# The invariance of Hochschild and cyclic homology under row extensions

 Dia 2018-10-08 13:30:00-03:00 Hora 2018-10-08 13:30:00-03:00 Lugar Salón de seminarios del piso 14, CMAT

### The invariance of Hochschild and cyclic homology under row extensions

#### Piotr Hajac(Instituto de Matemática de la Academia Polaca de Ciencias)

Goodwillie’s theorem states that the periodic cyclic homology is invariant under nilpotent extensions. We introduce a special type of nilpotent extensions of unital algebras (called row extensions) for which we prove a stronger result: the invariance of Hochschild and cyclic homology. The row extensions appear in abundance. They are always H-unital but generically non-unital and noncommutative. A very specific type of a row extension appears naturally in the construction of the Chern-Galois character. If P is an algebra with a principal coaction, and B is its coaction-invariant subalgebra, then the Chern-Galois character factors through the row extension of B by the nilpotent ideal consisting of the invariant universal differential one-forms on P. When P is a principal comodule algebra, one can identify this ideal with the kernel of the multiplication map restricted to the algebra of the associated Ehresmann-Schauenburg quantum groupoid. Based on joint work with Tomasz Maszczyk.