  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 $D_{gcd}(\bf b)$ for a degree
$\mathbf{b}\in \A$. We show that the homology of $D_{gcd}(\bf b)$ determines
the $i$-Betti numbers of degree $\bf b$. 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.

Commutative Algebra;Algebraic Geometry