a homomorphic image of "M" will also contain a homomorphic image of "K", "K" being the "most general" abelian group containing a homomorphic image of "M".
Explicit constructions.
To construct the Grothendieck group "K" of a commutative monoid "M", one forms the Cartesian product formula_10. The two coordinates are meant to represent a positive part and a negative part, so formula_11 corresponds to formula_12 in "K".
Addition on formula_13 is defined coordinate-wise:
formula_14.
Next one defines an equivalence relation on formula_10, such that formula_11 is equivalent to formula_17 if, for some element "k" of "M", "m"1 + "n"2 + "k" = "m"2 + "n"1 + "k" (the element "k" is necessary because the cancellation law does not hold in all monoids). The equivalence class of the element ("m"1, "m"2) is denoted by [("m"1, "m"2)]. One defines "K" to be the set of equivalence classes. Since the addition operation on "M" × "M" is compatible with our equivalence relation, one obtains an addition on "K", and "K" becomes an abelian group. The identity element of "K" is [(0, 0)], and the inverse of [("m"1, "m"2)] is [("m"2, "m"1)]. The homomorphism formula_18 sends the element "m" to [("m", 0)].
Alternatively, the Grothendieck group "K" of "M" can also be constructed using generators and relations: denoting by formula_19 the free abelian group generated by the set "M", the Grothendieck group "K" is the quotient of formula_20 by the subgroup generated by formula_21. (Here +′ and −′ denote the addition and subtraction in the free abelian group formula_20 while + denotes the addition in the monoid "M".) This construction has the advantage that it can be performed for any semigroup "M" and yields a group which satisfies the corresponding universal properties for semigroups, i.e. the "most general and smallest group containing a homomorphic image of "M"". This is known as the "group completion of a semigroup" or "group of fractions of a semigroup".
Properties.
In the language of category theory, any universal construction gives rise to a functor; one thus obtains a functor from the category of commutative monoids to the category of abelian groups which sends the commutative monoid "M" to its Grothendieck group "K". This functor is left adjoint to the forgetful functor from the category of abelian groups to the category of commutative monoids.
For a commutative monoid "M", the map "i" : "M" → "K" is injective if and only if "M" has the cancellation property, and it is bijective if and only if "M" is already a group.
Example: the integers.
The easiest example of a Grothendieck group is the construction of the integers formula_23 from the (additive) natural numbers formula_24.
First one observes that the natural numbers (including 0) together with the usual addition indeed form a commutative monoid formula_25 Now when one uses the Grothendieck group construction one obtains the formal differences between natural numbers as elements "n" − "m" and one has the equivalence relation
formula_26 for some formula_27.
Now define
formula_28
This defines the integers formula_23. Indeed, this is the usual construction to obtain the integers from the natural numbers. See "Construction" under Integers for a more detailed explanation.
Example: the positive rational numbers.
Similarly, the Grothendieck group of the multiplicative commutative monoid formula_30 (starting at 1) consists of formal fractions formula_31 with the equivalence 
formula_32 for some formula_33
which of course can be identified with the positive rational numbers.
Example: the Grothendieck group of a manifold.
The Grothendieck group is the fundamental construction of K-theory. The group formula_34 of a compact manifold "M" is defined to be the Grothendieck group of the commutative monoid of all isomorphism classes of vector bundles of finite rank on "M" with the monoid operation given by direct sum. This gives a contravariant functor from manifolds to abelian groups. This functor is studied and extended in topological K-theory.
Example: The Grothendieck group of a ring.
The zeroth algebraic K group formula_35 of a (not necessarily commutative) ring "R" is the Grothendieck group of the monoid consisting of isomorphism classes of finitely generated projective modules over "R", with the monoid operation given by the direct sum. Then formula_36 is a covariant functor from rings to abelian groups.
The two previous examples are related: consider the case where formula_37 is the ring of complex-valued smooth functions on a compact manifold "M". In this case the projective "R"-modules are dual to vector bundles over "M" (by the Serre–Swan theorem). Thus formula_35 and formula_34 are the same group.
Grothendieck group and extensions.
Definition.
Another construction that carries the name Grothendieck group is the following: Let "R" be a finite-dimensional algebra over some field "k" or more generally an artinian ring. Then define the Grothendieck group formula_40 as the abelian group generated by the set formula_41 of isomorphism classes of finitely generated "R"-modules and the following relations: For every short exact sequence
formula_42
of "R"-modules, add the relation
formula_43
This definition implies that for any two finitely generated "R"-modules "M" and "N", formula_44, because of the split short exact sequence
formula_45
Examples.
