Fiber Coefficients And Depth Of Fiber Cones

Let (R, m) be a Cohen-Macaulay local ring of dimension d＞0 having infinite residue field, I an m-primary ideal of R and K an ideal containing I. The fiber cone of I with respect to K is the graded algebra F_K(I)=(?)Iⁿ/KIⁿ. The graded algebra F_K(I) for K = m iscalled the fiber cone of I. When K = I, F_K(I)=G(I) is the associated graded ring of I.In this thesis, we present some bounds on the first and the second fiber coefficients of fiber cones under the assumptions that depth G(I)≥d-1 and r_L(I|K)＜∞, we also give some conditions which will force depth F_K(I)≥d-2. When depth G(I)≥d-1 and depth F_K(I)≥d-2, we study the properties of F_K(I)The main results of this thesis are following:Theorem 1 Let a₁,…,a_d-1∈I,a_d∈K be a Rees-superficial sequence for I and K such that a₁^*,…,a_d-1^*is a G(I)-regular sequence. Let r_L(I|K)＜∞, where L = (a₁,…,a_d), J = (a₁,…,a_d-1). Then(1)f₁(I,K)≤(?). If the equality occurs, then depth F_K(I)≥d-2;(2)f₁(I,K)≥(?). If the equality occurs, then depth F_K(I)≥d-1.Theorem 2 Let a₁,…,a_d-1∈I,a_d∈K be a Rees-superficial sequence for 7 and K such that a₁^*,…,a_d-1^* is a G(I)-regular sequence. Let A₀=K,r_L(I|K)＜∞, where L=(a₁,…,a_d),J=(a₁,…,a_d-1). Then(1)f₂(I,K)≤(?). The equality holds if and only if depthF_K(I)≥d-2;(2) f₂(I,K)≥(?)(n-1)λ(KIⁿ+L/L)+λ(R/K). If d≥3,K=(?)[(KI^k+(a₁,…,a_d-3):J^k), KI+(a₁,…,a_d-3)=(?)[(KI^k+1+(a₁,…,a_d-3)):J^k] and the above equalityholds, then depth F_K(I)≥d-1.Theorem 3 Let a₁,…,a_d-1∈I,a_d∈K be a Rees-superficial sequence for I and K. Let k be a positive integer such that KIⁿ∩L=JKI^n-1+a_dIⁿ for n≤k-1, and (?). Let depth G(I)≥d-1 and r:=r_L(I|K) Then(1) depth F_K(I)≥d-2; (2) If r＜∞, then(3) If r=∞, thenTheorem 4 Suppose that d≥2, J is a minimal reduction of I. Let k be a positive integer such that such that KIⁿ∩J=JKI^n-1 for n≤k-1 and (?). Let r=r_J^K(I).(1) If depth G(I)≥d-2, then depth F_K(I)≥d-1.(2) If depth G(I)≥d-1, thenTheorem 5 Let a₁,…,a_d-1∈I,a_d∈K be a Rees-superficial sequence for I and K such that a₁^*,…,a_d-1^* is a G(I)-regular sequence and a₁⁰,…,a_d-2⁰ is an F_K(I)-regular sequence. Let J=(a₁,…,a_d-1),L=(a₁…,a_d). Then(1) For all n≥1, the length (?) does not depend on L;(2) r_L(I|K) is independent of L;(3) r(I|K)=n_K(I)+d.Theorem 6 Let a₁,…,a_d-1∈I,a_d∈K be a Rees-superficial sequence for I and K. ThenIf depth G(I)≥d-1 and the equality occurs, then depth F_K(I)≥d-1.