Grothendieck group - Citizendia

In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the best possible way. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, commutativity is the ability to change the order of something without changing the end result In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation 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. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany In Mathematics, K-theory is a tool used in several disciplines

1 Universal property
2 Explicit construction
3 Generalization
4 Splitting principle
5 Example
6 References

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 property: 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 various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism

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. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, in the area of Category theory, a forgetful functor is a type of Functor. In Mathematics, the category Ab has the Abelian groups as objects and Group homomorphisms as Morphisms This is the prototype

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:

(m, n)

is meant to correspond to

m − n.

Addition is defined coordinate-wise:

(m₁, m₂) + (n₁, n₂) = (m₁ + n₁, m₂ + n₂).

Next we define an equivalence relation on M×M. We say that (m₁, m₂) is equivalent to (n₁, n₂) if, for some element k of M, m₁ + n₂ + k = m₂ + n₁ + k. 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. In Mathematics, K-theory is a tool used in several disciplines 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 bundles of finite rank on M with the monoid operation given by direct sum. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space

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 $\{x+'y-'(x+y)\mid x,y\in M\}$ .

Generalization

To apply the Grothendieck group to purely algebraic settings, it is useful to generalize it to the case of an essentially small 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 To do this, let $\mathcal A$ be an essentially small abelian category. Let F be the free abelian group generated by isomorphism classes of objects of the category. In Abstract algebra, a free abelian group is an Abelian group that has a "basis" in the sense that every element of the group can be written in (This is where the hypothesis of essential smallness is necessary; without it, F would not be a set. ) We will impose some relations on F. Call R the subgroup of F generated as follows: For each exact sequence 0→A→B→C→0 in $\mathcal A$ , the element

[A] + [C] - [B]

is in R. Then the Grothendieck group $K_0({\mathcal A})$ is F/R.

K₀ of an abelian category has a similar universal property to K₀ of a commutative monoid. We make a preliminary definition: A function χ from isomorphism classes of objects of an abelian category $\mathcal A$ to an abelian group A is called additive if, for each exact sequence 0→A→B→C→0, we have χ(A) + χ(C) - χ(B) = 0. Then, for any additive function χ: $\mathcal A$ →A, there is a unique abelian group homomorphism f: $K_0{\mathcal A}$ →A such that χ factors through f and the map that takes each object of $\mathcal A$ to the element representing its isomorphism class in $K_0({\mathcal A})$ .

This universal property makes $K_0({\mathcal A})$ the 'universal receiver' of generalized Euler characteristics. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant In particular, for every bounded complex of objects in ${\mathcal A}$

$\cdots \to 0 \to 0 \to A^n \to A^{n+1} \to \cdots \to A^{m-1} \to A^m \to 0 \to 0 \to \cdots$

we have a canonical element

$[A^*] = \sum_i (-1)^i [A^i] = \sum_i (-1)^i [H^i (A^*)] \in K_0.$

In fact the Grothendieck group was originally introduced for the study of Euler characteristics. In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology.

Splitting principle

The relationship between K₀ of a commutative monoid and K₀ of an abelian category comes from the splitting principle. In Mathematics, the splitting principle is a technique used to reduce questions about Vector bundles to the case of Line bundles In the theory of vector According to the splitting principle, we can always take an exact sequence 0→A→B→C→0 and find a closely related exact sequence in which the middle term splits, that is, it is the direct sum of the other two terms. Because of this, the Grothendieck group of the commutative monoid of vector bundles on a smooth manifold is the same as the Grothendieck group of the abelian category of vector bundles on that same smooth manifold.

K₀ is often defined for a ring or for a ringed space. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on The usual construction is as follows: For a not necessarily commutative ring R, one lets the abelian category $\mathcal A$ be the category of all finitely generated projective modules over the ring. In Mathematics, particularly in Abstract algebra and Homological algebra, the concept of projective module over a ring R is a more flexible generalisation For a ringed space (X,O_X), one lets the abelian category $\mathcal A$ be the category of all coherent sheaves on X. In Mathematics, especially in Algebraic geometry and the theory of Complex manifolds coherent sheaves are specific class of sheaves having This makes K₀ into a functor.

There is another Grothendieck group of a ring or a ringed space which is sometimes useful. The Grothendieck group G₀ of a ring is the Grothendieck group associated to the category of all finitely generated modules over a ring. Similarly, the Grothendieck group G₀ of a ringed space is the Grothendieck group associated to the category of all quasicoherent sheaves on the ringed space. G₀ is not a functor, but nevertheless it carries important information.

Example

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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Thus, for a vector space V the class $[V] = [k dim(V)]$ in $K 0 (V e c t f i n)$ . Moreover for an exact sequence

$0 \to k^l \to k^m \to k^n \to 0$

m = l + n, so

[k l + n] = [k l] + [k n] = (l + n)[k].

Thus $[V] = d i m (V)[k]$ , the Grothendieck group $K 0 (V e c t f i n)$ is isomorphic to ${\mathbb Z}$ and is generated by [k]. Finally for a bounded complex of finite dimensional vector spaces $V *$ ,

[V *] = χ(V *)[k]

where $χ$ is the standard Euler characteristic defined by

$\chi(V^*)= \sum_i (-1)^i {\rm dim\ } V = \sum_i (-1)^i{\rm dim\ } H^i(V^*)$

References

Grothendieck group on PlanetMath
Michael F. PlanetMath is a free, collaborative online Mathematics Encyclopedia. Atiyah, K-Theory, (Notes taken by D. W. Anderson, Fall 1964), published in 1967, W. A. Benjamin Inc. , New York.

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