Abstract
We show that direct summands of certain additive functors arising as bifunctors with a fixed argument in an abelian category are again of that form whenever the fixed argument has finite length or, more generally, satisfies the descending chain condition on images of nested endomorphisms. In particular, this provides a positive answer to a conjecture of M. Auslander in the case of categories of finite modules over artin algebras. This implies that the covariant Ext functors are the only injectives in the category of defect-zero finitely presented functors on such categories.
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Martsinkovsky, A. On Direct Summands of Homological Functors on Length Categories. Appl Categor Struct 24, 421–431 (2016). https://doi.org/10.1007/s10485-015-9403-1
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DOI: https://doi.org/10.1007/s10485-015-9403-1
Keywords
- Functor category
- Finitely presented functor
- Fitting lemma
- Length category
- Descending chain condition on images of nested endomorphisms
- Injective object
- Homological functor
- Hom modulo an ideal