This is a continuation of the previous article. Throughout this article, *R* is an artinian ring (and hence noetherian) and all modules are finitely-generated. Let *K*(*R*) be the Grothendieck group of all finitely-generated *R*-modules; *K*(*R*) is the free abelian group generated by [*M*] for simple modules *M*.

Now let *P*(*R*) be the Grothendieck group of all finitely-generated *projective R*-modules. Thus, *P*(*R*) is the free abelian group generated by [*P*] for finitely-generated *P*, modulo [*P*] = [*Q*] + [*Q’*] for each short exact 0 → *Q’* → *P* → *Q* → 0. By the lemma here, this means *Q* is a direct summand of *P* so *P* ≅ *Q*⊕*Q’*.

Theorem. The group P(R) is the free abelian group generated by [P] for indecomposable finitely-generated projective modules P. Furthermore, if in P(R) for projective modules then

**Proof**

By Krull-Schmidt’s theorem, every finitely-generated module *P* is uniquely written as a direct sum of indecomposable modules; if *P* is projective, so is each direct summand. Hence [*P*] is a sum of [*Q*] for indecomposable projective *Q*. Furthermore, each short exact 0 → *Q’* → *P* → *Q* → 0 gives a decomposition *P* ≅ *Q*⊕*Q’* so [*P*] = [*Q*] + [*Q’*] if and only if the modules on the LHS and RHS match after decomposition. The general case of *r*>2 follows by induction on *r*. ♦

Now we define a map:

which takes a projective module *P* to its class [*P*] in *K*(*R*). Note that this is a well-defined group homomorphism. The next map we define is:

which is given by , where *J* := *J*(*R*) is the Jacobian radical of *R*. Note that this is well-defined since a short exact sequence 0 → *Q’* → *P* → *Q* → 0 of projective modules splits to give *P* ≅ *Q*⊕*Q’* and so *P*/*JP* ≅ (*Q*/*JQ*) ⊕ (*Q’*/*JQ’*), and we get [*P*/*JP*] = [*Q*/*JQ*] + [*Q’*/*JQ’*] in *K*(*R*). Furthermore, we saw earlier that gives a bijection between projective indecomposable modules and simple modules. Since *P*(*R*) is freely generated by the projective indecomposable modules while *K*(*R*) is freely generated by the simple modules, we have:

Theorem. The map is a group isomorphism.

The map can be represented by a matrix with integer entries; let’s compute this for a simple example.

## Example

Let *R* be the ring of upper 3 × 3 matrices with real entries. We thus see that:

is a direct sum of indecomposable projective modules. On the other hand, *K* is isomorphic to the column of 3-vectors, which has a composition series: Denoting the consecutive factors by *A*, *B*, *C*, we see that in *K*(*R*), we have [*I*] = [*A*], [*J*] = [*A*]+[*B*], [*K*] = [*A*]+[*B*]+[*C*], so the matrix is:

**Exercise**

Calculate the corresponding matrix for the ring:

## Pairing

Next, consider the pairing given by:

We claim that this is well-defined; indeed for the first term, an exact sequence of projective modules splits as *P* ≅ *Q*⊕*Q’* so Hom(*P*, *M*) is the direct sum of Hom(*Q*, *M*) and Hom(*Q’*, *M*), and the RHS gives:

On the other hand, if 0 → *M’* → *M* → *M”* → 0 is an exact sequence of modules, then since *P* is projective, so is the resulting:

and we get as well.