Defining a sheaf of differential operators I have two questions related to the coordinate-free definition of $\mathcal{D}$-modules provided in Ginzburg's notes (see pages 24-25):

*

*Exercise 2.1.13 says that Diff($M,M$) is almost-commutative, i.e. that the associated graded ring $A$ is commutative. But the zeroth graded piece is simply $A$-linear homomorphisms from $M$ to $M$, in which composition $f\circ g$ need not be commutative in general. What am I missing?

*We define $\mathcal{D}(\mathcal{M},\mathcal{N})$ (for $\mathcal{M},\mathcal{N}$ coherent) on $X$ by defining it on the basis of open affines of $X$ by
$$\Gamma(U,\mathcal{D}(\mathcal{M},\mathcal{N}))=\text{Diff}_{\mathcal{O}(U)}(\mathcal{M}(U),\mathcal{N}(U)).$$ How do the details work out? I'm confused, in particular, about how to construct the restriction maps
$$\rho_{VU}:\text{Diff}_{\mathcal{O}(U)}(\mathcal{M}(U),\mathcal{N}(U))\to \text{Diff}_{\mathcal{O}(V)}(\mathcal{M}(V),\mathcal{N}(V)).$$
Is the coherence hypothesis on $\mathcal{M},\mathcal{N}$ needed/used here?

Thanks.
 A: I'm not an expert on $\mathcal{D}$-modules, but hopefully I can help a bit.

For 1, you're probably right. If $M = N$ is also equal to the base ring $A$, then yes, $\operatorname{Diff}(M,M)$ will be an almost commutative ring, sometimes denoted $\mathcal{D}(X)$ if $X$ is the affine scheme $X = \operatorname{Spec}A$ (of course using the sheaf definition in your question 2 you get a sheaf for any scheme).
Ginzburg himself in "Differential operators and BV structures in noncommutative geometry" points out that there is a subtlety here (see Defs. 2.1.1 and 2.1.2, Rem. 2.1.12): you can do what Ginzburg did in his notes that you linked (which matches EGA IV$_4$ §16.8; see 16.8.1), or you can define $\operatorname{Diff}_0(M,M)$ as the image of $A$ in $\operatorname{Hom}_A(M,M)$. In the latter case, you have $\operatorname{Diff}_0(M,M) = A$ and the almost commutativity follows by the same proof as for $\mathcal{D}(X)$.
In the former case, as you pointed out, you don't get almost commutativity. I would say asking around on MathOverflow (Why are their two definitions as listed in Ginzburg? How do researchers distinguish them? When is one better than the other?) or even just e-mailing Ginzburg himself is not a bad idea…

For 2, there are a couple of ways to do this. Let me first sketch what one might do after learning sheaves à la Hartshorne, using (quasi-)coherence. For each affine open subset $U \subset X$, let $\mathcal{D}_U(\mathcal{M},\mathcal{N})$ be the $\mathcal{O}_U$-module associated to $\operatorname{Diff}(\mathcal{M}(U),\mathcal{N}(U))$ (see §II.5 in Hartshorne). This gives you the maps $\rho_{VU}$ that you wanted, provided $V$ is contained in some affine open subset.
Now we take what we defined locally to get a sheaf defined on all of $X$, using the "Glueing Sheaves" exercise in Hartshorne (Exc. II.1.22): you can glue together the $\mathcal{D}_{U_\alpha}(\mathcal{M},\mathcal{N})$ defined on an open cover $\{U_\alpha\}$ if you have isomorphisms $\varphi_{\alpha\beta} \colon \mathcal{D}_{U_\alpha}(\mathcal{M},\mathcal{N})\rvert_{U_\alpha \cap U_\beta} \to \mathcal{D}_{U_\beta}(\mathcal{M},\mathcal{N})\rvert_{U_\alpha \cap U_\beta}$ satisfying the cocycle condition. So you can try to make sense of what the association above using associated sheaves really means, and then check the cocycle condition.
But who wants to do that. The easier way to see it is by noting $\operatorname{Diff}(M,N)$ is a sub-vector space of $\operatorname{Hom}_k(M,N)$ (by "additive" Ginzburg means $k$-additive, I believe). What follows is an adaptation from SGAIII, Exp. VII$_\mathrm{A}$.
Let $p \colon X \to \operatorname{Spec} k =: S$ be the structure morphism for $X/S$. Give $\mathcal{M},\mathcal{N}$ the structure of $p^{-1}\mathcal{O}_{S}$-modules via the map $p^{-1}\mathcal{O}_{S} \to \mathcal{O}_X$ obtained by adjunction from the structure morphism (in this case $p^{-1}\mathcal{O}_{S}$ is probably just the constant sheaf $k$). Then, the way you defined $\operatorname{Diff}(M,N)$, we can construct $\operatorname{Diff}(\mathcal{M},\mathcal{N})$ as the subsheaf of $\operatorname{\mathscr{H}\!om}_{p^{-1}\mathcal{O}_S}(\mathcal{M},\mathcal{N})$ whose sections on an open set $V$ contain morphisms whose restrictions to $U \cap V$ are in $\operatorname{Diff}_{\mathcal{O}(U \cap V)}(\mathcal{M}(U \cap V),\mathcal{N}(U \cap V))$ for each basis open set $U$. The advantage of this over the construction above is that we found a globally defined sheaf that locally agrees with the $\mathcal{D}_{U_\alpha}(\mathcal{M},\mathcal{N})$, hence is exactly what we would've gotten from the glueing construction by the uniqueness statement in Hartshorne, Exc. II.1.22.
