Depth And Stanley Depth Of The Facet Ideals Of Certain Simplicial Complexes

In this paper,we mainly study one class of monomial ideal,and we compute the depth and Stanley depth of the quotient ring of it.Let K be a,field and S = K[X1,…,Xn]the polynomial ring over K in n variables.Let In,d ?(x1x(2,1)…x(d,1),x(d-k+1,1)x(d-k+2,1)…x(2d-k,1),…,x(n1-d+1,1)…x(n1,1),x1x(2,2)…x(d,2),x(d-k+1,2)x(d-k+2,2)…x(2d-k,2),…,x(n2-d+l,2)…x(n2,2),x1x(2,s')· · · x(d,s'),x(d-k+l,s')x(d-k+2,s')· · · x(2d-k,s'),…x(ns1-d+1,s')…x(ns',s')).is a squarefree monornial ideal in S,where n =?i=1 s' ni-s'+1.Then we discuss the depth and Stanley depth of the quotient ring of In,d in two cases.Case 1.Let d>2k + 1,we show that sdepth(In,d)= depth(In,d),and compute the depth and Stanley depth of S/In,d.Case 2.Let d = 2k.When K>1,we give a precise formula for the depth and Stanley depth of S/In,d.When k>1,t ?2,we also provide a better lower bound for the depth and Stanley depth of S/In,dt.