Local basis criterion for Leray-Hirsch classes

On a trivialized fiber bundle over $U$, assume the fiber has finite-dimensional total cohomology over a field. The Künneth theorem identifies the total-space cohomology with a finite free graded module over $H^{\ast }(U)$. Classes restricting to a basis on every fiber give an invertible change of generators: its degree-zero coefficient blocks are invertible on each path component, while its positive-base-degree terms strictly decrease fiber degree and are therefore nilpotent. Inverting the diagonal blocks and then using a finite geometric series proves that multiplication by the chosen classes is an isomorphism over $U$.