# Bar versus Koszul

## Seminar/Forum

107
Peter Hall
In the first part of the talk I will introduce noncommutative differential forms following Cuntz and Quillen’s paper “Algebra extensions and nonsingularity”. This is one way of talking about some of the structures on the Bar complex $$B$$, which together with the Koszul complex $$K$$ gives the two standard resolutions of the diagonal over any ring. As we all know from elementary homological algebra, two projective resolutions of the same module must be homotopy equivalent, but although “everyone knows” the map $$K \rightarrow B$$ it seems the explicit formula for the inverse map $$B \rightarrow K$$ is less well-known. In the second part of the talk I will present some old work of Shepler-Witherspoon and myself with Carqueville which gives the formula in terms of divided difference operators.