COMPUTATION OF SOME EXAMPLES OF GROTHENDIECK GROUPS 1. Grothendieck Group of abelian Categories Let A be an abelian category. The Grothendieck group of A is an abelian group generated by the isomorphism classes [A] of objects A of A; subject to the relations [A] = [A0] + [A00] whenever 0 ! A0 ! A ! A00 ! 0 is an exact sequence in A: This group can be constructed as follows. Let F (A) be the free abelian group generated by isomorphism classes of objects [A] of A and R(A) be the subgroup of F (A) generated by elements [A] − [A0] − [A00] when 0 ! A0 ! A ! A00 ! 0 is exact. The quotient group F (A)=R(A) is the Grothendieck group K(A): The image of [A] in K(A) is still denoted by [A]: Using the exact sequence 0 ! A1 ! A1 ⊕ A2 ! A2 ! 0; we obtain [A1 ⊕ A2] = [A1] + [A2]: Given an abelian category A; we denote ob A the class of objects of A: Let G be an abelian group. A function f : ob A ! G is said to be additive if f(A) = f(A0) + f(A00) for any exact sequence 0 ! A0 ! A ! A00 ! 0: One can verify that an additive function f : ob A ! G induces a group homomorphism f : K(A) ! G: Let us compute some examples. Proposition 1.1. Let VectF be the category of finite dimensional vector spaces over a field ∼ F: Then K(VectF ) = Z: Proof. Two finite dimensional F -vectors spaces are isomorphic if and only if they have the same dimension. In other words, let V and W be finite dimensional F -vector spaces. Then [V ] = [W ] if and only if dimF V = dimF W: We define an additive function : ob VectF ! Z≥0;V 7! dimF V: For any exact sequences of K-vector spaces, 0 ! V 0 ! V ! V 00 ! 0; we have 0 00 dimF V = dimF V + dimF V : Hence induces a group homomorphism : K(VectF ) ! Z defined by ([V ] − [W ]) = dimF V − dimF W: This is well-defined. In fact, this is a group isomorphism. To see this, ([V ] − [W ]) = 0; if and only if dimF V = dimF W for any representative V of [V ] and W of [W ]: We find [V ] = [W ]: Hence [V ]−[W ] = 0 in K(VectF ): We prove that is a monomorphism. To show that it is surjective, we simply use the fact that ([F n] − [F m]) = n − m for any n; m ≥ 0: This completes the proof of our assertion. Proposition 1.2. Let Ab be the category of finitely generated abelian groups. Then ∼ K(Ab) = Z: 1 2 COMPUTATION OF SOME EXAMPLES OF GROTHENDIECK GROUPS Proof. If G is a finitely generated abelian group, then there exists n ≥ 0 and d1; ··· ; dr such that ∼ n G = Z ⊕ Zd1 ⊕ · · · ⊕ Zdr : Here Zd = Z=dZ: Hence we obtain r X [G] = n[Z] + [Zdi ]: i=1 Let f : Z ! Z be the homomorphism defined by f(x) = dx: Then Im f = dZ and ker f = 0: These give us a short exact sequence 0 ! Z ! Z ! Zd ! 0; This shows that [Z] = [Z] + [Zd]; and thus [Zd] = 0: Notice that if we denote Gt the torsion ∼ n part of G; then G=Gt is torsion free and thus free, and G=Gt = Z : Let us consider a map : ob Ab ! Z≥0;G ! rank(G=Gt): If 0 ! G0 ! G ! G00 ! 0 is an exact sequence of finitely generated abelian groups, we obtain an exact sequence of free modules 0 0 00 00 0 ! G =Gt ! G=Gt ! G =Gt ! 0 which implies that 0 0 00 00 rank(G=Gt) = rank(G =Gt) + rank(G =Gt ): In other words, is an additive function. We obtain a group homomorphism : K(Ab) ! Z: Notice that if (G) = 0; then G is torsion and thus [G] = 0: Hence is in fact a group n m isomorphism; surjectivity can be proved by taking ([Z ] − [Z ]) = n − m: This also shows that K(Ab) is the free abelian group generated by [Z]: Proposition 1.3. Let A be the category of finite abelian groups. Then K(A) is the free abelian group generated by f[Zp]: p is a primeg: Proof. Any finite abelian group A has a composition series A = An ⊃ · · · ⊃ A1 ⊃ A0 = 0 ∼ 1 such that Ai=Ai−1 = Zpi for some prime pi: By induction , we can show that n X [A] = [Ai=Ai−1]: i=1 Pn Hence [A] = i=1[Zpi ]: This shows that elements of K(A) is generated by f[Zp]: p is a primeg: Let rp : ob A ! Z to be rp(A) = the number of Ai=Ai−1 isomorphic to Zp: This is a well-defined function (independent of choice of composition series) by the Jordan Holder theorem. Then rp induces a well-defined group homomorphism rp : K(A) ! Z by rp([Zq]) = δpq for any primes p; q: Here we use rp for its induced map. Claim the set f[Zp]: p is a primeg is Z-linearly independent. 1 We leave it to the readers as an exercise. You may consider the exact sequence 0 ! Ai−1 ! Ai ! Ai=Ai−1 ! 0: COMPUTATION OF SOME EXAMPLES OF GROTHENDIECK GROUPS 3 Pr Suppose i=1 ai[Zpi ] = 0 for distinct prime numbers p1; ··· ; pr: Then r ! n X X rpk ai[Zpi ] = aiδik = ak: i=1 i=1 We obtain that ak = 0 since rp(0) = 0 for all prime p: This proves the linear independence of f[Zp]: p is a primeg: We complete the proof of our assertion. An object A in an abelian category A is simple if A has no proper subobject 2. We say that A has finite length if there exists a composition series A = An ⊃ An−1 ⊃ · · · ⊃ A0 = 0 of sub objects of A with each Ai=Ai−1 simple. Theorem 1.1. Let A be an abelian category such that every object of A has finite length. Then the Grothendick group K(A) is a free abelian group with basis f[S]: S is simpleg: Proof. The proof is similar to the proof of Proposition 1.3. 2A subobject of an object A in an abelian category is a monomorphism j : A0 ! A:.