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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.

Grothendieck group of a commutative monoid

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

i:M→Kdisplaystyle i:Mto K

such that for any monoid homomorphism

f:M→Adisplaystyle f:Mto A

from the commutative monoid M to an abelian group A, there is a unique group homomorphism

g:K→Adisplaystyle g:Kto A

such that

f=g∘i.displaystyle f=gcirc 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 M×M.displaystyle Mtimes M. The two coordinates are meant to represent a positive part and a negative part, so (m₁, m₂) corresponds to m₁ − m₂ in K.

Addition on M × M 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→Kdisplaystyle i:Mto 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),+′)displaystyle (Z(M),+') the free abelian group generated by the set M, the Grothendieck group K is the quotient of Z(M)displaystyle Z(M) by the subgroup generated by (x+′y)−′(x+y)∣x,y∈Mdisplaystyle (x+'y)-'(x+y)mid x,yin M. (Here +′ and −′ denote the addition and subtraction in the free abelian group Z(M)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; 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 Zdisplaystyle mathbb Z from the natural numbers N.displaystyle mathbb N . First one observes that the natural numbers (including 0) together with the usual addition indeed form a commutative monoid (N,+).displaystyle (mathbb 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

n−m∼n′−m′⇔n+m′=n′+mdisplaystyle n-msim n'-m'Leftrightarrow n+m'=n'+m.

Now define

∀n∈N:{n:=[n−0]−n:=[0−n]displaystyle forall nin mathbb N :qquad begincasesn:=[n-0]\-n:=[0-n]endcases

This defines the integers Zdisplaystyle 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.

The Grothendieck group is the fundamental construction of K-theory. The group K0(M)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.

The zeroth algebraic K group K0(R)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 K0displaystyle K_0 is a covariant functor from rings to abelian groups.

The two previous examples are related: consider the case where R is the ring C∞(M)displaystyle C^infty (M) 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 K0(R)displaystyle K_0(R) and K0(M)displaystyle K_0(M) 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 G0(R)displaystyle G_0(R) as the abelian group generated by the set [X]displaystyle Xin Rmathrm -Mod of isomorphism classes of finitely generated R-modules and the following relations: For every short exact sequence

0→A→B→C→0displaystyle 0to Ato Bto Cto 0

of R-modules add the relation

[A]−[B]+[C]=0displaystyle [A]-[B]+[C]=0

Note that the proposed definition of Grothendieck group G0(R)displaystyle G_0(R) is well-defined. Let R be an Artinian ring, and suppose Mdisplaystyle M and M′displaystyle M' are isomorphic finitely generated R-modules. Then there exists the following short exact sequence

0→0→M→M′→0displaystyle 0to 0to Mto M'to 0

The exact sequence hence implies that [0]−[M]+[M′]=0displaystyle [0]-[M]+[M']=0. Since [0]=0displaystyle [0]=0, it follows that [M]=[M′]displaystyle [M]=[M']. The proposed definition also implies that for any two finitely generated R-modules M and N, [M⊕N]=[M]+[N]displaystyle [Moplus N]=[M]+[N]. This follows from the given split short exact sequence.

0→M→M⊕N→N→0displaystyle 0to Mto Moplus Nto Nto 0

Examples

Let K be a field. Then the Grothendieck group G0(K)displaystyle G_0(K) is an abelian group generated by symbols [V]displaystyle [V] for any finite dimensional K-vector space V. In fact, G0(K)displaystyle G_0(K) is isomorphic to Zdisplaystyle mathbb Z whose generator is the element [K]displaystyle [K]. Here, the symbol [V]displaystyle [V] for a finite K-vector space V is defined as [V]=dimK⁡Vdisplaystyle [V]=dim _KV, the dimension of the vector space V. Suppose we have the following short exact sequence of K-vector spaces.