Let "K" be a field. Then the Grothendieck group formula_46 is an abelian group generated by symbols formula_47 for any finite-dimensional "K"-vector space "V". In fact, formula_46 is isomorphic to formula_23 whose generator is the element formula_50. Here, the symbol formula_47 for a finite-dimensional "K"-vector space "V" is defined as formula_52, the dimension of the vector space "V". Suppose one has the following short exact sequence of "K"-vector spaces.
formula_53
Since any short exact sequence of vector spaces splits, it holds that formula_54. In fact, for any two finite-dimensional vector spaces "V" and "W" the following holds:
formula_55
The above equality hence satisfies the condition of the symbol formula_47 in the Grothendieck group.
formula_57
Note that any two isomorphic finite-dimensional "K"-vector spaces have the same dimension. Also, any two finite-dimensional "K"-vector spaces "V" and "W" of same dimension are isomorphic to each other. In fact, every finite "n"-dimensional "K"-vector space "V" is isomorphic to formula_58. The observation from the previous paragraph hence proves the following equation:
formula_59
Hence, every symbol formula_47 is generated by the element formula_50 with integer coefficients, which implies that formula_46 is isomorphic to formula_23 with the generator formula_50.
More generally, let formula_23 be the set of integers. The Grothendieck group formula_66 is an abelian group generated by symbols formula_67 for any finitely generated abelian groups "A". One first notes that any finite abelian group "G" satisfies that formula_68. The following short exact sequence holds, where the map formula_69 is multiplication by "n".
formula_70
The exact sequence implies that formula_71, so every cyclic group has its symbol equal to 0. This in turn implies that every finite abelian group "G" satisfies formula_68 by the fundamental theorem of finite abelian groups.
Observe that by the fundamental theorem of finitely generated abelian groups, every abelian group "A" is isomorphic to a direct sum of a torsion subgroup and a torsion-free abelian group isomorphic to formula_73 for some non-negative integer "r", called the rank of "A" and denoted by formula_74 . Define the symbol formula_67 as formula_76. Then the Grothendieck group formula_66 is isomorphic to formula_23 with generator formula_79 Indeed, the observation made from the previous paragraph shows that every abelian group "A" has its symbol formula_67 the same to the symbol formula_81 where formula_74. Furthermore, the rank of the abelian group satisfies the conditions of the symbol formula_67 of the Grothendieck group. Suppose one has the following short exact sequence of abelian groups:
formula_42
Then tensoring with the rational numbers formula_85 implies the following equation.
formula_86
Since the above is a short exact sequence of formula_85-vector spaces, the sequence splits. Therefore, one has the following equation.
formula_88
On the other hand, one also has the following relation; for more information, see Rank of an abelian group.
formula_89
Therefore, the following equation holds:
formula_90
Hence one has shown that formula_66 is isomorphic to formula_23 with generator formula_79
Universal Property.
The Grothendieck group satisfies a universal property. One makes a preliminary definition: A function formula_94 from the set of isomorphism classes to an abelian group formula_95 is called "additive" if, for each exact sequence formula_42, one has formula_97 Then, for any additive function formula_98, there is a "unique" group homomorphism formula_99 such that formula_94 factors through "formula_101" and the map that takes each object of formula_102 to the element representing its isomorphism class in formula_103 Concretely this means that formula_101 satisfies the equation formula_105 for every finitely generated formula_106-module formula_107 and formula_101 is the only group homomorphism that does that.
Examples of additive functions are the character function from representation theory: If formula_106 is a finite-dimensional formula_110-algebra, then one can associate the character formula_111 to every finite-dimensional formula_106-module formula_113 is defined to be the trace of the formula_110-linear map that is given by multiplication with the element formula_115 on formula_107.
By choosing a suitable basis and writing the corresponding matrices in block triangular form one easily sees that character functions are additive in the above sense. By the universal property this gives us a "universal character" formula_117 such that formula_118.
If formula_119 and formula_106 is the group ring formula_121 of a finite group formula_122 then this character map even gives a natural isomorphism of formula_123 and the character ring formula_124. In the modular representation theory of finite groups, formula_110 can be a field formula_126 the algebraic closure of the finite field with "p" elements. In this case the analogously defined map that associates to each formula_127-module its Brauer character is also a natural isomorphism formula_128 onto the ring of Brauer characters. In this way Grothendieck groups show up in representation theory.
This universal property also makes formula_40 the 'universal receiver' of generalized Euler characteristics. In particular, for every bounded complex of objects in formula_130
formula_131
one has a canonical element
formula_132
In fact the Grothendieck group was originally introduced for the study of Euler characteristics.
Grothendieck groups of exact categories.
A common generalization of these two concepts is given by the Grothendieck group of an exact category formula_133. Simply put, an exact category is an additive category together with a class of distinguished short sequences "A" → "B" → "C". The distinguished sequences are called "exact sequences", hence the name. The precise axioms for this distinguished class do not matter for the construction of the Grothendieck group.
The Grothendieck group is defined in the same way as before as the abelian group with one generator ["M"&amp;hairsp;] for each (isomorphism class of) object(s) of the category formula_133 and one relation
formula_135
for each exact sequence
formula_136.
Alternatively and equivalently, one can define the Grothendieck group using a universal property: A map formula_137 from formula_133 into an abelian group "X" is called "additive" if for every exact sequence formula_136 one has formula_140; an abelian group "G" together with an additive mapping formula_141 is called the Grothendieck group of formula_133 iff every additive map formula_137 factors uniquely through formula_144.
Every abelian category is an exact