Here is a description of an ideal that plays an important part in the construction of projective Reed-Muller codes. The use of Eagon-Northcott complex which is a generalisation of the Koszul complex gives us a method to compute dimensions of projective Reed-Muller codes. Moreover a calculus of dimensions gives us a combinatoric identity. This communication is issued from a paper admitted in the Journal of Pure and Applied Algebra and we have adjoined a straightforward and subtle proof of the combinatoric identity given by Michel Quercia.