On simple A-multigraded minimal resolutions

Let A be a semigroup whose only invertible element is 0. For an A-homogeneous ideal we discuss the notions of simple i-syzygies and simple minimal free resolutions of R/I. When I is a lattice ideal, the simple 0-syzygies of R/I are the binomials in I. We show that for an appropriate choice of bases every A-homogeneous minimal free resolution of R/I is simple. We introduce the gcd-complex ∆gcd(b) for a degree b ∈ A. We show that the homology of ∆gcd(b) determines the i-Betti numbers of degree b. We discuss the notion of an indispensable complex of R/I. We show that the Koszul complex of a complete intersection lattice ideal I is the indispensable resolution of R/I when the A-degrees of the elements of the generating R-sequence are incomparable.