Depth And Stanley Depth Of The Quotient Rings Of Some Classes Of Monomial Ideals

This thesis studies six classes of squarefree monomial ideals.By the one-to-one correspondence between the squarefree monomial ideals and the simplicial complexes,we use some combinatorial properties of simplicial complexes to study these ideals,and compute the depth and Stanley depth of the quotient rings of them.In Chapter 3,let S1 = K[x1,...,xn]be a polynomial ring over a field K and In,d =(x1x2 …xd,xd-k+1xd-k+2 …x2d-k,...,xn-d+1xn+d+2 …xn).When d≥ 2k-1,we prove that sdepth(In,d)≥ depth(In,d)and compute the depth and Stanley depth of S1/In,st for all t>1.When d =2k,we also prove that sdepth(In,d)≥ depth(In,d)and give lower bounds for the depth and Stanley depth of S1/In,dt for all t ≥ 1.In Chapter 4,let S2 =K[xi,...,xn]be a polynomial ring over K and Jn,d =(x1 …xd,xd-k+1… x2d-k,...,xn-2d+2k+1…xn-d+2k,xn-d+k+1…xnx1…xk).When d≥2k + 1,we prove that sdepth(Jn,d)>depth(Jn,d)and give the precise formulas to compute the depth and Stanley depth of S2/In,d.When d = 2k,we prove that sdepth(Jn,d)≥ depth(Jn,d).We also discuss the the depth and Stanley depth of S2/Jn,d and compute the depth and Stanley depth of Jn,d/In,d.In Chapter 5,let S3 = K[x1,...,xl,xl+1,...,xn1,1,...,xl+1,...,xns,s]be a poly-nomial ring over K and Il,d＝(?)(x1…xlxl-l,i…xd,ixd-k+1,ixd-k+2,i…x2d-k,j,...,xni-d+1,ixni-d-2.i…xni,i).When d ≥ 2k + I and l ≤ d-k-1,we prove that sdepth(Il,d)≥ depth(Il,d)and give the precise formulas to compute the depth and Stanley depth of S3/Il,d t for all t ≥ 1.When d = 2k = 2l,we compute the depth of S3/Il,d,and give an upper bound and a lower bound for the Stanley depth of S3/Il,d.In Chapter 6,let S4 =K[x1,...,xk,xk+1,1,...,xn1,1,...,xk+1,s,...,xns,s]be a polynomial ring over K and Jk,d =(?)(x1 …xkxk|1,i…xd,ixd k|1,i…x2d k,i,...,xni-2d+2k+1,i …xni-d+2k,i,xni-+k+1,i…xni,ix1…xk).When d ≥ 2k+1,we prove that sdepth(Jk,d)≥ depth(Jk,d)and give the precise formulas to compute the depth and Stanley depth of S4/Jk,d.When d = 2k,we compute the depth of S4/Jk,d,and give an upper bound and a lower bound for the Stanley depth of S4/Jk,d.In Chapter 7,we denote αj:=(?)(di-ki),βj:=(?)(di-ki)+ dj,2 ≤ j ≤ r and let α1 = 0,β1 = d1.Let S5 = K[x1,...,xβr]be a polynomial ring over K and Ir =(xα1+1 … xβ1,xα2+1…xβ2,...,xαr+1…xβr).We prove that sdepth(Ir)≥depth(Ir)and compute the depth and Stanley depth of S5/Irt for all t ≥ 1.In Chapter 8,let S6 = K[x1,...,xl,xl|1,1,...,xn1,1,...,xl|1,s,...,xns,s,y1,...,yl,yl|1,1,...,ym1,1,...,yl+1,r,...,ymr,r,zl-1,...,zq],and Is,r =(?)(x1…xlxl+1,i…xd,i,xd-k+1,i…x2d-k,i,...,xni-d+1,i …xni,i)+(?)(y1 …ylyl+1,i…yd,i,yd-k-1,i … y2d-k,i,...,ymi-d+1,i …ymi,i)+(x1 …xlzl+1…zd,zd-k+1 …z2d-k,...,zq-l-2d+k+1 …zq+l-d+k,zq+l-d+1…zqy1…yl).We prove that sdepth(Is,r)≥ depth(Ir)and give the precise formulas to compute the depth and Stanley depth of S6/Is,r t for all t ≥ 1.