Modules over rings of differential operators.

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### D-modules, deRham spaces and microlocalization

Given a variety (or scheme, or stack, or presheaf on the category of rings), some geometers, myself included, like to study D-modules. The usual definition of a D-module is as sheaves of modules over ...

**21**

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**2**answers

428 views

### Equivalence of “Weyl Algebra” and “Crystalline” definitions of rings of differential operators between modules?

Let $B$ be a commutative $A$-algebra, and let $M$, $N$ be two $B$-modules. We can talk about the set of $A$-linear module homomorphisms $M \to N$, i.e. the set $\text{Hom}_A(M, N)$. Differential ...

**19**

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**4**answers

2k views

### Classification of PDE

Recently I have been attending a course on PDE's. I was totally ignorant of the subject and wasn't that motivated to be honest. But I was intrigued and felt I had to take the course seriously both for ...

**17**

votes

**3**answers

2k views

### How do I compare the different notions of Fourier transform for sheaves?

There is a close but not perfect relationship between algebraic D-modules on C^n, constructible sheaves on C^n in the analytic topology, and \ell-adic sheaves on an n-dimensional vector space over a ...

**14**

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1k views

### D-modules and Algebraic Solutions of PDEs

I am not certain if this is a complete question and I fear it might be shot down. Anyway, I try to pose it. My question is in connection to using D-modules to study PDEs (and systems of PDEs). When ...

**14**

votes

**2**answers

556 views

### Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map
$D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...

**12**

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918 views

### Examples for D-affine varieties?

A variety is called $\mathcal D$- affine if the global section functor induces an equivalence between quasi-coherent $\mathcal D$-modules and modules over $\Gamma(X,\mathcal D)$.
It is easy to see ...

**12**

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292 views

### Epsilon factors - a la Beilinson - What is it?

I understand, to some extent, Tate's thesis. Could somebody explain perhaps what are the epsilon factors in Beilinson's works, such as "$\epsilon$-factors for Gauss-Manin determinants", or ...

**11**

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1k views

### The algebraic Version of Riemann Hilbert Correspondence

It is well known that if I have a differentialable manifold (holomorphic maniford) $M$, then I have a functor from the categroy of vector bundles on $M$ with flat connections to the categroy of local ...

**11**

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1k views

### D-modules supported on the nilpotent cone

I am reading some papers which involve D-modules on a Lie algebra g, which are supported on the nilpotent cone n. They are equivariant for the action of G. (In particular, I consider g=sl_n).
It ...

**11**

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551 views

### Microlocalizing Hochschild homology

A recent paper of Ben-Zvi and Nadler gives a general formalism for understanding "dimensions" in sheaf theories. Without getting too far into details, amongst other things, this formalism allows us ...

**11**

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**1**answer

656 views

### $\mathcal{D}$-quasi-isomorphisms and coherent $\Omega$-modules

Let $X$ be a smooth $\mathbf{C}$-scheme of finite type. In section 7.2 of their preprint "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld define an ...

**11**

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**1**answer

1k views

### What is the recent development of D-module and representation theory of Kac-Moody algebra?

I just started to collect the papers of this field and know little things. So if I make stupid mistake, please correct me.
It seems that there are several approaches to localize Kac-Moody algebra(in ...

**11**

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**1**answer

671 views

### On the definition of regularity

In the litterature on D-modules, there are many definitions of regularity of holonomic D-modules.
(1) Bernstein first defines regularity on a curve then says a holonomic D-module is regular if its ...

**10**

votes

**1**answer

1k views

### The Infinitesimal topos in positive characteristic

This question was inspired by and is somewhat related to this question.
In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie des schemas" ...

**9**

votes

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768 views

### Does the Riemann-Hilbert Correspondence work at the DG level?

let $X$ denote a smooth complex algebraic variety. Let $D_{rh}(X)$ denote the category of regular holonomic $D$-modules on $X$ and $D_{rh}^b(D(X))$ denote the bounded derived category of $D$-modules ...

**9**

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**1**answer

362 views

### Bernstein-Sato polynomial (one variable)

Let $R = \mathbb{C}[x_1,...,x_n]$, $p \in R$. There exists a monic (of lowest degree) $b_p(x) \in \mathbb{C}[s]$ and a differential operator $D(s)$ such that
$$b_p(s) p^s = D(x)p^{s+1}.$$
The ...

**9**

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**0**answers

336 views

### Duality in category O vs. Duality of D-modules

Hello,
I omit in the following all the words "derived, twisted, holonomic, finitely-generated...".
We have the Bernstein-Beilinson equivalence between the category of $N$-equivariant $D$-modules on ...

**8**

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**3**answers

1k views

### What is a twisted D-Module intuitively?

When I think about $\mathcal{D}$-Modules, I find it very often useful to envison them as vectorbundles endowed with a rule that decides whether a given section is flat. Or alternatively a notion of ...

**8**

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453 views

### Ring of differential operators of a quotient ring

All rings are assumed to have unity.
Let $k$ be a field. Recall the definition of Grothendieck's ring of ($k$-linear) differential operators $D(R;k)$ of a commutative $k$-algebra $R$:
...

