Algebraic finite domination
Luke Steers · Research Portal (Queen's University Belfast) · 2017
Definition 0.3.3.Define a double complex as a collection of R-modules C a,b , indexed over Z 2 , and mapsThis convention provides that the boundaries anti-commute.This ensures that the obvious candidate for totalisation is a chain complex.Definition 0.3.4.Given a double complex C = {C a,b , d h , d v }, define the totalisation of the double complex, Tt(C), as the complex with module at degree k a+b=k C a,b and boundary consisting ofRemark 0.3.5.For a double complex C, the boundary map of Tt(C) sat-Both of these definitions follow [HQ15], so that we can lift the following result from it for later use.Lemma 0.3.6.Let f : C → D be a map of double complexes which are concentrated in finitely many columns.If f is a quasi-isomorphism on each column or on each row, then the induced map Tt(f ) : Tt(C) → Tt(D) is a quasi-isomorphism.Proof.Seen in [HQ15, Lemma 2.2.2].