$D_X$ 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?
 A: 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 $X$.
A: Keerthi Madapusi Sampath already answered the question in the comments, so I'll just add some remarks.  
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.
