In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid 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 the more general construction in category theory, introduced by Alexander Grothendieck in his fundamental work of the mid-1950s that resulted in the development of K-theory, which led to his proof of the Grothendieck-Riemann-Roch theorem. This article treats both constructions.
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Motivation
Given a commutative monoid M, we want to construct "the most general" abelian group K that arises from M 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.
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 of a commutative monoid M, one forms the Cartesian product
M×M.(The two coordinates are meant to represent a positive part and a negative part: (m1, m2) is meant to correspond to the element m1 − m2 in K.)
Addition on MxM 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 (the element k is necessary because the cancellation law does not hold in all monoids). The equivalence class of the element (m1, m2) is denoted by [(m1, m2)]. We define K to be the set of equivalence classes. Since the addition operation on M×M is compatible with our equivalence relation, we obtain an addition on K, and K becomes an abelian group. The identity element of K is [(0, 0)], and the inverse of [(m1, m2)] is [(m2, m1)]. The homomorphism i : M→K sends the element m to [(m, 0)].
Alternatively, the Grothendieck group K of M can also be constructed using generators and relations: denoting by (Z(M),+') the free abelian group generated by the set M, the Grothendieck group K is the quotient of Z(M) by the subgroup generated by
Properties
In the language of category theory, any universal construction gives rise to a functor; we thus obtain 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.
Examples: the integers, the Grothendieck group of a manifold and of a ring
The easiest example of a Grothendieck group is the construction of the integers Z from the natural numbers N. First one observes that the natural numbers (including 0) together with the usual addition indeed form a commutative monoid (N,+). Now when we use the Grothendieck group construction we obtain the formal differences between natural numbers as elements n - m and we have the equivalence relation
Now define
for all n ∈ N. This defines the integers Z. Indeed, this is the usual construction to obtain the integers from the natural numbers. See "Construction" under Integers for a more detailed explanation.
The Grothendieck group is the fundamental construction of K-theory. The group K0(M) 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.
The zeroth algebraic K group K0(R) 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 K0 is a covariant functor from rings to abelian groups.
The two previous examples are related: consider the case where R is the ring
Grothendieck group and extensions
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 G0(R) as the abelian group generated by the set
of R-modules add the relation
The abelian group defined by these generators and these relations is the Grothendieck group G0(R).
This group satisfies a universal property. We make a preliminary definition: A function χ from the set of isomorphism classes to an abelian group A is called additive if, for each exact sequence 0 → A → B → C → 0, we have
Examples of additive functions are the character function from representation theory: If R is a finite-dimensional k-algebra, then we can associate the character χV: R → k to every finite-dimensional R-module V: χV(x) is defined to be the trace of the k-linear map that is given by multiplication with the element x ∈ R on V.
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"
If k = C and R is the group ring C[G] of a finite group G then this character map even gives a natural isomorphism of G0(C[G]) and the character ring Ch(G). In the modular representation theory of finite groups k can be a field
This universal property also makes G0(R) the 'universal receiver' of generalized Euler characteristics. In particular, for every bounded complex of objects in R-mod
we have 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
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
for each exact sequence
Alternatively one can define the Grothendieck group using a similar universal property: An abelian group G together with a mapping
Every abelian category is an exact category if we just use the standard interpretation of "exact". This gives the notion of a Grothendieck group in the previous section if we choose
On the other hand, every additive category is also exact if we declare those and only those sequences to be exact that have the form
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].