On a sharp lower bound for the Tjurina number of zero-dimensional complete intersections

As is known, for isolated hypersurface singularities and complete intersections of positive dimension, the Milnor number is the least upper bound for the Tjurina number, i.e., $\tau \leqslant \mu$. In this paper we show that, for zero-dimensional complete intersections, the reverse inequality holds. The proof is based on properties of faithful modules over an Artinian local ring. We also exploit simple properties of the annihilator and the socle of the modules of Kähler differentials and derivations and the theory of duality in the cotangent complex of zero-dimensional singularities.