Betti Numbers, Grobner Basis And Syzygies For Certain Affine Monomial Curves

Abstract

Let e > 3 and mo,... ,me_i be positive integers with gcd(m0,... ,me_i) = 1, which form an almost arithmetic sequence, i.e., some e - 1 of these form an arithmetic progression. We further assume that m0,... ,mc_1 generate F := Σ e-1 I=0 Nmi minimally. Note that any three integers and also any arithmetic progression form an almost arithmetic sequence. We assume that 0 < m0 < • • • < me-2 form an arithmetic progression and n := mc-i is arbitrary Put p := e - 2. Let K be a field and XQ) ... ,Xj>, Y,T be indeterminates. Let p denote the kernel of the if-algebra homomorphism η: K[XQ, ..., XV) Y) -* K^T], defined by r){Xi) = Tm\.. .η{Xp) = Tmp, η](Y) = Tn. Then, p is the defining ideal for the affine monomial curve C in A^, defined parametrically by Xo = Trr^)...)Xv = T^}Y = T*. Furthermore, p is a homogeneous ideal with respect to the gradation on K[X0)... ,XP,F], given by wt(Z0) = mo, • • •, wt(Xp) = mp, wt(Y) = n. Let 4 := K[XQ> ...,XP) Y)/p denote the coordinate ring of C. With the assumption ch(K) = 0, in Chapter 1 we have derived an explicit formula for μ(DerK(A)), the minimal number of generators for the A-module DerK(A), the derivation module of A. Furthermore, since type(A) = μ(DerK(A)) — 1 and the last Betti number of A is equal to type(A), we therefore obtain an explicit formula for the last Betti number of A as well A minimal set of binomial generatorsG for the ideal p had been explicitly constructed by PatiL In Chapter 2, we show that the set G is a Grobner basis with respect to grevlex monomial ordering on K[X0)..., Xp, Y]. As an application of this observation, in Chapter 3 we obtain an explicit minimal free resolution for affine monomial curves in A4K defined by four coprime positive integers mo,.. m3, which form a minimal arithmetic progression. (Please refer the pdf file forformulas)