Linear Diophantine Equations and Local Cohomology

What can be said about the set E ~ of solutions in nonnegative integers to a system of linear equations with integer coefficients? For many purposes, such as those of linear programming, this question has been adequately answered. However, when this question is regarded from the vantage point of commutative algebra, many additional aspects arise. In particular, there is a natural way to associate with E ~ a graded module A ~ (over an appropriate graded commutative ring A), and one can ask for such standard information about A s as its depth, canonical module, etc. We will obtain such information by explicitly computing the Hilbert function of the local cohomology modules Hi(A ~) associated with A s (with respect to the irrelevant ideal of A) in terms of the reduced homology groups of certain polyhedral complexes associated with EL This method was suggested by some work of M. Hochster concerning polynomial rings modulo ideals generated by square-free monomials (unpublished by him but discussed in [Sts]), and I am grateful to him for making his ideas available to me. Similar techniques were employed by Goto and Watanabe [G-W] to study arbitrary affine semigroup rings, though they did not consider modules over them. As a consequence of our computations regarding Hi(A~), we can give a general "reciprocity theorem" (Theorem 4.2), whose statement does not involve commutative algebra, connecting the set E ~ of nonnegative integral solutions to the set of solutions in negative integers. This generalizes the results in [St2], where only a special class of equations was considered. It seems natural to find a purely combinatorial analogue to the algebraic results mentioned above. In Sect. 5 we discuss what we have accomplished along these lines, and offer a general ring-theoretic conjecture which would imply a much more definitive result. The following notation concerning sets will be used throughout.