The Grothendieck group

R. Virk

Department of Mathematics
University of Wisconsin
Madison, WI 53706

virk@math.wisc.edu

If $\mathcal{C}$ is an abelian category, we write $K(\mathcal{C})$ for the Grothendieck group of $\mathcal{C}$. This is the quotient of the free abelian group with generators the objects $M \in \mathcal{C}$, by the subgroup generated by elements $M_1 - M_2 + M_3$ for every short exact sequence

$$0 \to M_1 \to M_2 \to M_3 \to 0$$

in $\mathcal{C}$. If objects in $\mathcal{C}$ have finite length and unique composition factors we write $K(\mathcal{C})^*$ for the topological dual of $K(\mathcal{C})$; i.e. for the linear functions $f:K(\mathcal{C}) \to \mathbb{Z}$ such that $f(M) = 0$ for all but finitely many isomorphism classes of irreducible objects $M \in \mathcal{C}$. If $L \in \mathcal{C}$, let's write $[L]$ for its image in $K(\mathcal{C})$. Then as $L$ runs through the irreducible objects in $\mathcal{C}$, the elements $[L]$ form a basis of $K(\mathcal{C})$, and the functions

$$\delta_L:K(\mathcal{C}) \to \mathbb{Z}\qquad \delta_L(L') = \left... ...ong L'$, $L'$\ irreducible} \\ 1 & \mbox{if $L \cong L'$} \end{array} \right.$$

form a basis of $K(\mathcal{C})^*$. More generally, if $M \in \mathcal{C}$ and $L$ is an irreducible object in $\mathcal{C}$ write $[M:L]$ for the multiplicity of $L$ in a Jordan-Holder series of $M$, and extend this bilinearly to $[\quad : \quad ]:K(\mathcal{C}) \times K(\mathcal{C}) \to \mathbb{Z}$. Then for any $M \in K(\mathcal{C})$, write $\delta_M:K(\mathcal{C}) \to \mathbb{Z}$ for the function $N \mapsto [N:M]$. Now, if $F: \mathcal{C}\to \mathcal{C}'$ is an exact functor of abelian categories, we get an induced $\mathbb{Z}$-linear map $F:K(\mathcal{C}) \to K(\mathcal{C}')$, and we can define its transpose $F^*:K(\mathcal{C}')^* \to K(\mathcal{C})^*$ by $F^*f = fF$. Write $K(\mathcal{C})_{\mathbb{Q}} = K(\mathcal{C})\otimes \mathbb{Q}$. As $K(\mathcal{C})$ is a torsion free $\mathbb{Z}$-module, $K(\mathcal{C})_{\mathbb{Q}}$ is a $\mathbb{Q}$-vector space with distinguished sublattice $K(\mathcal{C})\subset K(\mathcal{C})_{\mathbb{Q}}$.

Proposition 0.0.1   Let $F:\mathcal{C}_1 \rightarrow \mathcal{C}_2$ and $G:\mathcal{C}_2\rightarrow \mathcal{C}_1$ be mutually adjoint exact functors between two Artinian abelian categories $\mathcal{C}_1$ and $\mathcal{C}_2$. Then $F$ and $G$ are mutual equivalences of categories if and only if they define mutually inverse isomorphisms on the level of Grothendieck groups.

Proof. For $M\in\mathcal{C}_2$, let $\varphi_M$ denote the adjunction map $M\to F_1F_2(M)$. Suppose $L\in\mathcal{C}_2$ is irreducible. Since $[L]=[F_1F_2(L)]$, we have that $\varphi_L$ is an isomorphism. Now proceed by induction on length. For arbitrary $M\in\mathcal{C}_2$ we have an exact sequence $0\to N\to M\to L \to 0$, with $N$ of lower length than $M$ and $L$ irreducible. This gives a commutative diagram

$$\xymatrix{ 0\ar[r]& N \ar[d]^{\varphi_N}\ar[r]& M\ar[d]^{\varphi_... ...L}\ar[r]&0 \\ 0\ar[r]& F_1F_2(N)\ar[r]& F_1F_2(M)\ar[r] & F_1F_2(L)\ar[r]& 0 }$$

The map $\varphi_N$ is an isomorphism by induction; $\varphi_L$ is an isomorphism by the previous argument. This forces $\varphi_M$ to be an isomorphism. $\qedsymbol$