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Wikipedia / Grothendieck group

In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. 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, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.

Grothendieck group of a commutative monoid

Motivation

is to be constructed by introducing inverse elements. It is characterized by a certain universal property.

does not have the cancellation property. In particular, in the case of a monoid operation denoted multiplicatively that has a zero element the Grothendieck group must be the trivial group (group with only one element), since one must have

.

Universal property

Let M be a commutative monoid. Its Grothendieck group is an abelian group K with a monoid homomorphism from M to an abelian group A, there is a unique group homomorphism

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 K.

is defined coordinate-wise:

  1. .

Next one defines an equivalence relation if, for some element k of M, m1 + n2 + k = m2 + n1 + k (the element k is necessary because the cancellation law does not hold in all monoids). The equivalence class of the element (m1, m2) is denoted by [(m1, m2)]. One defines K to be the set of equivalence classes. Since the addition operation on M × M is compatible with our equivalence relation, one obtains an addition on K, and K becomes an abelian group. The identity element of K is [(0, 0)], and the inverse of [(m1, m2)] is [(m2, m1 sends the element m to [(m, 0)].

Alternatively, the Grothendieck group K of M can also be constructed using generators and relations the free abelian group generated by the set M, the Grothendieck group K is the quotient by the subgroup generated while + denotes the addition in the monoid M.) This construction has the advantage that it can be performed for any semigroup M and yields a group which satisfies the corresponding universal properties for semigroups, i.e. the "most general and smallest group containing a homomorphic image of M. This is known as the "group completion of a semigroup" or "group of fractions of a semigroup".

Properties

In the language of category theory, any universal construction gives rise to a functor; one thus obtains a functor from the category of commutative monoids to the category of abelian groups which sends the commutative monoid M to its Grothendieck group K. This functor is left adjoint to the forgetful functor from the category of abelian groups to the category of commutative monoids.

For a commutative monoid M, the map i:M → K is injective if and only if M has the cancellation property, and it is bijective if and only if M is already a group.

Example: the integers

The easiest example of a Grothendieck group is the construction of the integers from the (additive) natural numbers 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

  1. .

Now define

. 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

with the equivalence

which of course can be identified with the positive rational numbers.

Example: the Grothendieck group of a manifold

The Grothendieck group is the fundamental construction of K-theory of a compact manifold M is defined to be the Grothendieck group of the commutative monoid of all isomorphism classes of vector bundles of finite rank on M with the monoid operation given by direct sum. This gives a contravariant functor from manifolds to abelian groups. This functor is studied and extended in topological K-theory.

Example: The Grothendieck group of a ring

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 is a covariant functor from rings to abelian groups.

is the ring of complex-valued smooth functions on a compact manifold M. In this case the projective R-modules are dual to vector bundles over M (by the Serre–Swan theorem 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 of isomorphism classes of finitely generated R-modules and the following relations: For every short exact sequence

of R-modules, add the relation

This definition implies that for any two finitely generated R-modules M and N, because of the split short exact sequence

Examples

Let K for any finite-dimensional K-vector space V is isomorphic for a finite-dimensional K-vector space V, the dimension of the vector space V. Suppose one has the following short exact sequence of K-vector spaces.

. In fact, for any two finite-dimensional vector spaces V and W the following holds:

in the Grothendieck group.

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. The observation from the previous paragraph hence proves the following equation:

.

for any finitely generated abelian groups A. One first notes that any finite abelian group G is multiplication by n.

The exact sequence, so every cyclic group has its symbol equal to 0. This in turn implies that every finite abelian group G by the fundamental theorem of finite abelian groups.

Observe that by the fundamental theorem of finitely generated abelian groups, every abelian group A is isomorphic to a direct sum of a torsion subgroup and a torsion-free abelian group for some non-negative integer r, called the rank of A Indeed, the observation made from the previous paragraph shows that every abelian group A of the Grothendieck group. Suppose one has the following short exact sequence of abelian groups:

Then tensoring implies the following equation.

-vector spaces, the sequence splits. Therefore, one has the following equation.

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

Therefore, the following equation holds:

Universal Property

is called additive, there is a unique factors through is the only group homomorphism that does that.

Examples of additive functions are the character function from representation theory is defined to be the trace-linear map.

By choosing a suitable basis and writing the corresponding matrices.

is the group ring of a finite group then this character map even gives a natural isomorphism. In the modular representation theory the algebraic closure of the finite field with p-module its Brauer character onto the ring of Brauer characters. In this way Grothendieck groups show up in representation theory.

the 'universal receiver' of generalized Euler characteristics. In particular, for every bounded complex

one has 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. 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 and one relation

for each exact sequence

  1. .

into an abelian group X; an abelian group G.

Every abelian category the category of finitely generated R. 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.)

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

  • Moreover, for an exact sequence

    m = l + n, so

    Thus

    Finally for a bounded complex of finite-dimensional vector spaces V*,

    is the standard Euler characteristic defined by

  • For a ringed space of all locally free sheaves over X is then defined as the Grothendieck group of this exact category and again this gives a functor.

  • to be the category of all coherent sheaves on X. This includes the special case (if the ringed space is an affine scheme being the category of finitely generated modules over a noetherian ring R is an abelian category and a fortiori an exact category so the construction above applies.

  • In the case where 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.

  • 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 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.

See also

References