Let K be a skew field and A = K @ A1 @ . . . a graded Kalgebra (both of them not necessarily commutative). We call A homogeneous (or standard) if it is generated by Al as a Kalgebra. A homogeneous Kalgebra A is Koszul if there exists a linear free resolution
of the residue field K Y A/A+ as an A-module. Here a, : A + K is the natural augmentation, the Fi's are considered graded left free Amodules whose basis elements have degree 0, and that the resolution is linear means the boundary maps a, , n 2 1, are graded of degree 1 (unless a,, = 0).
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 field K is always supposed to be commutative. For the first class of examples we replace the finite set of lattice points in a bounded polytope P C IR" by the intersection of P with a c-divisible subgroup of IRn (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 finitely 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 n U yields a Koszul semigroup ring R for every subgroup U of IR".
In fact, R has the form K + X A [ X ] for some Kalgebra A, and it turns out that K + X A ( 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 h [ X ] for arbitrary skew fields K and associative K-algebras A.