Artinian algebras and differential forms

TL;DR

This paper proves a conjecture about the non-vanishing of a specific submodule of Kähler differentials in certain finite-dimensional graded algebras, advancing understanding in algebraic geometry and commutative algebra.

Contribution

It confirms the conjecture for Gorenstein graded and standard graded algebras, providing new results in the theory of Artinian algebras and differential forms.

Findings

Conjecture holds for Gorenstein graded algebras.

Conjecture holds for standard graded algebras.

Advances understanding of Kähler differentials in Artinian algebras.

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.