Depth Of Quotient Rings Of Several Kinds Of Monomial Ideals’ Square

Let K be a field and R denote a polynomial ring over K with xi,x2,…,xn,as variables.In this paper,the depth lemma,the depth of quotient rings of path’s edge ideal and circle’s edge ideal,and other previous research results are used to discuss the depth of quotient rings of the edge ideals’square of the path,circle,wheel and other simple graph.We get the following results on R.In the second chapter,two important conclusions are obtained by calculating the depth of quotient rings of the edge ideals’ square of two special trees(including the edge ideals of paths).Firstly,if I2=(x1xt+1,…,xt-xt+1,xtxt+1,xt+1xt+2,…,xn-1xn)is a:monomial ideal of ring R,where t is a positive integer,then when n=t+1,t+2,it holds that depth(R/I22)=1;when n=t+3,depth(R/I22)≥1;when n≥t+4,depth(R/I22).[n-t/3].Secondly,let I=(x1xt+1,…,xtzt+1,xt+1xt+2,xt+2xt+3,…,xn-t-1xn-t,xn-txn-t+1,…,xn-txn)is a monomial ideal of ring R,where t>2 is a positive integer and n>2t+5.If n-2t≡1,0(mod 3),then depth(R/I2)=[n-2t+1/3];if n-2t ≡ 2(mod 3),then depth(R/I2)≥[n-2t+1/3].In the third chapter,we mainly give the depth of quotient rings of edge ideals’square of the circle and circle with whisker.If J3=(x1x2,x2x3,…,xn-1xn,xnx1)is a monomial ideal of the ring R,then we prove depth(R/J32)=[n-1/3]when n≥4.Suppose P=K[x1,…,xn,y1]is a polynomial ring,and J4=(x1x2,…,xn-1xn,xnx1,x1y1)is a monomial ideal of the ring P.If n≥4,then depth(P/J42)=[n-1/3].In the fourth chapter,we mainly calculate the depth of quotient ring of edge ideal’s square of wheel.Let K1=(x1x2,…,xn-2xn-1,xn-1x1)+(x1xn,x2xn,…,xn-2xn,xn-1xn)is a monomial ideal of the ring R,and we can prove depth(R/K12)=0 and depth(R/K13)=0.