Linear strand of edge ideals of comaximal graphs of commutative rings

Abstract

For an edge ideal I(G) of a simple graph G , we study the N -graded Betti numbers that appear in the linear strand of the minimal free resolution of I ( Γ ( Z n ) ) , where Γ ( Z n ) is the comaximal graph of the integral modulo ring Z n . We show that the extremal Betti number of I ( Γ ( Z n ) ) is ϕ ( n ) , where ϕ ( n ) is the Euler’s totient function and thereby we obtain a large class of edge ideals with even extremal Betti numbers. We find the regularity (Castelnuovo-Mumford) and the projective dimension of these ideals. Moreover, we exhibit explicit formulae that determine all the N -graded Betti numbers in the linear strand of the minimal free resolution of I ( Γ ( Z n ) ) for certain values of n .