Grothendieck group
Encyclopedia
In mathematics
, the Grothendieck group construction in abstract algebra
constructs an abelian group
from a commutative monoid
in the best possible way. It 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. The Grothendieck group is denoted by K or K0.
: 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
In the language of category theory
, the functor
that sends a commutative monoid M to its Grothendieck group N is left adjoint to the forgetful functor
from the category of abelian groups
to the category of commutative monoids.
The two coordinates are meant to represent a positive part and a negative part:
is meant to correspond to
Addition is defined coordinate-wise:
+ (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. It is easy to check that the addition operation is compatible with the equivalence relation. The identity element is now any element of the form (m, m), and the inverse of (m1, m2) is (m2, m1).
In this form, the Grothendieck group is the fundamental construction of K-theory
. The group K0(M) of a manifold
M is defined to be the Grothendieck group of the commutative monoid of all isomorphism classes of vector bundle
s of finite rank on M with the monoid operation given by direct sum. The zeroth algebraic K group K0(R) of a ring R is the Grothendieck group of the monoid consisting of isomorphism classes of projective modules over R, with the monoid operation given by the direct sum.
The Grothendieck group can also be constructed using generators and relations: denoting by (Z(M),+') the free abelian group generated by the set M, the Grothendieck group is the quotient of Z(M) by the subgroup generated by
.
be a finite dimensional algebra over some field 
or more generally an artinian ring
. Then define the Grothendieck group
as the group generated by the set (sic!) 
of isomorphism classes of finitely generated 
-modules and the following relations: For every exact sequence
of
-modules add the relation
The abelian group defined by this generators and this relations is the Grothendieck group
.
This group satisfies a universal property. We make a preliminary definition: A function
from the set of isomorphism classes to an abelian group 
is called additive if, for each exact sequence 
, we have 
. Then, for any additive function 
:R-mod
, there is a unique group homomorphism 
such that 
factors through f and the map that takes each object of 
to the element representing its isomorphism class in 
. Concretly this means that 
satisfies the equation 
for every finitely generated 
-module 
and 
is the only group homomorphism that does that.
Examples of additive functions are the character
function from representation theory
: If
is a finite dimensional 
-algebra, then we can associate the character 
to every finite dimensional 
-module 
: 
is defined to be the trace
of the
-linear map that is given by multiplication with the element 
on 
.
By choosing suitable basis and write 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"
such that 
.
If
and 
is the group ring
of a finite group
then this character map even gives a natural
isomorphism of
and the character ring 
. In the modular representation theory
of finite groups
can be a 
the algebraic closure
of the finite field
with
elements. In this case the analogously defined map that associates to each 
-module its Brauer character is also a natural isomorphism 
onto the ring of Brauer characters. In this way Grothendieck groups show up in representation theory.
This universal property also makes
the 'universal receiver' of generalized Euler characteristic
s. In particular, for every bounded complex of objects in
-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
. Simplified an exact category is an additive category together with a class of distinguished short sequences 
. 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.
It is defined in the same way as before as the abelian group with one generator
for each (isomorphism class of) object(s) of the category 
and one relation
for each exact sequence
.
Alternatively one can define the Grothendieck group using a similar universal property: An abelian group
together with a mapping 
is called the Grothendieck group of 
iff every "additive" map 
from 
into an abelian group 
("additive" in the above sense, i.e. for every exact sequence 
we have 
) factors uniquely through 
.
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
-mod the category of finitely generated 
-modules as 
. This is really abelian because 
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
with the canonical inclusion and projection morphisms. This procedure produces the Grothendieck group of the commutative monoid 
in the first sense (here 
means the "set" [ignoring all foundational issues] of isomorphism classes in 
.)
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
whenever there is a distinguished triangle 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Grothendieck group construction in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
constructs an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
from a commutative monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
in the best possible way. It takes its name from the more general construction in category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, introduced by Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
in his fundamental work of the mid-1950s that resulted in the development of K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
, which led to his proof of the Grothendieck-Riemann-Roch theorem. The Grothendieck group is denoted by K or K0.
