# Ring isomorphism in Leray-Serre spectral sequence

The Leray-Hirsch theorem: let $$k$$ be a field. Given a fibration $$F \to E \to B$$ with $$F, B$$ path connected and suppose system of local coefficient is zero and the following condition satisfied (a) $$H^n(B;k)$$ is finitely dimensional for each $$n$$ and (b) $$i^*:H^*(E;k) \to H^*(F;k)$$ is onto, here $$i$$ is the inclusion of fiber. Then $$H^*(E;k) \cong H^*(B;k)\otimes_k H^*(F;k)$$ as vector spaces.

My question is: What are some extra condition do we need so that above becomes ring isomorphism. In particular for the field $$k=\Bbb Z/2 \Bbb Z$$ what extra condition do we need.

Thank you very much for your kind help.

• The conditions of Proposition 8 are that $H_i(F;k) \to H_i(E;k)$ is injective for all $i \geq 0$ (or equivalently that $H^i(E;k) \to H^i(F;k)$ is surjective) and that each $H^i(F;k)$ and $H^i(B;k)$ are finite-dimensional. The only thing you haven't already assumed is that $H^i(F;k)$ is finite-dimensional. Once you assume that, you get what you want. – user98602 Jan 20 at 0:56
• @MikeMiller I got a translation of the above mentioned J P Serre paper. Thanks. – Shivani Sengupta Jan 20 at 15:53