Non-commutative geometry is developed as the differential calculus on modules over a ring

*A*when*A*is non-commutative. Note that, if*A=C(X)*is the commutative ring of smooth real functions on a manifold*X*, we are in the case of familiar differential geometry of vector bundles over*X*. If*A*is a graded commutative ring, one comes to supergeometry. However, one meets a difficulty if*A*is a non-commutative ring. In this case, higher order differential operators are ill defined. A problem lies in the fact that, in contrast with the case of a commutative or graded commutative ring, a multiplication p->ap fail to be a zero order differential operator if an element*a*of a ring*A*does not belong to its center.**References:**

G. Sardanashvily, Lectures on differentail geometry of modules and rings, arXiv: 0910.1515

I've always been curious, is there some relation between this approach using the commutative ring of smooth real functions on a manifold...and Morse theory?

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