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In 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 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, 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 geometr ...
. 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. Monoid ...
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 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 ...
, 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 a\ne b and ac=bc), 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 0.x=0 for every x\in M, 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 :x=1.x=(0^.0).x = 0^.(0.x)=0^.(0.0)=(0^.0).0=1.0=0 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 ...
i \colon M \to K satisfying the following universal property: for any monoid homomorphism f \colon M \to A 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) w ...
g \colon K \to A such that 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 M \times M. The two coordinates are meant to represent a positive part and a negative part, so (m_1, m_2) corresponds to m_1- m_2 in ''K''. Addition on M\times M is defined coordinate-wise: :(m_1, m_2) + (n_1,n_2) = (m_1+n_1, m_2+n_2). Next one defines an equivalence relation on M \times M, such that (m_1, m_2) is equivalent to (n_1, n_2) 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 does not hold in all monoids). The equivalence class of the element (''m''1, ''m''2) is denoted by ''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 ''m''1, ''m''2)is ''m''2, ''m''1) The homomorphism 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 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 (Z(M), +') the free abelian group 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 Z(M) 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 Z(M) 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, 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 mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
; 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 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 languag ...
s \Z 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 ...
s \N. First one observes that the natural numbers (including 0) together with the usual addition indeed form a commutative monoid (\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 :n - m \sim n' - m' \iff n + m' + k = n'+ m + k for some k \iff n + m' = n' + m. Now define :\forall n \in \N: \qquad \begin n := - 0\\ -n := - n\end 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.


Example: the positive rational numbers

Similarly, the Grothendieck group of the multiplicative commutative monoid (\N^*, \times) (starting at 1) consists of formal fractions p/q with the equivalence :p/q \sim p'/q' \iff pq'r = p'qr for some r \iff pq' = p'q 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 rat ...
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 geometr ...
. The group K_0(M) 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 ''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 p ...
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 K_0(R) 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 of ...
)
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 K_0 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 R = C^\infty(M) 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 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 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 K_0(R) and 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 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 ...
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 G_0(R) 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 ...
:0 \to A \to B \to C \to 0 of ''R''-modules, add the relation : - + = 0. This definition implies that for any two finitely generated ''R''-modules ''M'' and ''N'', \oplus N= + /math>, because of the
split Split(s) or The Split may refer to: Places * Split, Croatia, the largest coastal city in Croatia * Split Island, Canada, an island in the Hudson Bay * Split Island, Falkland Islands * Split Island, Fiji, better known as Hạfliua Arts, entertai ...
short exact sequence : 0 \to M \to M \oplus N \to N \to 0.


