Deformations of commutative Artinian algebras

We study deformations of Artinian algebras and zero-dimensional germs of varieties. A new approach to solving the problem of the existence of rigid Artinian algebras is presented; it is based on the use of the canonical duality in the cotangent complex. In particular, we show that there are no rigid Artinian algebras that are Gorenstein or almost complete intersections. The proof of the second statement uses an original method of calculating the tensor product of the conormal and canonical modules in terms of the torsion functor. In addition, we obtain simple estimates on the dimensions of the spaces of the lower and upper cotangent functors of Artinian algebras and describe some relations between them. Besides, for almost complete intersections we compute the homology and cohomology groups of higher degrees, consider several examples of nonsmoothable Artinian almost complete intersections and discuss some unusual properties of such algebras.