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Grothendieck group

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

The term Grothendieck group has a restricted and a more general meaning

In its restricted sense the Grothedieck group of a commutative monoid? (i.e. of a commutative semi-group with unit) AA is a specific presentation of its group completion, given as a certain group structure on a quotient of the Cartesian product A×AA \times A. This is such that applied to the additive monoid of natural numbers \mathbb{N} it produces the additive group of integers \mathbb{Z} represented by pairs of natural numbers (n +,n )(n_+, n_-) subject to an equivalence relation which identifies them with their difference n +n n_+ - n_-.

A vaguely similar procedure applied to isomorphism classes of certain Quillen exact categories happens to compute a group that is called the algebraic K-theory of these categories. There is a very simple very general nonsense behind this, that is described at K-theory.

Notably the Grothendieck group completion of the decategorification of the category of topological vector bundles on some (compact Hausdorff) topological space XX produces the group known as the topological K-theory of XX.

Given this one speaks more generally of the (algebraic) K-theory group of a suitable category (one presenting a stable (∞,1)-category in some way) as its Grothendieck group .

In that sense, the Grothendieck group of a (\infty-)category CC with a notion of cofibration sequences is the decategorification set K(C)K(C) equipped with a notion of addition that is encoded in these homotopy exact sequences.

We now first state the definition of “Grothendieck group completion” – which is really just the free group completion of an abelian monoid – and then the definition of Grothendieck group in the sense of algebraic K-theory. Notice that a priori both concepts are entirely independent constructions on different entities. But in various special case both can be applied to specific objects so as to produce the same result.

Of commutative monoids

Every abelian group is in particular a commutative monoid. The forgetful functor U:AbCMonU \colon Ab \to CMon from the category Ab ot the category CMon has a left adjoint FF (the correspnding free functor), the group completion functor

AbUFCMon Ab \underoverset {\underset{U}{\longrightarrow}} {\overset{F}{\longleftarrow}} {\bot} CMon

The Grothendieck group construction on commutative monoids (def. 1 below) is an explict presentation of this group completion functor.

For more details see at Grothendieck group of a commutative monoid.

Remark

The idea of the free group on an abelian monoid is a very simple algebraic idea that, at least for a cancellative monoid (so that the unit is monic and one can reasonably use the term ‘completion’) certainly predates Grothendieck. That the integers \mathbb{Z} is the group completion of the natural numbers \mathbb{N} goes back at least to Kronecker.

Definition

(Grothendieck group of a commutative monoid)

Let (A,+)(A,+) be a commutative monoid (i.e. a commutative semi-group).

On the Cartesian product of underlying sets A×AA \times A (the set of ordered pairs of elements in AA), consider the equivalence relation

((a +,a ) 1(b +,b ))(kA(a ++b +k=b ++a +k)) \big( (a_+, a_-) \sim_1 (b_+, b_-) \big) \;\Leftrightarrow\; \left( \underset{k \in A}{\exists} \left( a_+ + b_- + k = b_+ + a_- + k \right) \right)

or equivalently the equivalence relation

((a +,a ) 2(b +,b ))(k 1,k 2A((a ++k 1,a +k 1)=(b ++k 2,b +k 2))). \big( (a_+, a_-) \sim_2 (b_+, b_-) \big) \;\Leftrightarrow\; \left( \underset{k_1, k_2 \in A}{\exists} \left( (a_+ + k_1, a_- + k_1) = (b_+ + k_2, b_- + k_2) \right) \right) \,.

Write

G(A)(A×A)/ G(A) \coloneqq (A \times A)/\sim

for the set of equivalence classes under this equivalence relation. This inherits a binary operation

+:G(A)×G(A)G(A) + \;\colon\; G(A) \times G(A) \longrightarrow G(A)

by applying the addition in AA on representatives:

[a +,a ]+[b +,b ][a ++b +,a +b ]. [a_+,a_-] + [b_+,b_-] \coloneqq [ a_+ + b_+ , a_- + b_- ] \,.

This defines the structure of an abelian group

(G(A),+) (G(A),+)

and this is the Grothendieck group of AA.

