# 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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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 motivation 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).

In late 1960s W. Lawvere, with help from Tierney, extended the Grothendieck topoi to the theory of elementary topoi. This was not the only contribution of Lawvere in the 1960s. Lawvere promoted also the duality between spaces and dual objects which he calls quantity (cf. space and quantity). While Lawvere's impact has been deep, I object to the terminology: in physics a quantity is normally a single observable; physicist do not consider the algebra of all observables a quantity, but rather a field of quantities, or algebra of quantities. But never mind the terminology, Lawvere went on very deeply in presenting this point of view, which is really generalized noncommutative geometry. Of course, neither Grothendieck nor Lawvere did not pay that particular attention to reconstructing the differential geometry and measure theory from the study of operator algebras, what is the huge contribution of Connes, or from the study of noncommutative rings, which was implicit in Gabriel 1961 and more explicit with works of J. S. Golan, van Oystaeyen (and P. M. Cohn with his affine spectrum) and others in mid 1970s, working with spectra of noncommutative rings and noncommutative localization theory as a noncommutative analogue of Zariski topology. One should mention that sporadic appearance of operator algebras from the noncommutative geometry point of view is present to some extent in 1970s book of Semadeni on Banach spaces of continuous functions (MR296671), where he studies, among other topics, the noncommutative analogues of many topological properties of topological spaces; in less explicit form there are also works of Irving Segal which had a similar motivation.

Grothendieck says in his memoirs that the concept of abelian category as he promoted it in Tohoku is part of the same philosophy -- abelian categories, possibly with additional axioms like AB5 are sort of categories of sheaves of modules, and should be viewed as an idea which is sort of abelian/stable version of Grothendieck topoi. More precisely, in this line, there is a recent Nikolai Durov's concept of a vectoid. Pierre Gabriel, who was close to Grothendieck's school in his early days, had in his prophetic work of 1961 reconstruction theorem for schemes and study of subcategories and localizations in abelian categories which represent open or closed subschemes and so on. Gabriel's work is in fact the first big work in noncommutative algebraic geometry and his reconstruction theorem is really the basic motivation in algebraic flavour of the theorem. In a sense, Gabriel's work is an abelian version of some Grothendieck's basic ideas of topos theory (cf. noncommutative scheme for one of the modern ideas along that line of thought) and Grothendieck was well aware of the abelian direction of this thinking from the Tohoku times.

For a general vista, I recommend

• Pierre Cartier, A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry, Bull. Amer. Math. Soc. 38 (2001), 389-408, pdf.
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