Christophersen's problem for monomial algebras

Abstract

Christophersen's problem predicts that the connected component of the automorphism group of a finite-dimensional local algebra $A$ of dimension $\ell$ over an algebraically closed field of characteristic zero has dimension at least $\ell -1$, with equality if and only if $A$ is isomorphic to $\mathbf{k}[t]/(t^{\ell })$. We settle this problem in the class of monomial algebras. Using the combinatorial structure of the irredundant irreducible decomposition of a monomial ideal, we prove the predicted inequality and characterize the equality case. We further obtain stronger lower bounds depending on whether the defining monomial ideal is reducible or irreducible, and we determine all monomial algebras attaining each of these bounds.

13 pages, 1 figure