Chain complex

Direct sum of chain complexes

Let 𝐴 =(π΄βˆ™,πœ•βˆ™) and 𝐡 =(π΅βˆ™,πœ•β€²βˆ™) be chain complexes in an abelian category 𝖒. Then one may define the direct sum complex 𝐴 βŠ•π΅ by homology

β‹―πœ•π‘˜βˆ’1βŠ•πœ•β€²π‘˜βˆ’1β†β†β†β†β†β†β†β†β†β†β†β†π΄π‘˜βˆ’1πœ•π‘˜βŠ•πœ•β€²π‘˜β†β†β†β†β†β†β†β†β†π΄π‘˜βŠ•π΅π‘˜πœ•π‘˜+1βŠ•πœ•β€²π‘˜+1β†β†β†β†β†β†β†β†β†β†β†β†π΄π‘˜+1πœ•π‘˜+2βŠ•πœ•β€²π‘˜+2←←←←←←←←←←←←⋯

Properties

  1. 𝐴 βŠ•π΅ is exact iff 𝐴 and 𝐡 are exact.

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