An invertible cochain differential block and its candidate reduction

The block decomposition. Let $X^{\bullet }$ be a cochain complex in an additive category $\mathcal{A}$ (Complexes, homotopies and contractibility in an additive category) and fix an integer $n$. Suppose that in degrees $n$ and $n+1$ the objects of $X$ are given as biproducts $X^n=A\oplus U,X^{n+1}=B\oplus V,$ with the injections and projections of these biproducts fixed. With respect to these ordered decompositions write the differential at degree $n$ as the block matrix $d^n=\left(\begin{array}{cc}a& b\\ c& \phi \end{array}\right):A\oplus U\to B\oplus V,$ where the rows name $B,V$ and the columns name $A,U$: thus $a:A\to B$, $b:U\to B$, $c:A\to V$ and $\phi :U\to V$ are the four components, and composition of such block matrices is matrix multiplication (Composition of morphisms between finite biproducts is matrix multiplication).

The pivot. The block $\phi :U\to V$ is a pivot when it is an isomorphism, with two-sided inverse ${\phi }^{-1}:V\to U$. Only $\phi$ is assumed invertible; $a,b,c$ are arbitrary morphisms. The objects $A,U,B,V$ may themselves be biproducts of several objects, in which case $\phi$ is an invertible matrix of morphisms and is still required to be a single isomorphism $U\to V$; in particular $U$ and $V$ need not be nonzero or indecomposable.

The neighbours. With respect to the same decompositions write the neighbouring differentials as $d^{n-1}=\left(\begin{array}{c}p\\ q\end{array}\right):X^{n-1}\to A\oplus U,d^{n+1}=\left(\begin{array}{cc}r& s\end{array}\right):B\oplus V\to X^{n+2},$ so that $p:X^{n-1}\to A$, $q:X^{n-1}\to U$, $r:B\to X^{n+2}$ and $s:V\to X^{n+2}$ are the neighbouring components across the two pivot blocks.

The candidate reduction. The candidate reduction ${\bar{X}}^{\bullet }$ of $X^{\bullet }$ at the pivot $\phi$ is the collection of objects and morphisms ${\bar{X}}^j:=X^j(j\mathrm{\notin }\{n,n+1\}),{\bar{X}}^n:=A,{\bar{X}}^{n+1}:=B,$ with differentials ${\bar{d}}^j:=d^j(j\mathrm{\notin }\{n-1,n,n+1\}),{\bar{d}}^{n-1}:=p,{\bar{d}}^n:=a-b{\phi }^{-1}c,{\bar{d}}^{n+1}:=r\mathrm{.}$ The morphism $a-b{\phi }^{-1}c:A\to B$ is the Schur complement of the pivot $\phi$ in $d^n$. In words: the objects and arrows outside degrees $n,n+1$ are retained verbatim; the pivot blocks $U,V$ are discarded; and the differential at degree $n$ is replaced by its Schur complement, while the neighbouring differentials lose their components through the discarded blocks and keep $p$ and $r$.

Status of the construction. The words candidate and reduction are provisional: the definition alone does not assert that ${\bar{d}}^n{\bar{d}}^{n-1}=0$ or ${\bar{d}}^{n+1}{\bar{d}}^n=0$, that is, that ${\bar{X}}^{\bullet }$ is a cochain complex. The next lemma verifies this, using the identities $cp+\phi q=0$ and $rb+s\phi =0$ that follow from the square-zero composites $d^n{d}^{n-1}=0$ and $d^{n+1}{d}^n=0$ of the given complex.

Homological convention and scope. For a homological, degree-lowering complex the same formulas apply after the reindexing $n\mapsto -n$ of Complexes, homotopies and contractibility in an additive category: in the chain convention the pivot is the corresponding invertible block of $d_n$ and the reduced differential is again the Schur complement of that block, with the contracting homotopy of the discarded two-term complex acquiring degree $+1$. No sign is inserted. The construction uses only the additive structure; it assumes no abelian category, no exactness, no projectivity, no boundedness and no homology object.