0→V→T→W→0displaystyle 0to Vto Tto Wto 0

Since any short exact sequence of vector spaces splits, it holds that T≅V⊕Wdisplaystyle Tcong Voplus W. In fact, for any two finite dimensional vector spaces V and W the following holds.

dimK⁡(V⊕W)=dimK⁡(V)+dimK⁡(W)displaystyle dim _K(Voplus W)=dim _K(V)+dim _K(W)

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

[T]=[V⊕W]=[V]+[W]displaystyle [T]=[Voplus W]=[V]+[W]

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 K⊕ndisplaystyle K^oplus n. The observation from the previous paragraph hence proves the following equation.

[V]=[K⊕n]=n[K]displaystyle [V]=left[K^oplus nright]=n[K]

Hence, every symbol [V]displaystyle [V] is generated by the element [K]displaystyle [K] with integer coefficients, which implies that G0(K)displaystyle G_0(K) is isomorphic to Zdisplaystyle mathbb Z with the generator [K]displaystyle [K].

More generally, let Zdisplaystyle mathbb Z be the set of integers. The Grothendieck group G0(Z)displaystyle G_0(mathbb Z ) is an abelian group generated by symbols [A]displaystyle [A] for any finitely generated abelian groups A. We first note that any finite abelian group G satisfies that [G]=0displaystyle [G]=0. The following short exact sequence holds, where the map Z→Zdisplaystyle mathbb Z to mathbb Z is multiplication by n.

0→Z→Z→Z/nZ→0displaystyle 0to mathbb Z to mathbb Z to mathbb Z /nmathbb Z to 0

The exact sequence implies that [Z/nZ]=[Z]−[Z]=0displaystyle [mathbb Z /nmathbb 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 [G]=0displaystyle [G]=0 by the Fundamental Theorem of Finite Abelian groups.

Observe that by the Fundamental Theorem of Finitely Generated Abelian Groups, every abelian group is isomorphic to a direct sum of a torsion subgroup and a torsion-free abelian group isomorphic to Zrdisplaystyle mathbb Z ^r for some non-negative integer r. Note that the integer r is defined as the rank of the abelian group A. Define the symbol [A]displaystyle [A] as [A]=Rank(A)displaystyle [A]=Rank(A). Then the Grothendieck group G0(Z)displaystyle G_0(mathbb Z ) is isomorphic to Zdisplaystyle mathbb Z with generator [Z].displaystyle [mathbb Z ]. Indeed, the observation made from the previous paragraph shows that every abelian group A has its symbol [A]displaystyle [A] the same to the symbol [Zr]=r[Z]displaystyle [mathbb Z ^r]=r[mathbb Z ] where r=Rank(A)displaystyle r=Rank(A). Furthermore, the rank of the abelian group satisfies the conditions of the symbol [A]displaystyle [A] of the Grothendieck group. Suppose we have the following short exact sequence of abelian groups.

0→A→B→C→0displaystyle 0to Ato Bto Cto 0

Then tensoring with the rational numbers Qdisplaystyle mathbb Q implies the following equation.

0→A⊗ZQ→B⊗ZQ→C⊗ZQ→0displaystyle 0to Aotimes _mathbb Z mathbb Q to Botimes _mathbb Z mathbb Q to Cotimes _mathbb Z mathbb Q to 0

Since the above is a short exact sequence of Qdisplaystyle mathbb Q -vector spaces, the sequence splits. Therefore, we have the following equation.

dimQ⁡(B⊗ZQ)=dimQ⁡(A⊗ZQ)+dimQ⁡(C⊗ZQ)displaystyle dim _mathbb Q (Botimes _mathbb Z mathbb Q )=dim _mathbb Q (Aotimes _mathbb Z mathbb Q )+dim _mathbb Q (Cotimes _mathbb Z mathbb Q )

On the other hand, we also have the following relation. For more information, see: Rank of Abelian Group.

rank⁡(A)=dimQ⁡(A⊗ZQ)displaystyle operatorname rank (A)=dim _mathbb Q (Aotimes _mathbb Z mathbb Q )

Therefore, the following equation holds.

[B]=rank⁡(B)=rank⁡(A)+rank⁡(C)=[A]+[C]displaystyle [B]=operatorname rank (B)=operatorname rank (A)+operatorname rank (C)=[A]+[C]

Hence we have shown that G0(Z)displaystyle G_0(mathbb Z ) is isomorphic to Zdisplaystyle mathbb Z with generator [Z].displaystyle [mathbb Z ].

