Abelian-group model for spectral-sequence computations

For abelian groups A,B the homomorphisms form an abelian group under $(f+g)(a)=f(a)+g(a)$: additivity follows from commutativity in B, with zero map and pointwise negatives as identity and inverses. The group laws hold pointwise. Composition is bilinear because each map is additive [F2]. Identities and composites are homomorphisms by the same calculation, so this is a preadditive category.

The one-element group 0 is both initial and terminal. The product group $A\times B$ with coordinate operations has projections and injections a↦(a,0), b↦(0,b). A pair of maps T→A,B induces t↦(f(t),g(t)), uniquely by coordinates. Maps A→T and B→T induce $(a,b)\mapsto f(a)+g(b)$, uniquely since $(a,b)=(a,0)+(0,b)$. These are its product and coproduct properties. Iterating yields every finite biproduct, including the empty one 0.

For f:A→B, the subset $K=\{a:f(a)=0\}$ is a subgroup: f(0)=0 follows from f(0)=f(0)+f(0), and $f(a-a^{\mathrm{\prime }})=f(a)-f(a^{\mathrm{\prime }})$. Any homomorphism killed by f has values in K and factors uniquely through its inclusion. The image I={f(a):a∈A} is likewise a subgroup, since f(a)-f(a')=f(a-a').

For a subgroup I⊆B define b∼b' by b-b'∈I. Zero, negation and addition in I prove reflexivity, symmetry and transitivity. On the quotient set of [F5], define [b]+[c]=[b+c]; changing b,c by elements of I changes their sum by an element of I, so addition is well defined, and the group laws descend. A homomorphism h:B→T vanishes on I exactly when b↦h(b) is constant on each coset: one direction follows from h(b)-h(b')=h(b-b'), and the other by comparing i and 0 for i∈I. Thus [b]↦h(b) is the unique quotient homomorphism. With I=im f, this is a cokernel of f.

The kernel of B→B/I is exactly I since [b]=[0] iff b∈I. Consequently the categorical image of f is I, and its coimage is A/K. The canonical comparison [a]↦f(a) is well defined, is onto I by definition, and is injective since f(a)=f(a') iff a-a'∈K. Its inverse is the uniquely specified relation sending f(a) to [a], not a chosen representative function; it is additive because both maps preserve addition. Hence [F1] applies to steps 1.1–3.1 and proves the abelian-category assertion.

By [F3] the additive integers are an abelian group. If 2a=0 and a>0, positivity gives 2a>0; if a<0 it gives 2(-a)>0. Thus a=0, proving injectivity. Its image is 2ℤ and its cokernel is ℤ/2ℤ by step 3.1, with the congruence operations of [F4]. Division [F6] for divisors 2 and 4 gives exactly the distinct residue lists {0,1} and {0,1,2,3}. Their additions are modulo 2 and 4, so 1+1=0 in ℤ/2 and 1+1=2≠0, 2+2=0 in ℤ/4. Finite products have the coordinate operations of step 2.1.