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 ${\displaystyle a\neq b}$ and ${\displaystyle ac=bc}$), 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 ${\displaystyle 0.x=0}$ for every ${\displaystyle x\in M,}$ the Grothendieck group must be the trivial group (group with only one element), since one must have

${\displaystyle x=1.x=(0^{-1}.0).x=0^{-1}.(0.x)=0^{-1}.(0.0)=(0^{-1}.0).0=1.0=0}$

for every x.

Universal property

Let M be a commutative monoid. Its Grothendieck group is an abelian group K with a monoid homomorphism ${\displaystyle i\colon M\to K}$ satisfying the following universal property: for any monoid homomorphism ${\displaystyle f\colon M\to A}$ from M to an abelian group A, there is a unique group homomorphism ${\displaystyle g\colon K\to A}$ such that ${\displaystyle f=g\circ i.}$

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 ${\displaystyle M\times M}$. The two coordinates are meant to represent a positive part and a negative part, so ${\displaystyle (m_{1},m_{2})}$ corresponds to ${\displaystyle m_{1}-m_{2}}$ in K.

Addition on ${\displaystyle M\times M}$ is defined coordinate-wise:

${\displaystyle (m_{1},m_{2})+(n_{1},n_{2})=(m_{1}+n_{1},m_{2}+n_{2})}$.

Next one defines an equivalence relation on ${\displaystyle M\times M}$, such that ${\displaystyle (m_{1},m_{2})}$ is equivalent to ${\displaystyle (n_{1},n_{2})}$ 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)]. 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 [(m₁, m₂)] is [(m₂, m₁)]. The homomorphism ${\displaystyle i:M\to K}$ sends the element m to [(m, 0)].

Alternatively, the Grothendieck group K of M can also be constructed using generators and relations: denoting by ${\displaystyle (Z(M),+')}$ the free abelian group generated by the set M, the Grothendieck group K is the quotient of ${\displaystyle Z(M)}$ by the subgroup generated by ${\displaystyle \{(x+'y)-'(x+y)\mid x,y\in M\}}$. (Here +′ and −′ denote the addition and subtraction in the free abelian group ${\displaystyle Z(M)}$ 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 ${\displaystyle \mathbb {Z} }$ from the (additive) natural numbers ${\displaystyle \mathbb {N} }$. First one observes that the natural numbers (including 0) together with the usual addition indeed form a commutative monoid ${\displaystyle (\mathbb {N} ,+).}$ 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

${\displaystyle n-m\sim n'-m'\iff n+m'+k=n'+m+k}$ for some ${\displaystyle k\iff n+m'=n'+m}$.

Now define

${\displaystyle \forall n\in \mathbb {N}$ :\qquad {\begin{cases}n:=[n-0]\\-n:=[0-n]\end{cases}}}

This defines the integers ${\displaystyle \mathbb {Z} }$. 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 ${\displaystyle (\mathbb {N} ^{*},\times )}$ (starting at 1) consists of formal fractions ${\displaystyle p/q}$ with the equivalence

${\displaystyle p/q\sim p'/q'\iff pq'r=p'qr}$ for some ${\displaystyle r\iff pq'=p'q}$

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 ${\displaystyle K_{0}(M)}$ 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.

Example: The Grothendieck group of a ring

The zeroth algebraic K group ${\displaystyle K_{0}(R)}$ 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 ${\displaystyle K_{0}}$ is a covariant functor from rings to abelian groups.

The two previous examples are related: consider the case where ${\displaystyle R=C^{\infty }(M)}$ 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 ${\displaystyle K_{0}(R)}$ and ${\displaystyle K_{0}(M)}$ are the same group.

