On certain invariants of commutative Artinian algebras

This paper is devoted to the study of the relationship between the fundamental algebraic, analytic, and topological invariants of Artinian local algebras. Among other things, it is shown that the length of the modules of derivations and Kähler differentials of every local Artinian Gorenstein algebra is not greater than the length of the algebra reduced by one. The proof relies on the theory of duality in the cotangent complex of analytic algebras, the basic properties of faithful modules over Artinian rings, and the structure of annihilators and socles of the modules of derivations and Kähler differentials. As a result, it turns out that the Tyurina number of a smoothable zero-dimensional Gorenstein singularity cannot be less than its Milnor number, so the inequality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau greater-than-or-equal-to mu"> <mml:semantics> <mml:mrow> <mml:mi> τ </mml:mi> <mml:mo> ≥ </mml:mo> <mml:mi> μ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\tau \geq \mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> holds.