Obstruction Calculus for Functors of Artin Rings, I

In this paper we define and study obstruction theories for morphisms of functors of Artin rings. We prove the existence of a universal obstruction theory, and we give explicit criteria for completeness and for linearity. As applications, we extend several results in the literature, removing the finite-dimensionality of the tangent space and the existence of a vector space of obstructions from the assumptions.