EXAMPLES OF INFINITELY GENERATED KOSZUL ALGEBRAS

Let K be a skew field and A = K + A(1) +... a graded K - algebra (both of them not necessarily commutative). We call A homogeneous (or stand ard) if it is generated by Al as a K-algebra. A homogeneous K-algebra A is Koszul if there exists a linear free resolution [GRAPHICS] of the residue field K congruent to A/A(+) as an A-module. Here partial deri vative(o) : A --> K is the natural augmentation, the F-i's are conside red graded left free A - modules whose basis elements have degree 0, a nd that the resolution is linear means the boundary maps partial deriv ative(n), n greater than or equal to 1, are graded of degree 1 (unless partial derivative(n) = 0). The examples we will discuss in Section 1 are variants of the polytopal semigroup rings considered in BRUNS, GU BELADZE, and TRUNG [4]; in Section 1 the base field K is always suppos ed to be commutative. For the first class of examples we replace the f inite set of lattice points in a bounded polytope P subset of R-n by t he intersection of P with a c-divisible subgroup of R-n (for example R -n itself or Q(n)). It turns out that the corresponding semigroup ring s K[S] are Koszul, and this follows from the fact that K[S] can be wri tten as the direct limit of suitably re-embedded ''high'' Veronese sub rings of finitely generated subalgebras. The latter are Koszul accordi ng to a theorem of EISENBUD, REEVES, and TOTARO [5]. To obtain the sec ond class of examples we replace the polytope C by a cone with vertex in the origin. Then the intersection C boolean AND U yields a Koszul s emigroup ring R for every subgroup U of R-n In fact, R has the form K + X Lambda[X] for some K-algebra Lambda, and it turns out that K + X L ambda[X] is always Koszul (with respect to the grading by the powers o f X). Again we will use the ''Veronese trick''. In Section 2 we treat the construction K + X Lambda[X] for arbitrary skew fields K and assoc iative K-algebras Lambda. (See ANDERSON, ANDERSON, and ZAFRULLAH [1] a nd ANDERSON and RYCKEART [2] for the investigation of K + X Lambda[X] under a different aspect.) For them an explicit free resolution of the residue class field is constructed. This construction is of interest also when K and Lambda are commutative, and may have further applicati ons.