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 ${\displaystyle G_{0}(R)}$ as the abelian group generated by the set ${\displaystyle \{[X]\mid X\in R{\text{-mod}}\}}$ of isomorphism classes of finitely generated R-modules and the following relations: For every short exact sequence

${\displaystyle 0\to A\to B\to C\to 0}$

of R-modules, add the relation

${\displaystyle [A]-[B]+[C]=0.}$

This definition implies that for any two finitely generated R-modules M and N, ${\displaystyle [M\oplus N]=[M]+[N]}$, because of the split short exact sequence

${\displaystyle 0\to M\to M\oplus N\to N\to 0.}$

Examples

Let K be a field. Then the Grothendieck group ${\displaystyle G_{0}(K)}$ is an abelian group generated by symbols ${\displaystyle [V]}$ for any finite-dimensional K-vector space V. In fact, ${\displaystyle G_{0}(K)}$ is isomorphic to ${\displaystyle \mathbb {Z} }$ whose generator is the element ${\displaystyle [K]}$. Here, the symbol ${\displaystyle [V]}$ for a finite-dimensional K-vector space V is defined as ${\displaystyle [V]=\dim _{K}V}$, the dimension of the vector space V. Suppose one has the following short exact sequence of K-vector spaces.

${\displaystyle 0\to V\to T\to W\to 0}$

Since any short exact sequence of vector spaces splits, it holds that ${\displaystyle T\cong V\oplus W}$. In fact, for any two finite-dimensional vector spaces V and W the following holds:

${\displaystyle \dim _{K}(V\oplus W)=\dim _{K}(V)+\dim _{K}(W)}$

The above equality hence satisfies the condition of the symbol ${\displaystyle [V]}$ in the Grothendieck group.

${\displaystyle [T]=[V\oplus W]=[V]+[W]}$

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 ${\displaystyle K^{\oplus n}}$. The observation from the previous paragraph hence proves the following equation:

${\displaystyle [V]=\left[K^{\oplus n}\right]=n[K]}$

Hence, every symbol ${\displaystyle [V]}$ is generated by the element ${\displaystyle [K]}$ with integer coefficients, which implies that ${\displaystyle G_{0}(K)}$ is isomorphic to ${\displaystyle \mathbb {Z} }$ with the generator ${\displaystyle [K]}$.

More generally, let ${\displaystyle \mathbb {Z} }$ be the set of integers. The Grothendieck group ${\displaystyle G_{0}(\mathbb {Z} )}$ is an abelian group generated by symbols ${\displaystyle [A]}$ for any finitely generated abelian groups A. One first notes that any finite abelian group G satisfies that ${\displaystyle [G]=0}$. The following short exact sequence holds, where the map ${\displaystyle \mathbb {Z} \to \mathbb {Z} }$ is multiplication by n.

${\displaystyle 0\to \mathbb {Z} \to \mathbb {Z} \to \mathbb {Z} /n\mathbb {Z} \to 0}$

The exact sequence implies that ${\displaystyle [\mathbb {Z} /n\mathbb {Z} ]=[\mathbb {Z} ]-[\mathbb {Z} ]=0}$, so every cyclic group has its symbol equal to 0. This in turn implies that every finite abelian group G satisfies ${\displaystyle [G]=0}$ 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 ${\displaystyle \mathbb {Z} ^{r}}$ for some non-negative integer r, called the rank of A and denoted by ${\displaystyle r={\mbox{rank}}(A)}$ . Define the symbol ${\displaystyle [A]}$ as ${\displaystyle [A]={\mbox{rank}}(A)}$. Then the Grothendieck group ${\displaystyle G_{0}(\mathbb {Z} )}$ is isomorphic to ${\displaystyle \mathbb {Z} }$ with generator ${\displaystyle [\mathbb {Z} ].}$ Indeed, the observation made from the previous paragraph shows that every abelian group A has its symbol ${\displaystyle [A]}$ the same to the symbol ${\displaystyle [\mathbb {Z} ^{r}]=r[\mathbb {Z} ]}$ where ${\displaystyle r={\mbox{rank}}(A)}$. Furthermore, the rank of the abelian group satisfies the conditions of the symbol ${\displaystyle [A]}$ of the Grothendieck group. Suppose one has the following short exact sequence of abelian groups:

${\displaystyle 0\to A\to B\to C\to 0}$

Then tensoring with the rational numbers ${\displaystyle \mathbb {Q} }$ implies the following equation.

${\displaystyle 0\to A\otimes _{\mathbb {Z} }\mathbb {Q} \to B\otimes _{\mathbb {Z} }\mathbb {Q} \to C\otimes _{\mathbb {Z} }\mathbb {Q} \to 0}$

Since the above is a short exact sequence of ${\displaystyle \mathbb {Q} }$-vector spaces, the sequence splits. Therefore, one has the following equation.

