The following topic came up in a discussion with my mentor recently. Since the material is somewhat general and well-known, but relevant to my project area, I decided to write this post partially to help myself understand it better.

** Definition **

Consider an abelian category . Then:

Definition 1TheGrothendieck groupof is the abelian group defined via generators and relations as follows: is generated by symbols for each , and by relations for each exact sequence

Note here that if are isomorphic, then in by considering the exact sequence

The Grothendieck group has an important **universal property**:

Proposition 2To give a function for an abelian group satisfying for each exact sequence as in (1) (i.e. anEuler-Poincaré map, is equivalent to giving a group-homomorphism .

This property, which follows from the definition, determines the Grothendieck group up to isomorphism as the unique group making the above result valid.

** The Grothendieck Group of Representations **

Let be a finite group. Then consider the category defined as follows: the objects of are the finite-dimensional representations of , and morphisms are -module homomorphisms (also called intertwining operators).

Then:

Proposition 3The Grothendieck group is the abelian group generated by the irreducible characters.

Suppose the irreducible representations are , corresponding to characters . We have a group-homomorphism sending a sum to (this being well-defined as the characters are linearly independent). This is surjective since we can decompose a representation as a direct sum of irreducibles. We define the inverse by using the above universal property. First, define the Euler-Poincaré map by sending ; this is valid by the previous post, since each object can be decomposed **uniquely** into irreducibles. One then gets a map . These two maps between and (and vice versa) are checked to be inverse to each other.

This result can be generalized to semisimple abelian categories.

** The Eilenberg Swindle **

Suppose an abelian category admits infinite direct sums. Then I claim:

Theorem 4.

This is proved using the Eilenberg swindle. Given , we show that . But

we thus have an exact sequence

which gives us the relation .

The Eilenberg swindle can also be stated in a slightly different form.

The notion of Grothendieck group is useful in modular representation theory—which works in fields other than , having nonzero characteristic not relatively prime to . Then, it’s possible to say that certain maps between categories may not be surjective, but they are “mostly” surjective in that the induced map on Grothendieck groups is so. We’ll probably use Grothendieck groups more later.

July 12, 2009 at 3:10 am

My understanding was that the Grothendieck group construction was most generally defined on a commutative semigroup. The construction you give can be broken down into two steps, both of which satisfy a universal property: turn an abelian category into a commutative semigroup (also a monoid), then turn the semigroup into an abelian group.

I mention this because the simplest example of the Grothendieck group construction is quite familiar: it’s the construction of the integers from the positive integers.

July 12, 2009 at 3:58 am

Seconding what Qiaochu said — Grothendieck would be saddened that you’re not thinking in full generality. 😛

I’ll have to double-check this when I get a chance, but I believe that the construction of the ring of virtual species from the semiring of species works along essentially the same lines as the (general) Grothendieck group construction. (Question: is the category of combinatorial species — equivalently, I guess, the functor category , where is the category of sets with bijections as morphisms — abelian? If so, is the Grothendieck group of it isomorphic to the additive group of virtual categories?)

July 12, 2009 at 4:28 am

As far as I know, the coproduct and product are disjoint union and Cartesian product in the species sense, so no. (I would be very surprised if this wasn’t the case.) Nonetheless, disjoint union does get us a monoidal category which is commutative up to isomorphism, which is, I believe, essentially how the virtual species construction works.

July 12, 2009 at 3:27 pm

Thanks for the comments! I am learning quite a bit from these posts and discussions. However, how does one make the abelian category into a commutative semigroup? One could use direct sums, but then one wouldn’t get enough relations for the exact sequences as above (unless the category is semisimple).

July 12, 2009 at 6:31 pm

Hmm. If it’s not enough to quotient by natural isomorphism, then perhaps it’s best to regard the categorical Grothendieck group construction as a “categorification” of the semigroup one.

September 14, 2011 at 9:39 pm

Hi Akhil,

I was reading some of your posts, and did not understand your comment above (“but then one wouldn’t get enough relations for the exact sequences as above”). Did you mean to say we would quotient less by the semigroup construction since some exact sequence doesn’t split?

September 20, 2011 at 12:06 am

Sorry for the slow reply. That’s essentially what I meant.