Examples of Infinitely Generated Koszul Algebras

The examples we will discuss in Section 1 are variants of the polytopal semigroup rings considered in Bruns, Gubeladze, and Trung [4]; in Section 1 the base eld K is always supposed to be commutative. For the rst class of examples we replace the nite set of lattice points in a bounded polytope P R, by the intersection of P with a c-divisible subgroup of R, (for example R, itself or Q,). It turns out that the corresponding semigroup rings K[S] are Koszul, and this follows from the fact that K[S] can be written as the direct limit of suitably re-embedded `high'Veronese subrings of nitely generated subalgebras. The latter are Koszul according to a theorem of Eisenbud, Reeves, and Totaro [5]. To obtain the second class of examples we replace the polytope C by a cone with vertex in the origin. Then the intersection C \ U yields a Koszul semigroup ring R for every subgroup U of R,. In fact, R has the form K + X [X] for some K-algebra, and it turns out that K + X [X] is always Koszul (with respect to the grading by the powers of X). Again we will use the `Veronese trick'. In Section 2 we treat the construction K + X [X] for arbitrary skew elds K and associative K-algebras. (See Anderson, Anderson, and Zafrullah [1] and Anderson and Ryckeart [2] for the investigation of K + X [X] under a di erent aspect. ) For them an explicit free resolution of the residue class eld is constructed. This construction is of interest also when K and are commutative, and may have further applications.