Why is the ring of Grothendieck differential operators bad when $X$ is singular?

Question

$\DeclareMathOperator\Diff{Diff}$Suppose for simplicity that $X$ is affine, it is then possible to define $\Diff(X)$ — the ring of Grothendieck differential operators. When $X$ is smooth, then

Definition. the category of $D$-modules on $X$ is defined to be modules over $\Diff(X)$. (Category 1)

However, when $X$ is singular, this is not the right category to consider. One usually follows Kashiwara's approach:

Definition. choose a closed embedding $X\hookrightarrow V$ and define $D$-modules be to modules over $\Diff(V)$ such that are (set-theoretically) supported $X$. (Category 2)

The usual reason I heard for why to consider the second category is that $\Diff(X)$ behaves badly when $X$ is singular, and specifically people will point out that $\Diff(X)$ is not Noetherian. (Noetherian = left + right.) For example, this is the case when $X$ is the 'cubic cone' [BGG72].

However, I am no longer satisfied with this answer because of the following:

(1), when $X$ is a curve then $\Diff(X)$ is Noetherian. [SS88]
(2), when $X=V/G$ a quotient singularity then $\Diff(X)$ is Noetherian.

But in these cases, one still considers Category 2 for these $X$. So it has to be the case that, in general and in these cases, $\Diff(X)$ is bad not just because it is not Noetherian, it is also bad for other reasons. So my question is:

Question: Why do we work in category 2 in the situations above. Or a better questions, what is bad about $\Diff(X)$ besides not being Noetherian.

Note my question is not how to work in category 2, but why it fails badly if we work in category 1 in situations (1) and (2).

It is worth to remark that:

in (1), if the curve is cuspidal then category 1 $\cong$ category 2. [SS88] generalised in [BZN04]
in (2), if the $X=\mathbb{C}^2/(\mathbb{Z}/2\mathbb{Z})$ then category 1 $\cong$ category 2 (I think this is true, but do please correct me if I am wrong.)

[BGG72] I. N. Bernˇste ̆ın, I. M. Gel’fand, and S. I. Gel’fand. Differential operators on a cubic cone. Uspehi Mat. Nauk, 27(1(163)):185–190, 1972.

[BZN04] David Ben-Zvi and Thomas Nevins. Cusps and D-modules. Journal of the American Mathematical Society, 17.1:155–179, 2004

[SS88] S. P. Smith and J. T. Stafford. Differential operators on an affine curve. Proc. London Math. Soc. (3), 56(2):229–259, 1988.

Noted later: actually it is not true that on $X=\mathbb{C}^2/(\mathbb{Z}/2\mathbb{Z})$ then category 1 $\cong$ category 2, sorry for the confusion.

Answer 1

There probably can be many answers to this question, but here is one:

We want the category of $D$-modules to behave like categories of sheaves in other sheaf theories. In the complex setting, we want the Riemann-Hilbert correspondence, an equivalence of categories between constructible sheaves in the analytic topology and regular holonomic $D$-modules, and over an arbitrary base field, we want them to have similar behavior. Why do we want this? All these sheaf theories are expected to be various shadows of the category of motives, and probably motives are the really interesting thing we want to study, so we want our theories to be similar enough that they capture motives.

Anyways, for the categories of constructible sheaves (algebraic/analytic), there is an equivalence of categories between sheaves on $X$ and sheaves on $V$ supported on $X$.

So if we want the category of $D$-modules to have similar behavior, we must use category 2!

It's possible we could use some other definition and prove the equivalence with a full subcategory of $D$-modules on $V$ supported on $X$. But since we know this is what we want, we might as well take it to be true by definition.

So, the thing that is bad about the category 1 is simply that it is not 2. We absolutely can use definition 1 in the special cases where it agrees with 2, but doing so might leave us less prepared for the general case.

Answer 2

One thing that's wrong with the Grothendieck definition on singular varieties is the same thing that's wrong with defining the tangent space at a singular point rather than the tangent complex - it's not sufficiently derived (ie it's a strange truncated notion of the "true" derived object), which is why several different attempts to define D-modules give different answers there (the one quoted by OP, the notion of modules over vector fields - ie the algebra of differential operators generated by first order ones, which needn't be the full algebra in the singular case, and crystals - which amounts to the "option 2" one).

If you define D-modules in a derived fashion, as is done e.g. in the book of Gaitsgory-Rozenblyum, then order is restored: all the different natural notions you might come up with agree [here it's important to be in characteristic zero, a whole lot more interesting stuff happens in positive characteristic]. If you replace the naive notion of the sheaf vector fields by the tangent complex and consider modules for it, or define the Grothendieck differential operators derivedly (as a groupoid algebra for the de Rham groupoid of X), or define crystals (option 2), you get the same notion.