Additive functors preserve chosen Gaussian cancellations

Complex and pivot. Since $d^{n+1}{d}^n=0$ in $X^{\bullet }$, [L4] gives $F(d^{n+1})F(d^n)=F(d^{n+1}{d}^n)=F(0_{X^n,X^{n+2}})$, which is the zero morphism by [L5]; thus $F(X^{\bullet })$ is a cochain complex. Likewise $F(\phi )F({\phi }^{-1})=F(\phi {\phi }^{-1})=F(1_V)=1_{F(V)}$ and $F({\phi }^{-1})F(\phi )=1_{F(U)}$, so $F(\phi )$ is invertible with the displayed inverse.

Image matrices. By [L5] the $F$-images of the injections and projections of $X^n=A\oplus U$ and $X^{n+1}=B\oplus V$ exhibit $F(X^n)$ as $F(A)\oplus F(U)$ and $F(X^{n+1})$ as $F(B)\oplus F(V)$. Writing $d^n=i_{B}a{p}_A+i_{B}b{p}_U+i_{V}c{p}_A+i_V\phi p_U$ with the biproduct structure maps [L2], additivity of $F$ on hom-groups, preservation of composition and the biproduct relations give $F(d^n)=F(i_B)F(a)F(p_A)+F(i_B)F(b)F(p_U)+F(i_V)F(c)F(p_A)+F(i_V)F(\phi )F(p_U)$, whose matrix with respect to the image decompositions is $\left(\begin{array}{cc}F(a)& F(b)\\ F(c)& F(\phi )\end{array}\right)$ by the matrix convention of [L5]. The same computation applies to $d^{n-1}$ and $d^{n+1}$, giving the image neighbouring components $F(p),F(q),F(r),F(s)$.

Retract identities are preserved. Applying [L4] to the identities of [L1] and using [L5] for the zero morphisms: $F(p)F(\mathrm{ı})=F(p\mathrm{ı})=F(1_{{\bar{X}}^{\bullet }})=1_{F({\bar{X}}^{\bullet })}$; $F(d)F(h)+F(h)F(d)=F(dh+hd)=F(1_{X^{\bullet }}-\mathrm{ı}p)=1_{F(X^{\bullet })}-F(\mathrm{ı})F(p)$; $F(p)F(h)=F(ph)=F(0)=0$; $F(h)F(\mathrm{ı})=F(h\mathrm{ı})=0$; and $F(h{)}^2=F(h^2)=F(0)=0$. Since $p,\mathrm{ı}$ are cochain maps, $F(p),F(\mathrm{ı})$ are cochain maps by [L4].

The contractible summand and the isomorphism. By [L3] and [L4], $F(T)F(T^{-1})=F(T{T}^{-1})=F(1)=1$ and $F(T^{-1})F(T)=1$, so $F(T)$ is an isomorphism of complexes with inverse $F(T^{-1})$; and $F(K)$ has objects $F(U),F(V)$ in degrees $n,n+1$, vanishing terms elsewhere with zero differentials, and differential $F(\phi )$, with $F(\phi )F({\phi }^{-1})=1_{F(V)}$ and $F({\phi }^{-1})F(\phi )=1_{F(U)}$ from step 1.1, so $F({\phi }^{-1})$ is a contracting homotopy for $F(K)$.

The reduction of $F(X^{\bullet })$ is $F({\bar{X}}^{\bullet })$. By step 1.2 the reduction problem for $F(X^{\bullet })$ in degrees $n,n+1$ is the image matrix $\left(\begin{array}{cc}F(a)& F(b)\\ F(c)& F(\phi )\end{array}\right)$ with pivot $F(\phi )$, and by step 1.1 that pivot is invertible; [L4] gives $F(a)-F(b)F({\phi }^{-1})F(c)=F(a-b{\phi }^{-1}c)=F(\bar{d})$, and applying $F$ to the remaining data of [L2] gives objects $F(A),F(B)$ in degrees $n,n+1$, neighbouring arrows $F(p),F(r)$ and the unchanged images of the outside objects and arrows. Hence the candidate reduction of $F(X^{\bullet })$ at this pivot is exactly $F({\bar{X}}^{\bullet })$, its differential in degree $n$ being the image $F(\bar{d})$ of the Schur complement.

Conclusion. Step 1.1 shows that $F(X^{\bullet })$ is a complex with invertible pivot $F(\phi )$, step 1.2 computes the image matrices, and step 2.2 identifies the reduction of $F(X^{\bullet })$ with $F({\bar{X}}^{\bullet })$, which is clause 2 and the matrix assertion of clause 1. Step 1.3 verifies all five strong deformation retract identities for $F(p),F(\mathrm{ı}),F(h)$, and step 2.1 shows that $F(K)$ is contractible via $F({\phi }^{-1})$ and that $F(T)$ is an isomorphism, which is clause 3. Only additivity, preservation of finite biproducts and preservation of zero morphisms are used, so no exactness hypothesis enters; if $\mathcal{B}$ is abelian, [L6] applied to the image cancellation identified in step 2.2 gives inverse homology maps. This compares the homology of the two image complexes, not the image under $F$ of a homology object in $\mathcal{A}$. ∎