On Certain Gluing of semigroup rings and indispensable resolution of semigroup rings

Abstract: In this paper, our aim is twofold: First, by using the technique of gluing semigroups, we give infinitely many families of a projective closure with the Cohen-Macaulay (Gorenstein) property. Also, we give an effective technique for constructing large families of one dimensional Gorenstein local rings associated to monomial curves, which supports Rossi question, saying that every Gorenstein local ring has a non-decreasing Hilbert function. In the second part, we study strong indispensable minimal free resolutions of semigroup rings, focusing on the operation of the join of affine semigroups, which provide class of examples supporting Charalambous and Thoma question on the class of lattice ideal which has a strong indispensable free resolution.