Algebraic Shifting and Graded Betti Numbers

Let S = K[x1, . . . , xn] denote the polynomial ring in n variables over a field K with each deg xi = 1. Let ∆ be a simplicial complex on [n] = {1, . . . , n} and I∆ ⊂ S its Stanley–Reisner ideal. We write ∆e for the exterior algebraic shifted complex of ∆ and ∆c for a combinatorial shifted complex of ∆. Let βii+j(I∆) = dimK Tori(K, I∆)i+j denote the graded Betti numbers of I∆. In the present paper it will be proved that (i) βii+j(I∆e) ≤ βii+j(I∆c) for all i and j, where the base field is infinite, and (ii) βii+j(I∆) ≤ βii+j(I∆c) for all i and j, where the base field is arbitrary. Thus in particular one has βii+j(I∆) ≤ βii+j(I∆lex ) for all i and j, where ∆lex is the unique lexsegment simplicial complex with the same f -vector as ∆ and where the base field is arbitrary.