Example about endomorphism ring of an indecomposable object, which have non-trivial idempotent element in an additive category.

Question

Recently I read some about Krull-Remak-Schmidt category.

If $A$ is an additive category in which every idempotent splits, every object is the biproduct of finitely many indecomposable objects and the endomorphism ring of every indecomposable object is local, then $A$ is KRS.

If we consider the category of modules, then M is indecomposable if and only if End(M) has trivial idempotent elements. And every small Abelian category can be embedded to a category of module.

Hence, I want to find an example about endomorphism ring of an indecomposable object, which have non-trivial idempotent element in an additive category.

Answer 1

You can take the free additive category on an idempotent. This is the additive category whose objects are finite direct sums of a single object $M$ satisfying

$$\text{End}(M) \cong \mathbb{Z}[e]/(e^2 - e) \cong \mathbb{Z} \times \mathbb{Z}.$$

The only nontrivial idempotents in this endomorphism ring are $e$ and $1-e$ and they don't split, so $M$ is indecomposable despite having an idempotent endomorphism; this is because a splitting, if it existed, would write $M$ as the direct sum of two objects with endomorphism ring $\mathbb{Z}$ but this category contains no objects with this endomorphism ring.

(More abstractly, since this is the free additive category on an idempotent, if this idempotent split then every idempotent in every additive category would split. So if there is any counterexample, this must be one.)

The idempotent completion of this category is the additive category whose objects are finite direct sums of two objects $M_e, M_{1-e}$ (the free splitting of $e$) with endomorphism ring $\mathbb{Z}$ and no nonzero homomorphisms between them, and so can be identified with the category of pairs of finite free $\mathbb{Z}$-modules. The original category is the subcategory of this one where the two modules have the same rank.

If we consider the category of modules, then M is indecomposable if and only if End(M) has trivial idempotent elements.

You mean only trivial idempotent elements. This is true in any additive category in which idempotents split, including any additive category which has either finite limits or finite colimits. This includes abelian categories but also others.