Examples

Let ''K'' be a field. Then the Grothendieck group G_0(K) is an abelian group generated by symbols /math> for any finite-dimensional ''K''-
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V''. In fact, G_0(K) is isomorphic to \Z whose generator is the element /math>. Here, the symbol /math> for a finite-dimensional ''K''-vector space ''V'' is defined as = \dim_K V, the dimension of the vector space ''V''. Suppose one has the following short exact sequence of ''K''-vector spaces. :0 \to V \to T \to W \to 0 Since any short exact sequence of vector spaces splits, it holds that T \cong V \oplus W . In fact, for any two finite-dimensional vector spaces ''V'' and ''W'' the following holds: :\dim_K(V \oplus W) = \dim_K(V) + \dim_K(W) The above equality hence satisfies the condition of the symbol /math> in the Grothendieck group. : = \oplus W= + /math> 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 K^. The observation from the previous paragraph hence proves the following equation: : = \left K^ \right= n /math> Hence, every symbol /math> is generated by the element /math> with integer coefficients, which implies that G_0(K) is isomorphic to \Z with the generator /math>. More generally, let \Z be the set of integers. The Grothendieck group G_0(\Z) is an abelian group generated by symbols /math> for any finitely generated abelian groups ''A''. One first notes that any finite abelian group ''G'' satisfies that = 0. The following short exact sequence holds, where the map \Z \to \Z is multiplication by ''n''. :0 \to \Z \to \Z \to \Z /n\Z \to 0 The
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 ...
implies that Z /n\Z= Z- Z= 0, so every
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
has its symbol equal to 0. This in turn implies that every finite abelian group ''G'' satisfies = 0 by the fundamental theorem of finite abelian groups. Observe that by the
fundamental theorem of finitely generated abelian groups In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
, every abelian group ''A'' is isomorphic to a direct sum of a torsion subgroup and a
torsion-free abelian group In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only e ...
isomorphic to \Z^r for some non-negative integer ''r'', called the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
of ''A'' and denoted by r = \mbox(A) . Define the symbol /math> as = \mbox(A). Then the Grothendieck group G_0(\Z) is isomorphic to \Z with generator Z Indeed, the observation made from the previous paragraph shows that every abelian group ''A'' has its symbol /math> the same to the symbol Z^r= r Z/math> where r = \mbox(A). Furthermore, the rank of the abelian group satisfies the conditions of the symbol /math> of the Grothendieck group. Suppose one has the following short exact sequence of abelian groups: :0 \to A \to B \to C \to 0 Then tensoring with the rational numbers \Q implies the following equation. :0 \to A \otimes_\Z \Q \to B \otimes_\Z \Q \to C \otimes_\Z \Q \to 0 Since the above is a short exact sequence of \Q-vector spaces, the sequence splits. Therefore, one has the following equation. :\dim_\Q (B \otimes_\Z \Q ) = \dim_\Q (A \otimes_\Z \Q) + \dim_\Q (C \otimes_\Z \Q ) On the other hand, one also has the following relation; for more information, see Rank of an abelian group. :\operatorname(A) = \dim_\Q (A \otimes_\Z \Q ) Therefore, the following equation holds: : = \operatorname(B) = \operatorname(A) + \operatorname(C) = + /math> Hence one has shown that G_0(\Z) is isomorphic to \Z with generator Z


Universal Property

The Grothendieck group satisfies a universal property. One makes a preliminary definition: A function \chi from the set of isomorphism classes to an abelian group X is called ''additive'' if, for each exact sequence 0 \to A \to B \to C \to 0, one has \chi(A)-\chi(B)+\chi(C)= 0. Then, for any additive function \chi: R\text \to X, there is a ''unique'' group homomorphism f:G_0(R) \to X such that \chi factors through ''f'' and the map that takes each object of \mathcal A to the element representing its isomorphism class in G_0(R). Concretely this means that f satisfies the equation f( =\chi(V) for every finitely generated R-module V and f is the only group homomorphism that does that. Examples of additive functions are the character function from
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 studies modules over these abstract algebraic structures. In essen ...
: If R is a finite-dimensional k-algebra, then one can associate the character \chi_V: R \to k to every finite-dimensional R-module V: \chi_V(x) is defined to be the trace of the k-
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
that is given by multiplication with the element x \in R on V. By choosing a suitable
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
and writing the corresponding
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
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" \chi: G_0(R)\to \mathrm_K(R,K) such that \chi( = \chi_V. If k=\Complex and R is the
group 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 the set of elements of the giv ...
\Complex /math> of a finite group G then this character map even gives a
natural isomorphism In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
of G_0(\Complex and the character ring Ch(G). In the
modular representation theory Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field ''K'' of positive characteristic ''p'', necessarily a prime number. As well as ...
of finite groups, k can be a field \overline, the
algebraic 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. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
of the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
with ''p'' elements. In this case the analogously defined map that associates to each k /math>-module its Brauer character is also a natural isomorphism G_0(\overline \to \mathrm(G) onto the ring of Brauer characters. In this way Grothendieck groups show up in representation theory. This universal property also makes G_0(R) the 'universal receiver' of generalized Euler characteristics. In particular, for every
bounded complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
of objects in R\text :\cdots \to 0 \to 0 \to A^n \to A^ \to \cdots \to A^ \to A^m \to 0 \to 0 \to \cdots one has a canonical element : ^*= \sum_i (-1)^i ^i= \sum_i (-1)^i ^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 \mathcal. 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 \mathcal and one relation : = 0 for each exact sequence :A\hookrightarrow B\twoheadrightarrow C. Alternatively and equivalently, one can define the Grothendieck group using a universal property: A map \chi: \mathrm(\mathcal)\to X from \mathcal into an abelian group ''X'' is called "additive" if for every exact sequence A\hookrightarrow B\twoheadrightarrow C one has \chi(A)-\chi(B)+\chi(C)=0; an abelian group ''G'' together with an additive mapping \phi: \mathrm(\mathcal)\to G is called the Grothendieck group of \mathcal iff every additive map \chi: \mathrm(\mathcal)\to X factors uniquely through \phi. Every
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 ...
is an exact category if one just uses the standard interpretation of "exact". This gives the notion of a Grothendieck group in the previous section if one chooses \mathcal := R\text the category of finitely generated ''R''-modules as \mathcal. This is really abelian because ''R'' was assumed to be artinian (and hence
noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
) in the previous section. On the other hand, every additive category is also exact if one declares those and only those sequences to be exact that have the form A\hookrightarrow A\oplus B\twoheadrightarrow B with the canonical inclusion and projection morphisms. This procedure produces the Grothendieck group of the commutative monoid (\mathrm(\mathcal),\oplus) in the first sense (here \mathrm(\mathcal) means the "set" gnoring all foundational issuesof isomorphism classes in \mathcal.)


