Normal Polytopes, Triangulations, and Koszul Algebras

This paper is devoted to the algebraic and combinatorial properties of polytopal semigroup rings defined as follows. Let P be a lattice polytope in R n , i. e. a poly-tope whose vertices have integral coordinates, and K a field. Then one considers the embedding ι : R n → R n+1 , ι(x) = (x, 1), and defines S P to be the semigroup generated by the lattice points in ι(P); the K-algebra K[S P ] is called a polytopal semigroup ring. Such a ring can be characterized as an affine semigroup ring that is generated by its degree 1 elements and coincides with its normalization in degree 1. The first question to be asked about K[S P ] is whether it is normal, and a geometric or combinatorial characterization of normality is certainly the most important problem in the theory of polytopal semigroup rings. (By a theorem of Hochster [18], the normality of K[S P ] implies the Cohen–Macaulay property.) However, it is by no means clear whether such a characterization exists. The best known upper approximation to normality is the existence of a unimodular lattice covering (that is, a covering by lattice simplices of normalized volume 1). In Section 1 we show that the homothetic images cP of an arbitrary lattice polytope have such a covering for c 0. The existence of a unimodular covering is derived from a unimodular triangulation of the unit n-cube. The second ring-theoretic question we are interested in is the Koszul property: a graded K-algebra R is called Koszul if K has a linear free resolution as an R-module. (The resolution is linear if all the entries of its matrices are forms of degree 1; see Backelin and Fröberg [3] for a discussion of the basic properties of Koszul algebras.) It is immediate that a Koszul algebra is generated by its degree 1 component and is defined by degree 2 relations. (Though these properties do in general not imply that R is Koszul, no counterexample seems to be known among the semigroup rings.) A sufficient condition for the Koszul property is the existence of a Gröbner basis of degree 2 elements for the defining ideal of R (for example, see [9]). An algebraic approach to the multiples cP yields that K[S cP ] is normal for c ≥ dim P − 1, a Koszul algebra for c ≥ dim P , and …