PID finite-generation transfer for simply connected base and fiber

If $M$ and $G$ are finitely generated $R$-modules, choose finite generating sets. Their pairwise elementary tensors generate $M{\otimes }_{R}G$, so the tensor product is finitely generated. By [F3], write $M$ as a finite sum of copies of $R$ and modules $R\mathrm{/}(a)$. A free summand has zero first Tor. The two-term free resolution $0\to R\stackrel{a}{\to }R\to R\mathrm{/}(a)\to 0$ identifies ${\mathrm{Tor}}_1^R(R\mathrm{/}(a),G)$ with $\ker (a:G\to G)$. This is a submodule of the finitely generated, hence Noetherian, module $G$, so it is finitely generated by [F4]. Finite additivity now makes ${\mathrm{Tor}}_1^R(M,G)$ finitely generated.

For $p+q\le N$, put $G=H_q(F;R)$. Both $H_p(B;R)$ and $H_{p-1}(B;R)$ occurring in [F2] are within the hypothesis, with negative degree interpreted as zero. Step 1.1 makes the tensor and Tor end terms finitely generated. Lifts of finitely many generators of the quotient, together with generators of the submodule, generate the middle term, so [F2] makes every $E_{p,q}^2$ in this triangle finitely generated.

Every later $E_{p,q}^r$ is a quotient of a submodule of $E_{p,q}^2$. By [F4] the submodule is finitely generated, and its quotient is generated by the images of those generators. Thus every stable term of total degree at most $N$ is finitely generated.

For fixed $n\le N$, [F1] gives a finite filtration of $H_n(E;R)$ with those stable terms as successive quotients. Starting with zero, repeatedly lift a finite generating set of the next quotient and adjoin it to generators of the preceding filtration term. Finite induction proves that $H_n(E;R)$ is finitely generated. Every field and $\mathbb{Z}$ is a commutative PID, giving the final specializations.

For $N=0$, the only page term is $H_0(B;R){\otimes }_R{H}_0(F;R)\cong R$, and the conclusion is $H_0(E;R)\cong R$. Empty base and fiber are excluded by simple connectivity. The zero module, zero homology groups, empty torsion decomposition, one generator, a free module, one cyclic torsion summand, and a field where every torsion summand is absent are all included in steps 1.1–4.1. Degenerate singular chains do not affect the homology modules. Both tensor and Tor terms, submodule and quotient directions, filtration endpoints, and $p=0$ and $q=0$ axes are checked. AC is used only through [F2] and its balanced Tor interpretation; no page representatives or generators are chosen simultaneously over an infinite family. There is no converse or claim above degree $N$.