Countable tower six term limit sequence

The map $P(B)\to P(C)$ is onto: choose a lift in each coordinate by [F2]. If $c\in L(C)$, lift it to $b\in P(B)$. Then ${\mathrm{\Delta }}_{B}b\in P(A)$ by commutation, so define $\mathrm{\partial }c=[{\mathrm{\Delta }}_{B}b]$. Replacing $b$ by $b+a$ changes the result by ${\mathrm{\Delta }}_{A}a$, hence leaves its class unchanged. Sum and scalar multiple lifts establish linearity.

The map $L(A)\to L(B)$ is injective coordinatewise. A compatible $B$ tuple whose image in $L(C)$ vanishes lies coordinatewise in $A$ and is still compatible. This proves exactness at $L(B)$ as well.

Suppose every $u_m$ is onto. By [F2] choose a right inverse as a set map for each $u_m$. For prescribed $y\in P(A)$ put $x_0=0$ and recursively choose $x_{m+1}$ with $u_m{x}_{m+1}=x_m-y_m$ using these sections. Then ${\mathrm{\Delta }}_{A}x=y$, proving $R(A)=0$. For prescribed $x_m\in A_m$, the same recursion with $y=0$ constructs later coordinates, while compositions of transitions determine earlier ones. This proves surjectivity of $L(A)\to A_m$. No sections are asserted linear.

Restriction to $m\ge N$ gives a bijection on compatible tuples: earlier coordinates are forced by the transitions. It is surjective on Delta cokernels because a tail representative can be extended by zero. If a full representative restricts to $\mathrm{\Delta }x$ on the tail, extend $x$ backwards by the finite recursion $x_m=y_m+u_m{x}_{m+1}$. The original representative is now a full Delta boundary. Hence restriction is also injective on cokernels and is linear. These identifications commute with tower morphisms.

A compatible $B$ lift of $c$ has zero connecting class. Conversely if $\mathrm{\partial }c=0$, a lift $b$ has ${\mathrm{\Delta }}_{B}b={\mathrm{\Delta }}_{A}a$ for some $a\in P(A)$; then $b-a$ is a compatible lift. Thus exactness holds at $L(C)$ in both directions.

A connecting class becomes zero in $R(B)$. Conversely if $a\in P(A)$ represents a class becoming zero there, write $a={\mathrm{\Delta }}_{B}b$. The image $c$ of $b$ is compatible and has $\mathrm{\partial }c=[a]$. This proves exactness at $R(A)$.

If $b\in P(B)$ maps to ${\mathrm{\Delta }}_{C}c$, lift $c$ to $t\in P(B)$ as in step 1.1. Then $b-{\mathrm{\Delta }}_{B}t\in P(A)$ represents the same $R(B)$ class. Every image from $R(A)$ conversely maps to zero in $R(C)$. Finally any representative in $P(C)$ lifts to $P(B)$, proving surjectivity at $R(C)$.

A morphism of the exact tower sequences sends a selected lift to a lift and commutes with Delta. Thus it commutes with $\mathrm{\partial }$; it plainly commutes with coordinate inclusions and quotient maps too. The entire sequence is natural, including its end terms.

The exactness assertions follow from steps 1.2–2.3 and naturality from step 3.1; steps 1.3 and 1.4 prove the two additional claims. Zero modules, zero maps where exactness permits them, and repeated or constant terms cause no exceptions. A single nonzero term followed by zeros has zero $L$ and $R$ by the tail assertion. A constant identity tower has $L=A_0$ and $R=0$. The index set is the nonempty set of natural numbers; no empty tower is being claimed. All infinite selections were explicitly made in steps 1.1 and 1.3 under AC.