Universal Property

Grothendieck group satisfies a universal property. We make a preliminary definition: A function χdisplaystyle chi from the set of isomorphism classes to an abelian group Adisplaystyle A is called additive if, for each exact sequence 0→A→B→C→0displaystyle 0to Ato Bto Cto 0, we have χ(A)−χ(B)+χ(C)=0.displaystyle chi (A)-chi (B)+chi (C)=0. Then, for any additive function χ:R-mod→Xdisplaystyle chi :Rtext-modto X, there is a unique group homomorphism f:G0(R)→Xdisplaystyle f:G_0(R)to X such that χdisplaystyle chi factors through f and the map that takes each object of Adisplaystyle mathcal A to the element representing its isomorphism class in G0(R).displaystyle G_0(R). Concretely this means that fdisplaystyle f satisfies the equation f([V])=χ(V)displaystyle f([V])=chi (V) for every finitely generated Rdisplaystyle R-module Vdisplaystyle V and fdisplaystyle f is the only group homomorphism that does that.

Examples of additive functions are the character function from representation theory: If Rdisplaystyle R is a finite-dimensional kdisplaystyle k-algebra, then we can associate the character χV:R→kdisplaystyle chi _V:Rto k to every finite-dimensional Rdisplaystyle R-module V:χV(x)displaystyle V:chi _V(x) is defined to be the trace of the kdisplaystyle k-linear map that is given by multiplication with the element x∈Rdisplaystyle xin R on Vdisplaystyle 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" χ:G0(R)→HomK(R,K)displaystyle chi :G_0(R)to mathrm Hom _K(R,K) such that χ([V])=χVdisplaystyle chi ([V])=chi _V.

If k=Cdisplaystyle k=mathbb C and Rdisplaystyle R is the group ring C[G]displaystyle mathbb C [G] of a finite group Gdisplaystyle G then this character map even gives a natural isomorphism of G0(C[G])displaystyle G_0(mathbb C [G]) and the character ring Ch(G)displaystyle Ch(G). In the modular representation theory of finite groups kdisplaystyle k can be a field Fp¯,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 k[G]displaystyle k[G]-module its Brauer character is also a natural isomorphism G0(Fp¯[G])→BCh(G)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 G0(R)displaystyle G_0(R) the 'universal receiver' of generalized Euler characteristics. In particular, for every bounded complex of objects in R-moddisplaystyle Rtext-mod

⋯→0→0→An→An+1→⋯→Am−1→Am→0→0→⋯displaystyle cdots to 0to 0to A^nto A^n+1to cdots to A^m-1to A^mto 0to 0to cdots

we have a canonical element

[A∗]=∑i(−1)i[Ai]=∑i(−1)i[Hi(A∗)]∈G0(R).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.

Grothendieck groups of exact categories

A common generalization of these two concepts is given by the Grothendieck group of an exact category Adisplaystyle mathcal A. 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 Adisplaystyle mathcal A and one relation

[A]−[B]+[C]=0displaystyle [A]-[B]+[C]=0

for each exact sequence

A↪B↠Cdisplaystyle Ahookrightarrow Btwoheadrightarrow C.

Alternatively one can define the Grothendieck group using a similar universal property: An abelian group G together with a mapping ϕ:Ob(A)→Gdisplaystyle phi :mathrm Ob (mathcal A)to G is called the Grothendieck group of Adisplaystyle mathcal A iff every "additive" map χ:Ob(A)→Xdisplaystyle chi :mathrm Ob (mathcal A)to X from Adisplaystyle mathcal A into an abelian group X ("additive" in the above sense, i.e. for every exact sequence A↪B↠Cdisplaystyle Ahookrightarrow Btwoheadrightarrow C we have χ(A)−χ(B)+χ(C)=0displaystyle chi (A)-chi (B)+chi (C)=0) factors uniquely through φ.

