Linear strand of edge ideals of comaximal graphs of commutative rings

For an edge ideal I(G) of a simple graph (Formula presented.) we study the (Formula presented.) -graded Betti numbers that appear in the linear strand of the minimal free resolution of (Formula presented.) where (Formula presented.) is the comaximal graph of the integral modulo ring (Formula presented.). We show that the extremal Betti number of (Formula presented.) is (Formula presented.) where (Formula presented.) 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 (Formula presented.) -graded Betti numbers in the linear strand of the minimal free resolution of (Formula presented.) for certain values of (Formula presented.).