Let $C$ be (a full) exact subcategory of the category of $R$-modules. We suppose that $C$ is essentially small. If the Grothendieck group $K\_{0}(C)=0$, what can be said about the higher groups $K\_{n}(C)=0$ ? Is there a non-trivial example of such exact subcategory $C$ ?



Exact subcategory with trivial Grothendieck group: what are the consequences and examples
This is a long comment more than an answer.


If you think of $K\_0$ as a universal domain for all kinds of functions that associate a "dimension" to a module, then $K\_0(\mathscr{C})=0$ means that at least some of the modules in $\mathscr{C}$ are "the worst kind of infinite dimensional". For example they are not of finite length (otherwise the length would be a nontrivial dimension function) not even over subrings of $R$, they are not finitely generated over any commutative subring (otherwise dimension at some prime would be nontrivial) etc.


Because of the Eilenberg-Swindle the category of *all* $R$-modules $\mathscr{C}:=R\mathsf{-Mod}$ is always an example with $K\_0(\mathscr{C})=0$. The offending modules are infinite direct sums $X\oplus X\oplus X \oplus \cdots$ because those fit into exact sequences $0\to A\to A \to X\to 0$ and $0\to X\to A\to A\to 0$ respectively so that $[X] =0$ in $K\_0(R\mathsf{-mod})$. Is this "non-trivial" in the sense of your question? (The "essentially small" requirement can obviously be satisfied by taking any sufficiently large exact subcategory of R-mod instead, say the exact category generated by all finite length modules as well as a swindle module for each isomorphism class)


Conversely every category with $K\_0(\mathscr{C})=0$ is similar in the sense that $[X]=0$ implies that $[X]$ is a finite sum of relations $\sum\_{i=1}^n \pm([B\_i]-[A\_i]-[C\_i])$ (viewed as an element in the free abelian group generated by isomorphism classes) with exact sequences $0\to A\_i\to B\_i\to C\_i\to 0$ so that by taking direct sums over all positive and all negative signs, we have two sequences $0\to A\to B\to C\to 0$ and $0\to A'\to B'\to C'\to 0$ and $X$ is a direct summand of one of the modules. This is somewhat similar to a "swindle sequence". And in this sense the existence of "swindle sequences" is the common feature of all examples.


