τ-Tilting theory and τ-slices

Comparing the module categories of an algebra and of the endomorphism algebra of a given support τ-tilting module, we give a generalization of the Brenner-Butler's tilting theorem in the framework of τ-tilting theory. Afterwards we define τ-slices and prove that complete slices of tilted algebras and local slices of cluster tilted algebras are examples of complete τ-slices. Then we apply this concept to the study of simply connected tilted algebras. Finally, we study the one-point extensions and the split-by-nilpotent extensions of an algebra with τ-slices.