Proposition 1

Tag 0DRH, permanent. Cite it as [Bourbaki, Tag 0DRH].

For any family of complexes $(C_i)_{i \in I}$ , the canonical homomorphisms

\bigoplus_{i \in I} H(C_i) \to H(\bigoplus_{i \in I} C_i), \quad H(\prod_{i \in I} C_i) \to \prod_{i \in I} H(C_i)

are bijective.

For every inductive filtering system of complexes $(C_i)_{i \in I}$ , the canonical homomorphism

\varprojlim_{i \in I} H(C_i) \to H(\varprojlim_{i \in I} C_i)

is bijective.

This results immediately from II, p. 14, cor. 1 to prop. 7, p. 11, cor. to prop. 5, and p. 91, prop. 3.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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