What is the largest subcategory $C$ of a module category over an Artin algebra, such that $C$ is Krull-Schmidt (and abelian)? Does $C$ exist?

Question

Let $A$ be an Artin algebra, $\text{Mod}\,A$ the category of $A$-modules and $\text{mod}\,A$ the category of finitely generated $A$-modules. It is well-known that $\text{mod}\,A$ is a Krull-Schmidt category. Can we find a larger full subcategory $\mathcal{C}$ of $\text{Mod}\,A$, such that it remains a Krull-Schmidt category? Is there a largest $\mathcal{C}$ and if yes, how does $\mathcal{C}$ look like? What happens if we additionally demand $\mathcal{C}$ to be an abelian category?

Answer 1

To answer the first question, there is a larger Krull-Schmidt category than $\text{mod}\,A$ unless $A$ has finite representation type.

Every indecomposable pure-injective module has local endomorphism ring, and an Artin algebra of infinite representation type has an infinitely generated indecomposable pure-injective module.

So you could take $\mathcal{C}$ to be the full subcategory of finite direct sums of pure-injective modules.

[The claims above about pure-injective modules can be found, for example, as Theorem 4.3.43 and Theorem 5.3.40 of

Prest, Mike, Purity, spectra and localisation., Encyclopedia of Mathematics and its Applications 121. Cambridge: Cambridge University Press (ISBN 978-0-521-87308-6/hbk). xxviii, 769 p. (2009). ZBL1205.16002.]

I doubt this is often maximal. As Uriya First pointed out in comments, the maximal Krull-Schmidt subcategory of $\text{Mod}\,A$ will consist of all finite direct sums of modules with local endomorphism ring.

I also doubt that it will often (or ever, maybe) be abelian, for the following reason.

If $\mathcal{C}$ is a subcategory of $\text{Mod}\,A$ containing $\text{mod}\,A$ that is abelian, then since $\mathcal{C}$ contains $A$ and $DA$ it follows that exact sequences in $\mathcal{C}$ are exact in $\text{Mod}\,A$, so $\mathcal{C}$ is closed under kernels and cokernels in $\text{Mod}\,A$.

Often this can be used to show that if $\mathcal{C}$ contains an infinitely generated module then it contains an infinite direct sum of simple modules, and so cannot be Krull-Schmidt. I don't know if this is always the case, but here's one illustrative case:

Suppose $A$ is local, and $\mathcal{C}$ is an abelian subcategory of $\text{Mod}\,A$ that contains an infinitely generated module $X$. Then if $\operatorname{rad}^iX$ is the last nonzero power of the radical, there is a map $X\to X$ whose cokernel is $X/\operatorname{rad}^iX$. By induction, $\mathcal{C}$ contains $X/\operatorname{rad}X$, which is an infinite direct sum of simple modules.