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Noncommutative differential forms

Boris Tsygan

Freitag, 20. November 2020 14:15Uhr

Starting with a ring (possibly noncommutative), how can one develop calculus in such a way that, if we start with the ring of functions on an algebraic variety, we get the usual calculus of differential forms? We will do this from the very beginning and without requiring any prior knowledge. Namely, we will start with the basic construction of noncommutative differential forms and explain what has to be added to get a nontrivial theory. We will recover Hochschild and cyclic homology of rings, both in their original version and in the version of Ginzburg and Schedler. We will also show the connection with crystalline cohomology and its generalisation to noncommutative rings.