Grothendieck groups of triangulated categories

Generalizing even further it is also possible to define the Grothendieck group for
triangulated categories In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy cate ...
. The construction is essentially similar but uses the relations 'X'''Y''+ 'Z''= 0 whenever there is a distinguished triangle ''X'' → ''Y'' → ''Z'' → ''X''


Further examples

* In the abelian category of finite-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s over a
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'', two vector spaces are isomorphic if and only if they have the same dimension. Thus, for a vector space ''V'' :: = \big k^ \big\in K_0 (\mathrm_). :Moreover, for an exact sequence ::0 \to k^l \to k^m \to k^n \to 0 :''m'' = ''l'' + ''n'', so ::\left k^ \right= \left k^l \right+ \left k^n \right= (l+n) :Thus :: = \dim(V) :and K_0(\mathrm_) is isomorphic to \Z and is generated by Finally for a bounded complex of finite-dimensional vector spaces ''V'' *, :: ^*= \chi(V^*) /math> :where \chi is the standard Euler characteristic defined by ::\chi(V^*)= \sum_i (-1)^i \dim V = \sum_i (-1)^i \dim H^i(V^*). * For a ringed space (X,\mathcal_X), one can consider the category \mathcal of all locally free sheaves over ''X''. 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,\mathcal_X), one can also define the category \mathcal A to be the category of all
coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
on ''X''. This includes the special case (if the ringed space is an affine scheme) of \mathcal being the category of finitely generated modules over a noetherian ring ''R''. In both cases \mathcal 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 G_0(R) (defined via short exact sequences of finitely generated modules) and 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 Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
''R''-modules. * There is another Grothendieck group 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. 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 module In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
s whose Grothendieck group is the ''q''-adic completion of the Grothendieck group of ''A''.


See also

*
Field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
* Localization *
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 ...
*
Atiyah–Hirzebruch spectral sequence In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet, ...
for computing topological K-theory


References

* Michael F. Atiyah, ''K-Theory'', (Notes taken by D.W.Anderson, Fall 1964), published in 1967, W.A. Benjamin Inc., New York. * . * * {{planetmath reference, urlname=GrothendieckGroup, title=Grothendieck group
The Grothendieck Group of Algebraic Vector Bundles; Calculations of Affine and Projective Space

Grothendieck Group of a Smooth Projective Complex Curve
Algebraic structures Homological algebra K-theory