This comes with a canonical homomorphism of semigroups

A Aη AA G(A) a AAA [a,0]. \array{ A &\overset{\phantom{A} \eta_A \phantom{A} }{\longrightarrow}& G(A) \\ a &\overset{\phantom{AAA}}{\mapsto}& [a,0] } \,.

Of stable \infty-categories

Fundamentally a Grothendieck group is something assigned to a stable (∞,1)-category. It is the group structure +:K(C)×K(C)K(C)+ : K(C)\times K(C) \to K(C) on the decategorification K(C)K(C) of CC defined by the rule that for every fibration sequence

AXB A \to X \to B

in CC the equivalence classes [A][A], [B][B] and [X][X] satisfy

[X]=[A]+[B]. [X] = [A] + [B] \,.

In particular the inverse [A]-[A] of an element [A][A] is the class of its loop space object ΩA\Omega A or equivalently of its delooping BA\mathbf{B} A called the suspension ΣA\Sigma A, since by definition the sequences

ΩA0A \Omega A \to 0 \to A

and

A0ΣA A \to 0 \to \Sigma A

are fibration sequence, so that

[A]+[ΩA]=0 [A] + [\Omega A] = 0

and

[A]+[ΣA]=0. [A] + [\Sigma A] = 0 \,.

But there are many ways to model a stable (∞,1)-category by an ordinary category. Essentially for each of these ways there is a seperate prescription for how to model the above general construction in terms of concrete 1-categorical constructions.

In particular from an

and a

one obtains the corresponding categories of chain complexes. These are stable (∞,1)-categories. Below we list presciptions for how to compute the Grothendieck/K-theory groups of these in terms of the underlying 1-categories.

Apart from the case of abelian categories, this requires some handle on the fibration sequences. A tool developed to handle exactly this for the purpose of computing Grothendieck/K-theory groups is the notion of a Waldhausen category. That provides the sufficient extra information to get a hand on the homotopy exact sequences.

of an abelian category

Let CC be an abelian category. The Grothendieck group or algebraic K-theory group of CC, denoted K(C)K(C), is the abelian group generated by isomorphism classes of objects of CC, with relations of the form

[X]=[A]+[B][X] = [A] + [B]

whenever there is a short exact sequence

0AXC0 0 \to A \to X \to C \to 0

of a Quillen exact category

An exact category CC in the present sense is a full subcategory of an abelian category C^\hat C such that the collection of all sequences 0AXB00 \to A \to X \to B \to 0 in CC that are exact sequences in C^\hat C has the property that for every exact sequence AXBA \to X \to B in C^\hat C with AA and BCB \in C also their “sum” XX is in CC.

Given an exact category CC with the inherited notion of exact sequences this way, the definition of its Grothendieck group is as above.

of a Waldhausen category

A Waldhausen category is a category with weak equivalences with an initial object – called 00 – and equipped with the notion of auxiliary morphism called cofibrations. These satisfy some axioms which are such that the ordinary 1-categorical cokernel of a cofibration AXA \hookrightarrow X , i.e. the ordinary pushout

A X 0 B \array{ A &\hookrightarrow& X \\ \downarrow && \downarrow \\ 0 &\to & B }

computes the desired homotopy pushout. (This is exactly dual to the reasoning by which one computes homotopy pullbacks in a category of fibrant objects. See there for details.)

Therefore in a Waldhausen category a cofibration sequence is a pushout sequence

AXB A \hookrightarrow X \to B

where the first morphism is a cofibration.

The Grothendieck/K-theory-group of the Waldhausen category CC is then, as before, on the decategorification K(C)K(C) the abelian group structure given by

[X]=[A]+[B] [X] = [A] + [B]

for all cofibration sequences as above.

Examples

These two examples illustrate a general fact: the Grothendieck group of a monoidal abelian category inherits a ring structure from the tensor product in this category, and thus becomes a ring, called the Grothendieck ring. See also the general discussion at decategorification.

References

  • Charles Weibel, The K-book: An introduction to algebraic K-theory (web)

    section 2: The Grothendieck group K 0K_0 (pdf)

See also

Revised on May 27, 2017 03:19:17 by Urs Schreiber (92.218.150.85)