On certain invariants of commutative Artinian algebras

The paper studies the relationships between the basic algebraic, topological and analytical invariants of Artinian algebras. Thus, among other things, we show that the length of the modules of derivations and Kahler differentials of every local Gorenstein algebra does not exceed the length of the Artinian algebra itself minus one. The proof is based on the theory of duality in the cotangent complex of analytic algebras, on the properties of faithful modules over an Artinian ring, and on a description of the structure of the annihilator and socle of modules of derivations and Kahler differentials of an Artinian algebra. In particular, it follows that the Tjurina number of every zero-dimensional Gorenstein singularity is not less than its Milnor number, i.e. the inequality $\tau \geqslant \mu$ holds.