Combinatorial Shifting And Graded Betti Numbers

Shifting operations were first considered by Erd(?)s,Ko and Rado,it is an important tool to study simplicial complexes. Let I C S be a squarefree monomial ideal,Ic is obtained from I under combinatorial shifting. When I S is a squarefree strongly stable ideal ,Ic = I.Therefore P and / have the same graded Betti numbers,projective dimension and regularity. In this paper,we study the relationship of the Betti numbers between Ic and I. In section 1,the concepts of combinatorial shifting and some related results are given. In section 2 we show that when J is a stable ideal,shiftij(J) is also a stable ideal, and when J is a stable ideal,we obtain G(shiftij(J)) from G(J) . In section 3,we show that when I is a squarefree stable ideal, shiftij(I)and I have the same graded Betti numbers,projective dimension and regularity, then Ic and I have the same graded Betti numbers,projective dimension and regularity. At last we apply the results we obtained to simplicial complexes.