# Grothendieck and Non-commutative Geometry?

When Grothendieck and his followers were working on their profound progress of algebraic geometry, did they ever consider non-commutative rings? Is there anyway evidence that Grothendieck foresaw the developments that would later come in non-commutative geometry or quantum group theory?

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Soft question tag seems to apply here. –  Adam Hughes Nov 30 '10 at 18:07
Ok. Edited accordingly. –  Abtan Massini Nov 30 '10 at 18:11
A friend of mine postulated the following: in English language calculus course and books, exponential processes, population growth or radioactivity, are often introduced under the heading "Growth and Decay" problems, see for example  math.dartmouth.edu/~klbooksite/3.02/302.html  The suggestion was that this is the reason for the choice of letter in Grothendieck K-theory.  I'm here all week. Don't forget to tip your waiter or waitress.  –  Will Jagy Nov 30 '10 at 19:02
The bigger the groan, the better they are! –  Todd Trimble Nov 30 '10 at 19:15
Will, that's awesome! Keep 'em coming :) –  Philip Brooker Nov 30 '10 at 22:15

No and yes, depending on the level of understanding. The consideration of noncommutative rings telling about geometry is almost nonexistent in Grothendieck's published opus. One of the exceptions is that he considered cohomologies for the possibly noncommutative sheaves of $\mathcal{O}$-algebras for commutative $\mathcal{O}$ (the latter is used in Semiquantum geometry). On the other hand, Grothendieck has been pioneer on abandoning the points of spaces as primary objects and promoting the category of sheaves over the space as defining the space. This is the point of view of topos theory which he invented; he noticed that the topological properties do not depend on a site but only on the associated topos of sheaves, and proposed a topos as a natural generalization of a topological space. Manin took Grothendieck's advice that one should consider the topos of sheaves as replacing the space, together with Serre's theorem that the category of quasicoherent modules determines a projective variety, as a mogivation to his approach to noncommutative geometry and quantum groups. The modern view of noncommutative geometry is that it is about the presentation of space via the structures consisting of all possible objects of some kind living on a space (algebra of functions, some structures consisting of cocycles, like category of vector bundles, category of sheaves, higher category of higher stacks).