Minimal graded free resolutions for monomial curves in 𝔸4 defined by almost arithmetic sequences. Issue 2 (1st February 2017) - Kumar Roy, Achintya; Sengupta, Indranath; Tripathi, Gaurab
ABSTRACT: Letm = ( m 0, m 1, m 2, n ) be an almost arithmetic sequence, i.e., a sequence of positive integers with gcd( m 0, m 1, m 2, n ) = 1, such that m 0 < m 1 < m 2 form an arithmetic progression, n is arbitrary and they minimally generate the numerical semigroup Γ = m 0 ℕ + m 1 ℕ + m 2 ℕ + n ℕ. Let k be a field. The homogeneous coordinate ring k [Γ] of the affine monomial curve parametrically defined by X 0 = t m 0, X 1 = t m 1, X 2 = t m 2, Y = t n is a graded R -module, where R is the polynomial ring k [ X 0, X 1, X 2, Y ] with the grading degX i : = m i, degY : = n . In this paper, we construct a minimal graded free resolution for k [Γ].