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Integer multiplication on rational complexes
PDF p. 6: Finite-coefficient comparison
Statement
Let C be a cochain complex of Q-vector spaces, with Q-linear differentials, and n a positive integer. Multiplication by n is an isomorphism of C as a complex of abelian groups and hence in the derived category D(Z). Here D(Z) is the category of cochain complexes of abelian groups with quasi-isomorphisms inverted; a quasi-isomorphism induces isomorphisms on all cohomology groups.
Facts & Assumptions
Given: C and n as in the statement.
Goal: Prove that multiplication by n on C is an isomorphism in D(Z).
Multiplication by n on each Q-vector space is invertible, with inverse multiplication by 1/n. Integer multiplication on rational vector spaces
Proof
By [F1], multiplication by 1/n is the inverse of multiplication by n in every degree. Q-linearity of the differentials makes multiplication by 1/n commute with them, so these inverses form a chain map.
The two chain maps are mutually inverse, and their images remain inverse in D(Z).
Depends on
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