Examples of Infinitely Generated Koszul Algebras

Abstract

of the residue field K ∼= A/A+ as an A-module. Here ∂0 : A → K is the natural augmentation, the Fi are considered graded left free A-modules whose basis elements have degree 0, and that the resolution is linear means the boundary maps ∂n, n ≥ 1, are graded of degree 1 (unless ∂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 ⊂ 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 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 ∩U yields a Koszul semigroup ring R for every subgroup U of R. In fact, R has the form K + XΛ[X] for some K-algebra Λ, and it turns out that K + XΛ[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 + XΛ[X] for arbitrary skew fields K and associative K-algebras Λ. (See Anderson, Anderson, and Zafrullah [1] and Anderson and Ryckeart [2] for the investigation of K + XΛ[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 Λ are commutative, and may have further applications.