$\tau$-Tilting Theory and $\tau$-Slices

Comparing the module categories of an algebra and of the endomorphism algebra of a given support $\tau$-tilting module, we give a generalization of the Brenner-Butler's tilting theorem in the framework of $\tau$-tilting theory. Afterwards we define $\tau$-slices and prove that complete slices of tilted algebras and local slices of cluster tilted algebras are examples of complete $\tau$-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 $\tau$-slices.