${\displaystyle \dim _{\mathbb {Q} }(B\otimes _{\mathbb {Z} }\mathbb {Q} )=\dim _{\mathbb {Q} }(A\otimes _{\mathbb {Z} }\mathbb {Q} )+\dim _{\mathbb {Q} }(C\otimes _{\mathbb {Z} }\mathbb {Q} )}$

On the other hand, one also has the following relation; for more information, see Rank of an abelian group.

${\displaystyle \operatorname {rank} (A)=\dim _{\mathbb {Q} }(A\otimes _{\mathbb {Z} }\mathbb {Q} )}$

Therefore, the following equation holds:

${\displaystyle [B]=\operatorname {rank} (B)=\operatorname {rank} (A)+\operatorname {rank} (C)=[A]+[C]}$

Hence one has shown that ${\displaystyle G_{0}(\mathbb {Z} )}$ is isomorphic to ${\displaystyle \mathbb {Z} }$ with generator ${\displaystyle [\mathbb {Z} ].}$

Universal Property

The Grothendieck group satisfies a universal property. One makes a preliminary definition: A function ${\displaystyle \chi }$ from the set of isomorphism classes to an abelian group ${\displaystyle X}$ is called additive if, for each exact sequence ${\displaystyle 0\to A\to B\to C\to 0}$, one has ${\displaystyle \chi (A)-\chi (B)+\chi (C)=0.}$ Then, for any additive function ${\displaystyle \chi :R{\text{-mod}}\to X}$, there is a unique group homomorphism ${\displaystyle f:G_{0}(R)\to X}$ such that ${\displaystyle \chi }$ factors through ${\displaystyle f}$ and the map that takes each object of ${\displaystyle {\mathcal {A}}}$ to the element representing its isomorphism class in ${\displaystyle G_{0}(R).}$ Concretely this means that ${\displaystyle f}$ satisfies the equation ${\displaystyle f([V])=\chi (V)}$ for every finitely generated ${\displaystyle R}$-module ${\displaystyle V}$ and ${\displaystyle f}$ is the only group homomorphism that does that.

Examples of additive functions are the character function from representation theory: If ${\displaystyle R}$ is a finite-dimensional ${\displaystyle k}$-algebra, then one can associate the character ${\displaystyle \chi _{V}:R\to k}$ to every finite-dimensional ${\displaystyle R}$-module ${\displaystyle V:\chi _{V}(x)}$ is defined to be the trace of the ${\displaystyle k}$-linear map that is given by multiplication with the element ${\displaystyle x\in R}$ on ${\displaystyle V}$.

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" ${\displaystyle \chi :G_{0}(R)\to \mathrm {Hom} _{K}(R,K)}$ such that ${\displaystyle \chi ([V])=\chi _{V}}$.

If ${\displaystyle k=\mathbb {C} }$ and ${\displaystyle R}$ is the group ring ${\displaystyle \mathbb {C} [G]}$ of a finite group ${\displaystyle G}$ then this character map even gives a natural isomorphism of ${\displaystyle G_{0}(\mathbb {C} [G])}$ and the character ring ${\displaystyle Ch(G)}$. In the modular representation theory of finite groups, ${\displaystyle k}$ can be a field ${\displaystyle {\overline {\mathbb {F} _{p}}},}$ the algebraic closure of the finite field with p elements. In this case the analogously defined map that associates to each ${\displaystyle k[G]}$-module its Brauer character is also a natural isomorphism ${\displaystyle G_{0}({\overline {\mathbb {F} _{p}}}[G])\to \mathrm {BCh} (G)}$ onto the ring of Brauer characters. In this way Grothendieck groups show up in representation theory.

This universal property also makes ${\displaystyle G_{0}(R)}$ the 'universal receiver' of generalized Euler characteristics. In particular, for every bounded complex of objects in ${\displaystyle R{\text{-mod}}}$

${\displaystyle \cdots \to 0\to 0\to A^{n}\to A^{n+1}\to \cdots \to A^{m-1}\to A^{m}\to 0\to 0\to \cdots }$

one has a canonical element

${\displaystyle [A^{*}]=\sum _{i}(-1)^{i}[A^{i}]=\sum _{i}(-1)^{i}[H^{i}(A^{*})]\in G_{0}(R).}$

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