In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Grothendieck group, or group of differences,
of a
commutative monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ar ...
is a certain
abelian group
In mathematics, 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 the order in which they are written. That is, the group operation is comm ...
. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a
homomorphic image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, introduced by
Alexander Grothendieck in his
proof of the
Grothendieck–Riemann–Roch theorem
In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is ...
, which resulted in the development of
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...
. This specific case is the
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
of
isomorphism class
In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other.
Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the stru ...
es of
objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
of an
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 prototypical example of an abelian category is the category of ab ...
, with the
direct sum as its operation.
Grothendieck group of a commutative monoid
Motivation
Given a commutative monoid , "the most general" abelian group that arises from is to be constructed by introducing
inverse element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
s to all elements of . Such an abelian group always exists; it is called the Grothendieck group of . It is characterized by a certain
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
and can also be concretely constructed from .
If does not have the
cancellation property
In mathematics, the notion of cancellative is a generalization of the notion of invertible.
An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that .
An ...
(that is, there exists and in such that
and
), then the Grothendieck group cannot contain . In particular, in the case of a monoid operation denoted multiplicatively that has a
zero element
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive identi ...
satisfying
for every
the Grothendieck group must be the
trivial group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
(
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
with only one element), since one must have
:
for every .
Universal property
Let ''M'' be a commutative monoid. Its Grothendieck group is an abelian group ''K'' with a
monoid homomorphism
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ar ...
satisfying the following universal property: for any monoid homomorphism
from ''M'' to an abelian group ''A'', there is a unique
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
wh ...
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
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
. The two coordinates are meant to represent a positive part and a negative part, so
corresponds to
in ''K''.
Addition on
is defined coordinate-wise:
:
.
Next one defines an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
on
, such that
is equivalent to
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
In mathematics, the notion of cancellative is a generalization of the notion of invertible.
An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that .
A ...
does not hold in all monoids). The
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of the element (''m''
1, ''m''
2) is denoted by
1, ''m''2)">''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
1, ''m''2)">''m''1, ''m''2)is
2, ''m''1)">''m''2, ''m''1) The homomorphism
sends the element ''m'' to
''m'', 0)
Alternatively, the Grothendieck group ''K'' of ''M'' can also be constructed using
generators and relations
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
: denoting by
the
free abelian group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
generated by the set ''M'', the Grothendieck group ''K'' is the
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of
by the
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
generated by
. (Here +′ and −′ denote the addition and subtraction in the free abelian group
while + denotes the addition in the monoid ''M''.) This construction has the advantage that it can be performed for any
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
''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
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, any
universal
Universal is the adjective for universe.
Universal may also refer to:
Companies
* NBCUniversal, a media and entertainment company
** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal
** Universal TV, a ...
construction gives rise to a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
; one thus obtains a functor from the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
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
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
to the
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sign ...
from the category of abelian groups to the category of commutative monoids.
For a commutative monoid ''M'', the map ''i'' : ''M'' → ''K'' is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
if and only if ''M'' has the cancellation property, and it is
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
if and only if ''M'' is already a group.
Example: the integers
The easiest example of a Grothendieck group is the construction of the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s
from the (additive)
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s
.
First one observes that the natural numbers (including 0) together with the usual addition indeed form a commutative monoid
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
:
for some
.
Now define
:
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.
Example: the positive rational numbers
Similarly, the Grothendieck group of the multiplicative commutative monoid
(starting at 1) consists of formal fractions
with the equivalence
:
for some
which of course can be identified with the positive
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s.
Example: the Grothendieck group of a manifold
The Grothendieck group is the fundamental construction of
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...
. The group
of a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
''M'' is defined to be the Grothendieck group of the commutative monoid of all isomorphism classes of
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 (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
s of finite rank on ''M'' with the monoid operation given by direct sum. This gives a
contravariant functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
from
manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
to abelian groups. This functor is studied and extended in
topological K-theory
In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early ...
.
Example: The Grothendieck group of a ring
The zeroth algebraic K group
of a (not necessarily
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
)
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'' is the Grothendieck group of the monoid consisting of isomorphism classes of
finitely generated projective modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over ''R'', with the monoid operation given by the
direct sum. Then
is a covariant functor from
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
to abelian groups.
The two previous examples are related: consider the case where
is the ring of
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued
smooth function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s on a compact manifold ''M''. In this case the projective ''R''-modules are
dual to vector bundles over ''M'' (by the
Serre–Swan theorem
In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout ...
). Thus
and
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
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
over some
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''k'' or more generally an
artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ...
. Then define the Grothendieck group
as the abelian group generated by the set
of isomorphism classes of finitely generated ''R''-modules and the following relations: For every
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
:
of ''R''-modules, add the relation
:
This definition implies that for any two finitely generated ''R''-modules ''M'' and ''N'',