It is well-know that the category of coherent D-modules over a smooth algebraic $k$-scheme is a Tannakian category. So it is equivalent to the category of finite representations of some affine group scheme G/k. My question is do we have the similar statement for quasi-coherent D-modules? I hope that the category of quasi-coherent D-modules is equivalent to the representation (not necessarily finite) category of some affine group schemes. Is that true, is there any reference for that?
I hope that any quasi-coherent D-module is the union of its coherent sub D-modules. If the answer to the above question is true then this is true. If the above is wrong. I still believe this is true. At least I think it is true for char$k$=0, where a quasi-coherent D-module is a quasi-coherent sheaf with a flat connection. If this is true, could you give me any reference?
Best Answer
Hi Lei, I think your second statement is false, even in char. $0$: Let $X=\mathbb{A}_k^1$ for some field $k$, and $U:=X\setminus \{0\}$. Denote the open immersion by $i$. Then $i_*\mathcal{O}_U$ is $\mathcal{O}_X$-quasi-coherent and not coherent. Consider the canonical connection on $i_*\mathcal{O}_U$: If $x$ is a coordinate on $\mathbb{A}^1_k$, then $\nabla(x):=dx$. This is a connection on $i_*\mathcal{O}_U$: if we plug in $\frac{1}{x}$ we get $-\frac{1}{x^2}dx\in i_*\mathcal{O}_U\otimes \Omega^1_{X/k}$. But the section $\frac{1}{x}$ is not contained in a $\mathcal{O}_X$-coherent sub $D_X$-module: The smallest sub $D_X$-module containing $\frac{1}{x}$ contains $\frac{1}{x^n}$ for all $n\in \mathbb{Z}$. This is not finitely generated over $\mathcal{O}_X$.
What is true however, is that on a smooth variety any $\mathcal{O}_X$-quasi-coherent $D_X$-module is the union of its $D_X$-coherent submodules, but these do not need to be $\mathcal{O}_X$-coherent. A reference for this is this:
D-Modules, Perverse Sheaves, and Representation Theory: Cor. 1.4.17
http://books.google.com/books?id=8ewkW5SC7DcC&lpg=PP1&dq=hotta%20takeuchi&pg=PA29#v=onepage&q&f=false)