 Then a pure Hodge A-structure of weight "n" is defined as before, replacing formula_16 with A.There are natural functors of base change and restriction relating Hodge A-structures and B-structures for A a subring of B.It was noticed by Jean-Pierre Serre in the 1960s based on the Weil conjectures that even singular (possibly reducible) and non-complete algebraic varieties should admit 'virtual Betti numbers'.