Exact subcategory with trivial Grothendieck group: what are the consequences and examples

Question

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 $C$ ?

Answer 1

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.

Answer 2

There is almost nothing one can deduce about the higher K-groups from just knowing $K_{0}(C)=0$.

As was pointed out in the comments, the restriction to modules over a general ring does not really restrict anything. Any exact category can (as explained) via the Quillen embedding be realized as an extension-closed subcategory of an abelian category and for the latter use Freyd-Mitchell (exactly as explained in the comments).

For producing nearly arbitrary examples: Let D be any idempotent complete exact category, e.g. any abelian category. Take (for example) the Tate category $\underleftrightarrow{\lim}D$ (an alternative notation is $\operatorname{Tate}(C)$) of Sho Saito's paper "On Previdi's delooping conjecture" (https://arxiv.org/abs/1203.0831). This is an exact category. By the previous remarks, it can be realized as a fully exact subcategory of a category of modules over a ring.

As proved in that paper, the nonconnective K-theory just shifts by one degree upwards (Theorem 1.2 of that paper). Since D was idempotent complete, the nonconnective K-theory will agree with usual Algebraic K-theory.

A direct computation will show that $K_0$ of this Tate category vanishes. So whatever K-groups you can find occurring in any idempotent complete exact category, you can make them appear (with a shift by one) in higher K-groups while simultaneously making $K_0$ vanish.

Such ideas were necessary to define nonconnective K-theory for exact categories in the first place, so you can alternatively find out about such constructs in Schlichting's paper "Delooping the K-theory of exact categories".

One does not have to use these Tate category constructions. There are others, e.g. the so-called Calkin category Ind(D)/D, which was used by Drinfeld in his paper on infinite-dimensional vector bundles (https://arxiv.org/pdf/math/0309155.pdf), see Section 3.3.1. The proofs for all these constructions rely on variants of Eilenberg swindles by the way, i.e. a machine to really make all K-groups vanish. One then re-assembles these contractible K-theory spectra in a way such that they become the loop space of an arbitrary input K-theory spectrum. That's the rough idea. In order to not having to do the "re-assembling" by hand (it's tough to exhibit bicartesian squares from scratch) one tries to make localization sequences "become" the squares one needs.

Finally, if your category $D$ happens to have a projective generator, you can also find such for the Tate category above (https://arxiv.org/pdf/1508.07880.pdf, Theorem 1.(2) for n=1). Thus, you can even realize all these examples simply as the projective modules over a certain ring $R$. This produces counter-examples even avoiding the Frey-Mitchell and Quillen embedding steps and one can reasonably explicitly describe these rings. By work of Wagoner, also various infinite matrix rings have these properties.