Let denote the polynomial ring in variables over a field with each . Let be a simplicial complex on and its Stanley-Reisner ideal. We write for the exterior algebraic shifted complex of and for a combinatorial shifted complex of . Let denote the graded Betti numbers of . In the present paper it will be proved that (i) for all and , where the base field is infinite, and (ii) for all and , where the base field is arbitrary. Thus in particular one has for all and , where is the unique lexsegment simplicial complex with the same -vector as and where the base field is arbitrary.