= Cohomological Künneth theorem over a field
{title2=$H^*(X\times F;k)\cong H^*(X;k)\otimes_kH^*(F;k)$}
If $k$ is a <field> and $H^*(F;k)$ has finite total dimension, the <cohomology cross product> gives the displayed graded-ring <isomorphism> for arbitrary spaces. Over a <field>, every chain complex splits into its <homology> with zero differential and contractible summands. The <Eilenberg–Zilber theorem> thus replaces the fiber chain complex by a bounded finite-dimensional complex; dualizing its tensor product with $C_*(X;k)$ gives a finite direct sum of shifted cochain complexes of $X$. Its cohomology is the displayed tensor product. Finiteness prevents the infinite-product obstruction that can occur for two infinite discrete spaces. The standard product formula and its finiteness hypothesis are discussed in https://pi.math.cornell.edu/~hatcher/AT/ATch3.pdf, Theorem 3.15.
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