On the non-existence of torus actions

In this paper we compute the higher left derived functors of the indecomposable functor, in certain degrees, for a general class of algebras. The techniques do not depend on the existence of a Projective extension sequence—a common method used to make such computations throughout the literature and as a result, we generalize all known computations of these higher derived functors. As a result of these calculations we have the following applications. First, we prove that certain torus actions on a Quasitoric manifold are restricted due to the combinatorial structure of the orbit implying the non-existence of certain torus actions. Second, we obtain a refinement of the class of equivariantly formal torus actions, splitting them into two classes. In the context of augmented simplicial algebras over a field, the results provide explicit computations of the cotangent complex. We also show very explicitly how one higher derived functor depends on another and in the case of Toric Topology, the generators of these derived functors are linked backed to the combinatorics of the orbit.