# $D_X$ algebras, $D_X$ schemes, connections

I have a basic question about the definition of $D_X$-algebras and schemes, which are defined in [BD2], Chiral algebras. I have some understanding of connections on a vector bundle, but I am not sure about connections in different contexts. Here $X$ is a scheme, of course.

Question 1: [BD2] defines a $D_X$ algebra as a commutative unital $O_X$ algebra equipped with an integrable connection along $X$. What precisely does an "integrable connection along X mean in this context?

Question 2: [BD2] defines a $D_X$ scheme as a X-scheme equipped with an integrable connection along $X$. Again, what does "integrable connection along X" mean in this context?

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In characteristic $0$, an integrable connection on an $\mathcal{O}_X$-module $M$ can be interpreted as infinitesimal descent data: for any two maps $f_1,f_2:Y\to X$ whose locus of agreement is defined by a nilpotent ideal, there is a canonical isomorphism $f_1^*M\rightarrow f_2^*M$. This interpretation of course makes sense for any kind of object over $X$, not just $\mathcal{O}_X$-modules. –  Keerthi Madapusi Pera May 22 '11 at 7:53
What is [BD2] refering to? –  J.C. Ottem May 22 '11 at 8:54
I supposed Beilinson Drinfeld Chiral Algebras ams.org/bookstore-getitem/item=COLL-51 –  Giuseppe Tortorella May 22 '11 at 10:12
I think these notes by Jacob Lurie are relavant: math.harvard.edu/~gaitsgde/grad%5F2009/SeminarNotes/… –  Lars May 23 '11 at 12:08

For question 1 it means that derivations of $O_X$ act on the algebra $A$ as derivations: $D(ab)=D(a)b+aD(b)$ while integrability means $[D_1,D_2]a=D_1(D_2 a)-D_2(D_1a)$, these conditions allow you to extend the action from derivations to a left action of the whole algebra $D_x$ of differential operators. This is the local case. For 2 I guess you patch these conditions to get a global version.

If you prefer a geometric picture: you have a fiber bundle $Y\to X$ where $Y$ is supplied with an involutive distribution whose planes project isomorphically to the tangent planes of $X$.

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The definitions in question are given in section 2.3.1, on page 80 of Beilinson and Drinfeld's Chiral algebras. The authors assume $X$ is smooth over the spectrum of a field $k$ of characteristic zero, but only bother to say so back on page 66 (and only mention $k$ on page 53). Their use of integrable connections along $X$ is a reference to Grothendieck's crystalline interpretation of connections, and they wrote their own sketchy introduction to this subject in section 7.10 of their unfinished book Quantization of Hitchin's Integrable System and Hecke Eigensheaves. The interpretation there is that schemes and commutative algebras of quasicoherent sheaves form fibered categories over the crystalline site $X_{cr}$, and the corresponding $D_X$-objects are Cartesian sections. This is just another way to put the infinitesimal descent data into a nice-looking package.
Just a remark: I think the classical reference for $\mathcal{D}$-schemes (from the old-school, rather than crystalline, point of view) is Beilinson-Bernstein "Proof of Jantzen conjectures," Section 1. At least, it's slightly less terse than "Chiral Algebras" is. –  Thomas Nevins May 23 '11 at 16:55
@Thomas: The term "$D$-scheme" introduced in that paper (section 1.4.2) has a different meaning from $D_X$-scheme discussed here. There, it is a scheme $X$ equipped with a $D$-algebra $A$ on $X$, not a scheme $Y$ over $X$ equipped with an integrable connection along $X$. –  S. Carnahan May 24 '11 at 4:12