L-homologies of double complexes

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The notion of L-homologies (of double complexes) as proposed in this paper extends the notion of classical horizontal and vertical homologies along with two other new homologies introduced in the homological diagram lemma called the salamander lemma. We enumerate all L-homologies associated with an object of a double complex and provide new examples of exact sequences. We describe a classification problem of these exact sequences. We study two poset structures on these L-homologies; one of them determines the trivialities of horizontal and vertical homologies of an object in terms of other L-homologies of that object, whereas the second structure shows the significance of the two homologies introduced in the salamander lemma. Finally, we prove the existence of a faithful amnestic Grothendieck fibration from the category of L-homologies to a category consisting of objects and morphisms of a given double complex.

Original languageEnglish
Pages (from-to)1600-1608
Number of pages9
JournalCommunications in Algebra
Issue number4
Publication statusPublished - 2021


  • Double complexes
  • exact sequences
  • Grothendieck fibrations
  • posets

ASJC Scopus subject areas

  • Algebra and Number Theory


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