where formula_30 is the fundamental class of the fiber formula_31. By properties of the Gysin Sequence, formula_32 is a multiplicative homomorphism; by induction, formula_33 is generated by elements with formula_34, where formula_35 must be zero, and hence where formula_32 must be surjective. It follows that formula_32 must always be surjective: by the universal property of polynomial rings, a choice of preimage for each generator induces a multiplicative splitting. Hence, by exactness, formula_38 must always be injective. We therefore have short exact sequences split by a ring homomorphism