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 A:=Rdisplaystyle mathcal A:=R-mod the category of finitely generated R-modules as Adisplaystyle mathcal A. 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 we declare those and only those sequences to be exact that have the form A↪A⊕B↠Bdisplaystyle Ahookrightarrow Aoplus Btwoheadrightarrow B with the canonical inclusion and projection morphisms. This procedure produces the Grothendieck group of the commutative monoid (Iso(A),⊕)displaystyle (mathrm Iso (mathcal A),oplus ) in the first sense (here Iso(A)displaystyle mathrm Iso (mathcal A) means the "set" [ignoring all foundational issues] of isomorphism classes in Adisplaystyle mathcal A.)

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].

Further examples

• In the abelian category of finite-dimensional vector spaces over a field k, two vector spaces are isomorphic if and only if they have the same dimension. Thus, for a vector space V
  

[V]=[kdim⁡(V)]∈K0(Vectfin).displaystyle [V]=left[k^dim(V)right]in K_0(mathrm Vect _mathrm fin ).

Moreover, for an exact sequence

0→kl→km→kn→0displaystyle 0to k^lto k^mto k^nto 0

m = l + n, so

[kl+n]=[kl]+[kn]=(l+n)[k].displaystyle left[k^l+nright]=left[k^lright]+left[k^nright]=(l+n)[k].

Thus

[V]=dim⁡(V)[k],displaystyle [V]=dim(V)[k],

and K0(Vectfin)displaystyle K_0(mathrm Vect _mathrm fin ) is isomorphic to Zdisplaystyle mathbb Z and is generated by [k].displaystyle [k]. Finally for a bounded complex of finite-dimensional vector spaces V*,

[V∗]=χ(V∗)[k]displaystyle [V^*]=chi (V^*)[k]

where χdisplaystyle chi is the standard Euler characteristic defined by

χ(V∗)=∑i(−1)idim⁡V=∑i(−1)idim⁡Hi(V∗).displaystyle chi (V^*)=sum _i(-1)^idim V=sum _i(-1)^idim H^i(V^*).

• For a ringed space (X,OX)displaystyle (X,mathcal O_X), one can consider the category Adisplaystyle mathcal A of all locally free sheaves over X. K0(X)displaystyle K_0(X) is then defined as the Grothendieck group of this exact category and again this gives a functor.

• For a ringed space (X,OX)displaystyle (X,mathcal O_X), one can also define the category Adisplaystyle mathcal A to be the category of all coherent sheaves on X. This includes the special case (if the ringed space is an affine scheme) of Adisplaystyle mathcal A being the category of finitely generated modules over a noetherian ring R. In both cases Adisplaystyle mathcal A is an abelian category and a fortiori an exact category so the construction above applies.

• In the case where R is a finite-dimensional algebra over some field, the Grothendieck groups G0(R)displaystyle G_0(R) (defined via short exact sequences of finitely generated modules) and K0(R)displaystyle K_0(R) (defined via direct sum of finitely generated projective modules) coincide. In fact, both groups are isomorphic to the free abelian group generated by the isomorphism classes of simple R-modules.

• There is another Grothendieck group G0displaystyle G_0 of a ring or a ringed space which is sometimes useful. The category in the case is chosen to be the category of all quasi-coherent sheaves on the ringed space which reduces to the category of all modules over some ring R in case of affine schemes. G0displaystyle G_0 is not a functor, but nevertheless it carries important information.

• Since the (bounded) derived category is triangulated, there is a Grothendieck group for derived categories too. This has applications in representation theory for example. For the unbounded category the Grothendieck group however vanishes. For a derived category of some complex finite-dimensional positively graded algebra there is a subcategory in the unbounded derived category containing the abelian category A of finite-dimensional graded modules whose Grothendieck group is the q-adic completion of the Grothendieck group of A.

References

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  Achar, Pramod N.; Stroppel, Catharina (2013), "Completions of Grothendieck groups", Bulletin of the London Mathematical Society, 45 (1): 200–212, arXiv:1105.2715, doi:10.1112/blms/bds079, MR 3033967.mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em.


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