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added to CartSp a section that lists lots of notions of (generalized) geometry modeled on this category.
You were asking here about some of your sharp observations about infinitesimal thickenings. I have tried to help with references to something what sounds very similar in noncommutative geometry. Did you make some comparison ?
I looked at it, but I don’t yet have a good understanding of the standard theory of differential bimodules that is being generalized there.
What’s the motivating example here?
The motivating example is the standard theory of regular differential operators (on complex analytic manifols, on rigid analytic varieties or for algebraic schemes). for Gorthendieck the motivation was the crystalline cohomology. The reg. dif. op. is the case of the endomorphism object of a ring object (say of the structure sheaf). There is a duality between the infinitesimals and regular differential operators. D-modules are the modules over the sheaf of regular differential operators. It is amazing that the local role which is in the commutative case give to the Weyl algebra in some interesting case of modern geometry (related to integrable systems of Calogero-Moser type) is played by the preprojective algebras, which have similar generators and relations. But the modules for the preprojective algebras correspond to A-infinity modules (actually $A_2$-modules) for the Weyl algebra; both approaches are studied by Yuri Berest. So somehow the higher categorical interpretation and noncommutativization of the algebra of differential operators meet. But somehow the preprojective algebras are a different noncommutative extension than those in work of Lunts and Rosenberg. But both have the property of localization: the rings of differential operators are compatible with a good class of localizations. Flat localizations having right adjoint in the case of Lunts and Rosenberg and stably flat maps of algebras in the case of preprojective algebras. Godo comparison is not known.
So what’s the basic example of differential bimodule that i should be thinking of? If I am thinking of a space $Spec A$, is it the $A$-bimodule $A \otimes A$?
No, the differential part of End A, what is the ring Diff A of regular differential operators on A, which is just a small part of all the Z-linear endomorphisms of A. For Spec of affine n-space it is the Weyl algebra.
in particular the A itself sits as 0-th order differential operators on A. You have filtration by the degree, a differential bimodule is the one which is equal to the union of its own differential filtration, called the differential part.
Ah, that helps! Thanks. Ah, now it’s obvious. The filtration is by “order of the differential opetor”. The condition says that the commutator of a differential operator of order $n$ with a multiplication operator is a differential operator of order $(n-1)$.
I’ll create differential bimodule in a moment, so that I won’t forget…
Oh, silly me, it already exists, of course. :-) But I add the standard example now.
The condition says that the commutator of a differential operator of order $n$ with a multiplication operator is a differential operator of order $(n-1)$.
Hard to believe that nobody noticed this as a characterising property to define the differential operators until Grothendieck entered the area.
Surely it was noticed (in a special case) in quantum mechanics? $[x,\frac{d}{dx}] = i\hbar$ :P
No, it was not. It is not only that the commutator is lower order differential operators but more importantly that no other operator except for differential operator of order n has that property. Moreover, the fact is true for differential operators on any variety/manifold.
(tongue removed from cheek)
What other sort of operators are considered in that setup then? What’s an example?
You look at Hom(E,F), of k’linear map of sheaves, where E and F are two O-modules. For this bimodule you can look its differential part. If E and F is simply O we have End(O) – just all linear operators. Its differential part are regular differential operators, but you can also look at differential operators in similar way from one vector bundle to another vector bundle as usually PDE people look, and do the same kind of Gorthendieck styleO-linearization at sheaf theoretic level. This is explained at the beginning of the book of Ogus and Berthelot on crystalline cohomology.
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