Artinian algebras and differential forms

Abstract

This article concerns commutative algebras over a field $k$ of characteristic zero which are finite dimensional as vectorspaces, and particularly those of such algebras which are graded. Here the term graded is applied to non-negatively graded algebras $A$ with $A_0$ reduced and finite dimensional. Thus the trivial grading $A=A_0$ is only allowed if $A$ is a product of finite field extensions of $k$. It has been conjectured (G. Corti\~nas, S. Geller, C. Weibel; The Artinian Berger Conjecture. Math. Zeitschrift {\bf 228} 3 (1998) 569-588) that for all finite dimensional algebras $A$ which are not principal ideal algebras (i.e. have at least one nonprincipal ideal), the following submodule of the K\"ahler differentials is nonzero: $$\bigcap{\ker(\Omega_A @>>>\Omega_B)}$$ Here the intersection is taken over all principal ideal algebras $B$ and all homomorphisms $A @>>>B$. In this paper we prove that the conjecture holds for both Gorenstein graded and standard graded algebras.