Universal property
In its simplest form, the Grothendieck group of a commutative monoid is the universal way of making that monoid into an abelian group. Let M be a commutative monoid. Its Grothendieck group N should have the following universal propertyUniversal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
: There exists a monoid homomorphism
- i:M→N
such that for any monoid homomorphism
- f:M→A
from the commutative monoid M to an abelian group A, there is a unique group homomorphism
- g:N→A
such that
- f=gi.
In the language of category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, the functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
that sends a commutative monoid M to its Grothendieck group N is left adjoint to the forgetful functor
Forgetful functor
In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...
from the category of abelian groups
Category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....
to the category of commutative monoids.
Explicit construction
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:
is meant to correspond to
- m − n.
Addition is defined coordinate-wise:
+ (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. It is easy to check that the addition operation is compatible with the equivalence relation. The identity element is now any element of the form (m, m), and the inverse of (m1, m2) is (m2, m1).
In this form, the Grothendieck group is the fundamental construction of K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
. The group K0(M) of a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
M is defined to be the Grothendieck group of the commutative monoid of all isomorphism classes of vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s of finite rank on M with the monoid operation given by direct sum. The zeroth algebraic K group K0(R) of a ring R is the Grothendieck group of the monoid consisting of isomorphism classes of projective modules over R, with the monoid operation given by the direct sum.
The Grothendieck group can also be constructed using generators and relations: denoting by (Z(M),+') the free abelian group generated by the set M, the Grothendieck group is the quotient of Z(M) by the subgroup generated by
.Grothendieck group and extensions
Another construction that carries the name "Grothendieck group" is the following: Let be a finite dimensional algebra over some field or more generally an artinian ringArtinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
. Then define the Grothendieck group
as the group generated by the set (sic!) of isomorphism classes of finitely generated -modules and the following relations: For every exact sequenceof
-modules add the relationThe abelian group defined by this generators and this relations is the Grothendieck group
.This group satisfies a universal property. We make a preliminary definition: A function
from the set of isomorphism classes to an abelian group is called additive if, for each exact sequence , we have . Then, for any additive function :R-mod, there is a unique group homomorphism such that factors through f and the map that takes each object of to the element representing its isomorphism class in . Concretly this means that satisfies the equation for every finitely generated -module and is the only group homomorphism that does that.Examples of additive functions are the character
Character (mathematics)
In mathematics, a character is a special kind of function from a group to a field . There are at least two distinct, but overlapping meanings...
function from representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
: If
is a finite dimensional -algebra, then we can associate the character to every finite dimensional -module : is defined to be the traceTrace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
of the
-linear map that is given by multiplication with the element on . By choosing suitable basis and write 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"
such that . If
and is the group ringGroup ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
of a finite groupFinite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
then this character map even gives a naturalNatural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
isomorphism of
and the character ring . In the modular representation theoryModular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...
of finite groups
can be a the algebraic closureAlgebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
of the finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
with
elements. In this case the analogously defined map that associates to each -module its Brauer character is also a natural isomorphism onto the ring of Brauer characters. In this way Grothendieck groups show up in representation theory.This universal property also makes
the 'universal receiver' of generalized Euler characteristicEuler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
s. In particular, for every bounded complex of objects in
-modwe have a canonical element
In fact the Grothendieck group was originally introduced for the study of Euler characteristics.
Grothendieck groups of exact categoriesExact categoryIn mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition...
A common generalization of these two concepts is given by the Grothendieck group of an exact categoryExact category
In mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition...
