Grothendieck group

In mathematics, the Grothendieck group construction constructs an abelian group from a commutative monoid in the most universal way, in the sense that any abelian group containing a homomorphic image of will also contain a homomorphic image of the Grothendieck group of . 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 additive inverses. 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".
Note that the existence of a zero element in the monoid runs counter to the inverse property, as the embedded zero element in "K" must have an inverse element formula_1 whose sum with 0 must simultaneously be 0 and 1, forcing formula_2. The general construction in the presence of zero elements always constructs the trivial group, as the only group which satisfies this equation.
Universal property.
Let "M" be a commutative monoid. Its Grothendieck group "K" is an abelian group with the following universal property: There exists a monoid homomorphism
such that for any monoid homomorphism
from the commutative monoid "M" to an abelian group "A", there is a unique group homomorphism
such that
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_7. The two coordinates are meant to represent a positive part and a negative part, so formula_8 corresponds to formula_9 in "K".
Addition on formula_10 is defined coordinate-wise:
Next one defines an equivalence relation on formula_7, such that formula_8 is equivalent to formula_14 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_15 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_16 the free abelian group generated by the set "M", the Grothendieck group "K" is the quotient of formula_17 by the subgroup generated by formula_18. (Here +′ and −′ denote the addition and subtraction in the free abelian group formula_17 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_20 from the (additive) natural numbers formula_21.
First one observes that the natural numbers (including 0) together with the usual addition indeed form a commutative monoid formula_22 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
Now define
This defines the integers formula_20. 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_27 (starting at 1) consists of formal fractions formula_28 with the equivalence 
Example: the Grothendieck group of a manifold.
The Grothendieck group is the fundamental construction of K-theory. The group formula_31 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_32 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_33 is a covariant functor from rings to abelian groups.
The two previous examples are related: consider the case where formula_34 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_32 and formula_31 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_37 as the abelian group generated by the set formula_38 of isomorphism classes of finitely generated "R"-modules and the following relations: For every short exact sequence
of "R"-modules add the relation
This definition implies that for any two finitely generated "R"-modules "M" and "N", formula_41, because of the split short exact sequence.
Examples.
Let "K" be a field. Then the Grothendieck group formula_43 is an abelian group generated by symbols formula_44 for any finite dimensional "K"-vector space "V". In fact, formula_43 is isomorphic to formula_46 whose generator is the element formula_47. Here, the symbol formula_44 for a finite "K"-vector space "V" is defined as formula_49, the dimension of the vector space "V". Suppose one has the following short exact sequence of "K"-vector spaces.
Since any short exact sequence of vector spaces splits, it holds that formula_51. In fact, for any two finite dimensional vector spaces "V" and "W" the following holds.
The above equality hence satisfies the condition of the symbol formula_44 in the Grothendieck group.
Note that any two isomorphic finite dimensional "K"-vector space has the same dimension. Also, any two finite dimensional "K"-vector space "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_55. The observation from the previous paragraph hence proves the following equation. 
Hence, every symbol formula_44 is generated by the element formula_47 with integer coefficients, which implies that formula_43 is isomorphic to formula_20 with the generator formula_47.
More generally, let formula_20 be the set of integers. The Grothendieck group formula_63 is an abelian group generated by symbols formula_64 for any finitely generated abelian groups "A". One first notes that any finite abelian group "G" satisfies that formula_65. The following short exact sequence holds, where the map formula_66 is multiplication by "n".
The exact sequence implies that formula_68, so every cyclic group has its symbol equal to 0. This in turn implies that every finite abelian group "G" satisfies formula_65 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_70 for some non-negative integer "r", called the rank of "A" and denoted by formula_71 . Define the symbol formula_64 as formula_73. Then the Grothendieck group formula_63 is isomorphic to formula_20 with generator formula_76 Indeed, the observation made from the previous paragraph shows that every abelian group "A" has its symbol formula_64 the same to the symbol formula_78 where formula_71. Furthermore, the rank of the abelian group satisfies the conditions of the symbol formula_64 of the Grothendieck group. Suppose one has the following short exact sequence of abelian groups.
Then tensoring with the rational numbers formula_82 implies the following equation.
Since the above is a short exact sequence of formula_82-vector spaces, the sequence splits. Therefore, one has the following equation.
On the other hand, one also has the following relation. For more information, see: Rank of Abelian Group.
Therefore, the following equation holds.
Hence one has shown that formula_63 is isomorphic to formula_20 with generator formula_76
Universal Property.
The Grothendieck group satisfies a universal property. One makes a preliminary definition: A function formula_91 from the set of isomorphism classes to an abelian group formula_92 is called "additive" if, for each exact sequence formula_93, one has formula_94 Then, for any additive function formula_95, there is a "unique" group homomorphism formula_96 such that formula_91 factors through "formula_98" and the map that takes each object of formula_99 to the element representing its isomorphism class in formula_100 Concretely this means that formula_98 satisfies the equation formula_102 for every finitely generated formula_103-module formula_104 and formula_98 is the only group homomorphism that does that.
Examples of additive functions are the character function from representation theory: If formula_103 is a finite-dimensional formula_107-algebra, then one can associate the character formula_108 to every finite-dimensional formula_103-module formula_110 is defined to be the trace of the formula_107-linear map that is given by multiplication with the element formula_112 on formula_104.
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_114 such that formula_115.
If formula_116 and formula_103 is the group ring formula_118 of a finite group formula_119 then this character map even gives a natural isomorphism of formula_120 and the character ring formula_121. In the modular representation theory of finite groups formula_107 can be a field formula_123 the algebraic closure of the finite field with "p" elements. In this case the analogously defined map that associates to each formula_124-module its Brauer character is also a natural isomorphism formula_125 onto the ring of Brauer characters. In this way Grothendieck groups show up in representation theory.
This universal property also makes formula_37 the 'universal receiver' of generalized Euler characteristics. In particular, for every bounded complex of objects in formula_127
one has a canonical element
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_130. 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"] for each (isomorphism class of) object(s) of the category formula_130 and one relation
for each exact sequence
Alternatively and equivalently, one can define the Grothendieck group using a universal property: A map formula_134 from formula_130 into an abelian group "X" is called "additive" if for every exact sequence formula_133 one has formula_137; an abelian group "G" together with an additive mapping formula_138 is called the Grothendieck group of formula_130 iff every additive map formula_134 factors uniquely through φ.
Every abelian category is an exact category if one just uses the standard interpretation of "exact". This gives the notion of a Grothendieck group in the previous section if one chooses formula_141-mod the category of finitely generated "R"-modules as formula_130. This is really abelian because "R" was assumed to be artinian and (hence noetherian) in the previous section.
On the other hand, every additive category is also exact if one declares those and only those sequences to be exact that have the form formula_143 with the canonical inclusion and projection morphisms. This procedure produces the Grothendieck group of the commutative monoid formula_144 in the first sense (here formula_145 means the "set" [ignoring all foundational issues] of isomorphism classes in formula_130.)
Grothendieck groups of triangulated categories.
Generalizing even further it is also possible to define the Grothendieck group for triangulated categories. The construction is essentially similar but uses the relations ["X"] - ["Y"] + ["Z"] = 0 whenever there is a distinguished triangle "X" → "Y" → "Z" → "X"[1].