[Paper Review] When stable short exact sequences define an exact structure on any additive category

Septimiu Crivei|arXiv (Cornell University)|Sep 15, 2012

Algebraic structures and combinatorial models18 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition for the stable short exact sequences—kernel-cokernel pairs with semi-stable kernels and cokernels—to define the maximal exact structure on an arbitrary additive category. Using the idempotent completion of the category and transfer of properties via a fully faithful functor, the authors show that this holds precisely when the original category is closed under pushouts and pullbacks relative to the maximal exact structure on the completion. The key contribution is a general criterion that extends the known result from weakly idempotent complete categories to arbitrary additive categories.

ABSTRACT

Rump has recently showed the existence of a unique maximal Quillen exact structure on any additive category. We study when this is given by the stable short exact sequences, i.e. kernel-cokernel pairs consisting of a semi-stable kernel and a semi-stable cokernel.

Motivation & Objective

• To determine when the class of stable short exact sequences forms the maximal exact structure on an arbitrary additive category.
• To resolve the open question of whether Rump's maximal exact structure coincides with the structure defined by stable short exact sequences.
• To extend previous results from weakly idempotent complete categories to general additive categories.
• To provide a general criterion based on closure under pushouts and pullbacks in the idempotent completion.

Proposed method

• Utilizing the additive idempotent (Karoubian) completion of an additive category, which is weakly idempotent complete.
• Constructing a fully faithful functor $ H: \mathcal{C} \to \widehat{\mathcal{C}} $ from the original category to its idempotent completion.
• Proving that $ H $ preserves stable short exact sequences, enabling transfer of properties from the completion to the original category.
• Introducing the notion of $ \mathcal{C} $ being closed under pushouts and pullbacks for $ (\widehat{\mathcal{C}}, \mathcal{E}_{\rm max}) $, which serves as the key criterion.
• Applying the criterion to show that the maximal exact structure on the category of chain complexes $ \mathbf{Ch}(\mathcal{C}) $ and on the category of projective spectra $ \mathbf{P}(\mathcal{C}) $ is given by degree-wise stable short exact sequences.
• Leveraging known results on exact subcategories and extension-closed subcategories to support the transfer of exact structures.

Experimental results

Research questions

• RQ1 Under what conditions do stable short exact sequences define the maximal exact structure on an arbitrary additive category?
• RQ2 Does Rump's maximal exact structure on an additive category coincide with the structure generated by stable short exact sequences?
• RQ3 Can the criterion for the maximal exact structure via stable short exact sequences be extended beyond weakly idempotent complete categories?
• RQ4 How does the idempotent completion of a category relate to the preservation and transfer of exact structures?
• RQ5 What conditions ensure that the maximal exact structure on $ \mathbf{Ch}(\mathcal{C}) $ or $ \mathbf{P}(\mathcal{C}) $ is induced by stable short exact sequences in $ \mathcal{C} $?

Key findings

• The stable short exact sequences define the maximal exact structure on an additive category $ \mathcal{C} $ if and only if $ \mathcal{C} $ is closed under pushouts and pullbacks for the maximal exact structure on its idempotent completion $ \widehat{\mathcal{C}} $.
• The canonical functor $ H: \mathcal{C} \to \widehat{\mathcal{C}} $ preserves stable short exact sequences, enabling the transfer of structural properties.
• The maximal exact structure on the category of chain complexes $ \mathbf{Ch}(\mathcal{C}) $ coincides with the structure induced by degree-wise stable short exact sequences in $ \mathcal{C} $, provided $ \mathcal{C} $ satisfies the closure condition.
• The maximal exact structure on the category of projective spectra $ \mathbf{P}(\mathcal{C}) $ is given by those short exact sequences for which each component is a stable short exact sequence in $ \mathcal{C} $, under the same closure assumption.
• The result generalizes previous theorems that required weak idempotent completeness, now applying to arbitrary additive categories.
• The closure condition—closure under pushouts and pullbacks relative to $ (\widehat{\mathcal{C}}, \mathcal{E}_{\rm max}) $—is often easier to verify than directly checking the obscuring axioms of Quillen exact categories.