Examples of infinitely generated Koszul algebras

Abstract

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 standard) 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 derivative(o) : A --> K is the natural augmentation, the F-i's are considered graded left free A - modules whose basis elements have degree 0, and that the resolution is linear means the boundary maps partial derivative(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, 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 subset of R-n by the 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 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 boolean AND U yields a Koszul semigroup 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 Lambda[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 Lambda[X] for arbitrary skew fields K and associative K-algebras Lambda. (See ANDERSON, ANDERSON, and ZAFRULLAH [1] and 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 applications.