Betti Numbers for Certain Cohen–macaulay Tangent Cones

Abstract

We compute Betti numbers for a Cohen–Macaulay tangent cone of a monomial curve in the affine $4$-space corresponding to a pseudo-symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.

On the Betti numbers of the tangent cones for Gorenstein monomial curves

“…Lemma 3.1 [13] Assume that the multiplicity of the monomial curve is n i . Suppose that the k-algebra ho- This is a very effective result to reduce the number of cases for finding the Betti numbers of the tangent cones.…”

“…Since it is very difficult to obtain a description of the differential in the resolution, we can get some information about the numerical invariants of the resolution such as Betti numbers. The i-th Betti number of an R -module M , β i (M ) , is the rank of the free modules appearing in the minimal free resolution of M where been addressed for Cohen-Macaulay tangent cone of a monomial curve in A (k) corresponding to a pseudosymmetric numerical semigroup in [13]. In this paper, we solve the problem for Cohen-Macaulay tangent cone of a monomial curve in A 4 (k) corresponding to a non-complete intersection symmetric numerical semigroup.…”

Minimal free resolutions of the tangent cones for Gorenstein monomial curves

“…Lemma 3.1. [13] Assume that the multiplicity of the monomial curve is n i . Suppose that the k-algebra homomorphism π i : R → R = k[x 1 , .…”

Free resolutions for the tangent cones of some homogeneous pseudo symmetric monomial curves

“…Construction of an explicit minimal free resolution of a finitely generated algebra is a difficult problem in general. This problem has been studied by many mathematicans, in particular for the homogeneous coordinate ring of an affine monomial curve in [1,5,6,[11][12][13]15].…”

“…They computed the Betti numbers by explicitly computing the minimal graded free resolution. Pseudo symmetric semigroup case is studied in [15] by showing that being homogeneous and being homogeneous type are equivalent for 4 generated pseudo symmetric monomial curves with Cohen-Macaulay tangent cones by computing the Betti sequences for nonhomogeneous case. Though in the homogeneous case the Betti sequence is 130 N. S ¸AH İN already known as (1,5,6,2), an explicit computation of minimal graded free resolutions were not given.…”