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!

  • $\begingroup$ Is there a reason to expect that the composition of two $D$-affine morphisms should be $D$-affine? For example, is an affine bundle over a projective space $D$-affine? $\endgroup$ – Sam Gunningham Feb 10 '13 at 2:43
  • $\begingroup$ Not sure how to answer your question, but I think the first definition is ``correct'', and I don't think you are missing something! $\endgroup$ – Sam Gunningham Feb 10 '13 at 2:45
  • $\begingroup$ @ Sam : Unless you take very specific bundle, I think they are not $D$-affine. That's my problem!!! Classical affiness behaves well with respect to composition, so I thought it should be the same for $D$-affiness... $\endgroup$ – Johan Feb 10 '13 at 11:47
  • $\begingroup$ Is there any reason why you don't define it in terms of the derived direct image $f_+$? That would seem to be the natural thing to do from a $D$-module perspective... $\endgroup$ – Ketil Tveiten Feb 12 '13 at 11:11

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!

  • 3
    $\begingroup$ It seems that the terminology "D-affine" is misleading in several ways. The main one is expressed in the last paragraph above. But also if one carefully accounts for the shift by spin structure/square root of the canonical bundle involved in quantization, you find that "D-affinity" is a form of ampleness of the square root of the anticanonical bundle --- ie it might better be called "D-Fano".. I think that intuition should lead to the correct expectations about compositions of D-affine morphisms. $\endgroup$ – David Ben-Zvi Feb 11 '13 at 21:07
  • 1
    $\begingroup$ What do you mean by "Projective spaces and flag varieties don't live in interesting families"? For example, the projectivization $P(W)$ of a rank n vector bundle W over base Y is a family of n-1-dimensional projective spaces that is often not a direct product. $\endgroup$ – Victor Protsak Feb 12 '13 at 0:29
  • $\begingroup$ @Victor: You are correct. Sorry, wasn't thinking straight... I had in mind the lack of deformations of projective space, but of course this doesn't imply what I was saying. $\endgroup$ – Sam Gunningham Feb 12 '13 at 1:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.