Hochschild cohomology of finite dimensional algebras is a fundamental homological invariant, used for instance to define and compute support varieties of modules or complexes. This needs a non-trivial assumption known as the (Fg) condition.

In this talk, we develop a general theory providing tools to compare Hochschild cohomology of two given algebras, validity of the (Fg) condition and support varieties theoretically in terms of connections between derived module categories, and (based on the theoretical comparisons) practically by a set of tools allowing to add or remove vertices or arrows or to glue algebras. In order to provide these tools, a new version of Hochschild cohomology, called small stable Hochschild cohomology, is introduced.

As applications, we reduce Hochschild cohomology, validity of the (Fg) condition and support varieties for quadratic monomial algebras, in particular gentle algebras, and for cluster tilted algebras to the well-known case of Nakayama algebras. This is a joint work with Steffen Koenig.