The Leray–Hirsch associated-graded isomorphism lifts without extension ambiguity

Put a summand $H^{q-\mathrm{\mid }{e}_i\mathrm{\mid }}(B;R)$ in filtration at least $p$ when its base class has Serre filtration at least $p$. By [F2], $p^{\ast }{a}_i$ has that filtration and cupping with the fixed class $e_i$ cannot decrease it. Hence the displayed $\mathrm{\Phi }$ is a filtered homomorphism, not merely a map invented on $E_{\mathrm{\infty }}$.

Fix total degree $q$. The first-quadrant bounds make both induced filtrations finite in that degree, while [F1] supplies exhaustivity, completeness, and the identification with the stable page. Under the hypothesis that $\mathrm{gr}\mathrm{\Phi }$ is an isomorphism, [F3] therefore says $\mathrm{\Phi }:H_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}^q\to H^q(E;R)$ is an isomorphism.

Applying step 2.1 in every degree proves the graded-module assertion. The inverse is obtained by the filtered-isomorphism lemma from kernels and cokernels along the finite filtration; no complements or splitting maps are chosen. Empty summand families and the zero ring give zero maps between zero modules, a one-step filtration reduces to the assumed graded isomorphism, and filtration endpoints are covered by finiteness. AC is used exactly through [A1] in the cohomological Serre suppliers.