Question

I am just a beginner in $D$-module, so this could be a stupid question, but I can't find an easy reference for it. I would like to define the notion of $D$-affine morphism. The most obvious way would be to say that a smooth morphism between smooth quasi-projective schemes: $$ f : X \rightarrow Y$$ is $D$-affine if the following two conditions hold:

1) the functor $f_* : Mod_{qc}(D_X) \rightarrow Mod(f_*D_X)$ is exact,

2) for any $M \in Mod_{qc}(D_X)$, we have $f_*M = 0 \Rightarrow M=0$.

Since I want things to be somehow a bit functorial, I would like that the composition of two $D$-affine morphisms is still $D$-affine. But it's not clear to me that the direct image of a quasi-coherent $D$-module by an $D$-affine morphism is again a quasi-coherent $D$-module.

One could also define $D$-affiness by the following conditions:

1) the functor $\int_{X/Y}^{0} : Mod_{qc}(D_X) \rightarrow Mod_{qc}(D_Y)$ is exact,

2) for any $M \in Mod_{qc}(D_X)$, we have $\int _{X/Y}^0 M = 0 \Rightarrow M=0$.

here $\int_{X/Y}^0$ denotes the $0$-th homology of the derived push-forward for $D$-modules. Since $f$ is smooth the $\int_{X/Y}^k$ vanish for $k<0$, so that the Leray spectral sequence for $D$-modules push-forward guarantees me that the composition of two smooth $D$-affine morphisms is again $D$-affine. But with this definition, my favourites examples (that is $f : G/P \rightarrow spec \mathbb{C}$, for $G/P$ a rational homogeneous space) are not $D$-affine anymore.

Is there something I am missing?

Thanks a lot!

Answer 1

The situation as I see it is as follows:

The first definition you give is the natural one. It implies that D-modules on the $X$ are given by sheaves of modules for the sheaf of algebras $f_\ast D_X$ on $Y$. However, I am curious if there are actually any interesting examples of such a morphism. Projective spaces and flag varieties don't live in interesting families... Is it then the case that any D-affine morphism is either affine or a product $X=Y\times Z$, where $Z$ is $D$-affine?

The second definition is far too weak. For example, under that definition, if a scheme $X$ is D-affine over a point then any D-module on $X$ is a local system. I think the only schemes that are D-affine in the second sense are finite collections of points.

As you remark, it is not true that D-affineness respects composition. For example, take $X$ to be the total space of $\mathcal O(1)$ living over $Y=\mathbb P^1$. Then $f:X\to Y$ is affine and $Y\to pt$ is D-affine. However, $f_\ast \mathcal O_X = \bigoplus _{n\geq 0} \mathcal O(-n)$, which has higher cohomologies. So $X$ is not D-affine (over a point).

To me D-affineness is a strange and mysterious thing. Flag varieties are D-affine for very different reasons than affine varieties. Perhaps it is not helpful to include both these things in the same definition. Being D-affine is somehow not a notion that is intrinsic to D-modules: I don't think it can be expressed just in terms of the de Rham stack $X_{dR}$ and D-module functors. It is defined in terms of the forgetful functor to $\mathcal O$-modules.

I hope this helps!