 Let formula_71 be homogeneous polynomials in formula_72, of respective degrees formula_73 Setting formula_74 one has the following exact sequences 

The additivity of Hilbert series implies thus 
A simple recursion gives 

This shows that the complete intersection defined by a regular sequence of polynomials has a codimension of , and that its degree is the product of the degrees of the polynomials in the sequence.Every graded module over a graded regular ring has a graded free resolution, meaning there exists an exact sequence
where the formula_79 are graded free modules, and the arrows are graded linear maps of degree zero.The additivity of Hilbert series implies that

If formula_81 is a polynomial ring, and if one knows the degrees of the basis elements of the formula_82 then the formulas of the preceding sections allow deducing formula_83 from formula_84 In fact, these formulas imply that, if a graded free module has a basis of homogeneous elements of degrees formula_85 then its Hilbert series is 

These formulas may be viewed as a way for computing Hilbert series.