Chain complex

Direct sum of chain complexes

Let 𝐴 =(𝐴∙,𝜕∙) and 𝐵 =(𝐵∙,𝜕′∙) be chain complexes in an abelian category 𝖢. Then one may define the direct sum complex 𝐴 ⊕𝐵 by #m/def/homology

⋯𝜕𝑘−1⊕𝜕′𝑘−1←←←←←←←←←←←←←←←𝐴𝑘−1𝜕𝑘⊕𝜕′𝑘←←←←←←←←←←𝐴𝑘⊕𝐵𝑘𝜕𝑘+1⊕𝜕′𝑘+1←←←←←←←←←←←←←←←𝐴𝑘+1𝜕𝑘+2⊕𝜕′𝑘+2←←←←←←←←←←←←←←←⋯

Properties

  1. 𝐴 ⊕𝐵 is exact iff 𝐴 and 𝐵 are exact.
Proof

Note that ker⁡(𝜕𝑘 ⊕𝜕′𝑘) =(ker⁡𝜕𝑘) ⊕(ker⁡𝜕′𝑘). Similarly im⁡(𝜕𝑘−1 ⊕𝜕′𝑘−1) =(im⁡𝜕𝑘−1) ⊕(im⁡𝜕′𝑘−1). From this the statement ^P1 is clear.

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