For a commutative ring $R$, let $\mathrm{mod-}R$ denote the category of finitely generated $R$-modules. Let $\mathcal C$ be an abelian, full, isomorphism closed (i.e. $M\cong N$ in $\mathrm{mod-}R$ and $M\in Ob(\mathcal C)\implies N\in Ob(\mathcal C)$) subcategory of $\mathrm{mod-}R$.
The Grothendieck group of $\mathcal C$, usually denoted by $G(\mathcal C)$, is defined as
$$\dfrac{\bigoplus_{X\in \operatorname{Ob}(\mathcal C)}\mathbb Z[X]}{\langle [A]-[B]+[C] \mid 0\to A \to B \to C\to 0 \text{ is exact} \rangle }$$
where $[X]$ denotes the isomorphism class of $X$.
Now if for every $M,N\in Ob(\mathcal C)$, we also have $M\otimes_R N\in Ob(\mathcal C)$, then the $\mathbb Z$-module
$$\bigoplus_{X\in \operatorname{Ob}(\mathcal C)}\mathbb Z[X]$$
can be given a $\mathbb Z$-algebra structure where the ring multiplication is induced by the tensor product. If every module in $\mathcal C$ is flat, then the $\mathbb Z$-submodule $${\langle [A]-[B]+[C] \mid 0\to A \to B \to C\to 0 \text{ is exact} \rangle }$$ of $\bigoplus_{X\in \operatorname{Ob}(\mathcal C)}\mathbb Z[X]$ is also an ideal, so the Grothendieck group actually becomes a commutative ring where the product structure is induced from tensor product.
My question is: What happens in case not every module of $\mathcal C$ necessarily flat? Does the Grothendieck group necessarily become a commutative ring where the product structure is induced from tensor product? I'm particularly interested in the case of $G(\mathrm{mod-}R)$: is it necessarily a commutative ring?
Best Answer
Suppose $R$ is a ring such that for every $R$-module $M$ there is a projective resolution $P_\bullet \to M$ of finite length (actually, this condition can be loosened quite a bit as I will note at the end). Then we can define a ring structure on $G(R)$ the Grothendieck group of finitely generated $R$-modules by defining the following product $$ [A] \cdot [B] = \sum_i (-1)^i \text{Tor}_i(A,B). $$ Note that if $P$ is a projective $R$-module, then $[A]\cdot[P] = [A \otimes P]$ since the $\text{Tor}_i(A,P)$ vanish for $i >0$.
The way to show this is a little painful. We need to show that
It is in my opinion easier to define this product as a map on $G(R)$ first, though tastes may vary. Fix a finitely generated $R$-module $M$ and let $$ - \cdot M : G(R) \to G(R),\; [A] \mapsto [A] \cdot M := \sum_i (-1)^i [\text{Tor}_i(A,M)]. $$
Then one can show that this map is independent of the choice of resolution by showing $\text{Tor}_i(-,M)$ is. The well-definedness of the map follows since for any short exact sequende $0 \to A \to B \to C \to 0$ we have a long exact sequence of Tor, which shows $[A] \cdot M - [B] \cdot M + [C] \cdot M = 0$. The symmetry follows from the symmetry of Tor, so $[A] \cdot B = [B] \cdot A$.
From this point it is therefore acceptable to write $[A] \cdot [B]$, and we can prove the ring axioms one by one. For this it is good to note that for an $R$-module $M$ with projective resolution $P_\bullet \to M$ we have $$ [M] \cdot [N] = \sum_i (-1)^i [\text{Tor}_i(M,N)] = \sum_i(-1)^i H_i(P_\bullet \otimes N) = \sum_i (-1)^i [P_i \otimes N]. $$
This computation involves the nontrivial lemma that in the Grothendieck group we have $\sum_i(-1)^i [C_i] = \sum_i (-1)^i [H_i(C_\bullet)]$ for any finite complex $C_\bullet$ in the category.
Then the ring axioms can easily be checked by using properties of the tensor product.
The requirement that every object in the category have a finite projective resolution is actually too strong. If every object has a finite resolution that is $F$-acyclic for a functor $F$ on the category, then it induces a derived morphism on Grothendieck groups. Actually, the resolution need not even be in the category itself. For example for categories of coherent sheaves we almost never have projective resolutions, but in some situations (details omitted on purpose) we may have locally free resolutions, which are $\text{Tor}_i(-, *)$ acyclic, so that we can define the product.