. Simplified an exact category is an additive category together with a class of distinguished short sequences . 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.It is defined in the same way as before as the abelian group with one generator
for each (isomorphism class of) object(s) of the category and one relationfor each exact sequence
.Alternatively one can define the Grothendieck group using a similar universal property: An abelian group
together with a mapping is called the Grothendieck group of iff every "additive" map from into an abelian group ("additive" in the above sense, i.e. for every exact sequence we have ) factors uniquely through .Every abelian category
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
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
-mod the category of finitely generated -modules as . This is really abelian because was assumed to be artinian and (hence noetherian) in the previous section.On the other hand every additive category
Additive category
In mathematics, specifically in category theory, an additive category is a preadditive category C such that all finite collections of objects A1,...,An of C have a biproduct A1 ⊕ ⋯ ⊕ An in C....
is also exact if we declare those and only those sequences to be exact that have the form
with the canonical inclusion and projection morphisms. This procedure produces the Grothendieck group of the commutative monoid in the first sense (here means the "set" [ignoring all foundational issues] of isomorphism classes in .) Grothendieck groups of triangulated categoriesTriangulated categoryA triangulated category is a mathematical category satisfying some axioms that are based on the properties of the homotopy category of spectra, and the derived category of an abelian category. A t-category is a triangulated category with a t-structure.- History :The notion of a derived category...
Generalizing even further it is also possible to define the Grothendieck group for triangulated categoriesTriangulated category
A triangulated category is a mathematical category satisfying some axioms that are based on the properties of the homotopy category of spectra, and the derived category of an abelian category. A t-category is a triangulated category with a t-structure.- History :The notion of a derived category...
. The construction is essentially similar but uses the relations
whenever there is a distinguished triangle .Examples
- The easiest example of the Grothendieck group construction is the construction of the integers from the natural numbers. First one observes that the natural numbers together with the usual addition indeed form a commutative monoid
.
- Now when we use the Grothendieck group construction we obtain the formal differences between natural numbers as elements
and we have the equivalence relation
.
- Now define
,
- for all
. This defines the integers 
. Indeed this is the usual construction to obtain the integers from the natural numbers. See "Construction" under Integers for a more detailed explanation.
- In the abelian category of finite dimensional vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s over a fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, two vector spaces are isomorphic if and only if they have the same dimension. Thus, for a vector space V the class 
in 
. Moreover for an exact sequence
-
- m = l + n, so
- Thus
, the Grothendieck group 
is isomorphic to 
and is generated by [k]. Finally for a bounded complex of finite dimensional vector spaces 
,
- where
is the standard Euler characteristic defined by
-
is often defined for a ringRing (mathematics)In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
. The usual construction is as follows: For a (not necessarily commutative) ring R, one defines the category
to be the category of all finitely generated projective moduleProjective moduleIn mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
s over the ring.
is then defined to be the Grothendieck group of 
. This gives a (contravariant) functor of 
.
- A special case of the above is the case where
is the ring 
of (say complex-valued) smooth functionSmooth functionIn mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
s an a compact manifold
. In this case the projective 
-modules are dualDual (category theory)In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...
to vector bundels over
(by the Serre-Swan theorem). The above construction thus reconstructs the zeroth topological K-theoryTopological K-theoryIn mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on general topological spaces, by means of ideas now recognised as K-theory that were introduced by Alexander Grothendieck...
group
, i.e. the Grothendieck group of the commutative monoid of (isomorphism classes) vector bundles over 
with addition being the direct sum. (This time 
is covariant functor of 
because of the duality in the intermediate step).
- A ringed-space-version of the latter example works as follows: Choose
to be the category of all locally free sheaves over 
. 
is again defined as the Grothendieck group of this category and again this gives a functor.
- For a ringed space
, one can also define the category 
to be the category of all coherent sheavesCoherent sheafIn mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...
on X. This includes the special case (if the ringed space is an affine scheme) of
being the category of finitely generated modules over a noetherian ring 
. In both cases 
is an abelian category and a fortiori an exact category so the construction above applies.
- In the special case where
is a finite dimensional algebra over some field this reduces to the Grothendieck group 
mentioned above.
- There is another Grothendieck group
of a ring or a ringed space which is sometimes useful. The category in the case is chosen to be the category of all quasicoherent sheaves on the ringed space which reduces to the category of all modules over some ring 
in case of affine schemes. 
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.