**8**

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**1**answer

544 views

### what is the connection between D-modules and coordinate bundles?

Fix $n$ and a field $k$ of characteristic zero. Let $G$ be the pro-algebraic group of automorphims of $k[[x_1,...x_n]]$. Let $G_0$ be the subgroup of automorphisms preserving the closed point (note ...

**7**

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2k views

### Beilinson-Bernstein and Koszul duality

For geometric representation theorists down here.
Consider the Beilinson-Bernstein theorem:
Functor of global sections establishes
the correspondence between twisted
D-modules with fixed ...

**7**

votes

**4**answers

2k views

### Gluing perverse sheaves?

It might be a stupid question.
How to glue perverse sheaves? I am considering the following example. Flag variety of $sl_2$, which is $P^1$. Consider category of perverse sheaves on $P^1$. Denoted ...

**7**

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96 views

### Differential operators between modules, $\mathcal{D}_A(M, M)$ necessarily a filtered, almost commutative ring?

See here. Does it follow immediately that $\mathcal{D}_A(M, M)$ as defined in the link is a filtered, almost commutative ring? How can I visualize this geometrically?

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**1**answer

502 views

### How does one interpret the naive t-structure on constructible sheaves as a t-structure on D-modules?

By the Riemann-Hilbert correspondence, there is an equivalence between
(1)
$\mathcal{D}\operatorname{-mod}(X)$
, the (derived) category of holonomic D-modules on a complex variety X, and
(2)
...

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votes

**3**answers

771 views

### Making D-modules on affine varieties more explicit

This is a follow up to my question about D-modules supported on the nilpotent cone. I got some good answers there but now I have a more basic question.
Consider an affine algebraic variety X, a ...

**7**

votes

**1**answer

233 views

### Distinguished triangle of closed - open partition, for D-modules

Hello,
I am sorry if this question is not appropriate for MO.
Suppose $X$ is the affine line, $i:Z\to X$ is the origin, and $j: U \to X$ is the complement to $Z$ in $X$.
I then have a distinguished ...

**7**

votes

**0**answers

428 views

### Status of Borho and Brylinski's irreducibility conjectures?

In Differential Operators on Homogeneous Spaces, III, Section 6.7, Borho and Brylinski make a long series of equivalent conjectures which they show are all equivalent. Perhaps the easiest statement ...

**6**

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848 views

### For quasi-coherent D-Modules

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 ...

**6**

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**1**answer

458 views

### l-adic Turrittin

What would be an l-adic analogue of the Turrittin-Levelt decomposition theorem?
Turrittin-Levelt is a structure theorem of meromorphic connections on a complex curve in the formal neighbourhood of a ...

**6**

votes

**1**answer

720 views

### Localization of vanishing cycles

Consider a regular holonomic D-module (or a perverse sheaf) $M$ on a smooth variety $X$. Let $f:X\to A^1$ be a polynomial (or holomorphic) function.
Question: Is it true that the $\lambda \in A^1$ ...

**6**

votes

**2**answers

371 views

### D-module that is coherent as O-module

Suppose that $X$ is an algebraic variety over $\mathbb C$, not necessarily smooth. Is it still true that each $\mathcal D_X$-module ($\mathcal D_X$ is of course the sheaf of differential operators) ...

**6**

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**1**answer

954 views

### Computation of vanishing cycles

Here's the problem I'm looking at:
$F$ is a perverse sheaf (or a regular holonomic D-module, or even a mixed hodge module) on $\mathbb{C}^2$ stratified by $z_1 = 0$, $z_2=0$. It can be caracterized ...

**6**

votes

**0**answers

524 views

### Recommendation textbooks on D-module

I am going to take part in a seminar on D-modules and applications, the textbooks that will be used are : D-modules, Perverse Sheaves, and Representation Theory, A Primer of Algebraic D-Modules
...

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228 views

### Non-characteristic maps (ala D-modules)

I am trying to understand a `well known' fact (see Kashiwara'sIntroduction to microlocal analysis page 63, remark 4.8) about non-characteristic morphisms. Here is ...

**5**

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780 views

### Confusion about D-modules and functors

Given a morphism $f:X\rightarrow Y$ between smooth complex varieties, one can define functors from the bounded derived category with holonomic cohomology on $Y$ to the same category on $X$. The ...

**5**

votes

**2**answers

895 views

### Relation between characteristic variety and support of D-Module

Given a coherent $\mathcal{D}_X$-Module $M$, one can assign to it its characteristic variety
$$ch(M)\subseteq T^*X$$
or one could look at its support
$$supp(M)\subseteq X$$
as an $\mathcal {O}_X$ ...

**5**

votes

**1**answer

650 views

### $\mathcal{D}$-modules of level m

My question is regarding the definition of $\mathcal{D}$-modules of level $m$ given in this paper. As an example, let $X=\mathbb{A}^1$ over $S=\text{Spec }\overline{\mathbb{F}_p}$; I was told that a ...

**5**

votes

**1**answer

432 views

### Are these notions of strongly equivariant D-modules equivalent?

It seems that there are two notions of strongly equivaraint $D_X$- Modules and I would like to know if they are equivalent, or at least how they are related.
Let $\rho: G\times X \rightarrow X$ be an ...

**5**

votes

**1**answer

712 views

### Relation between holonomic D-modules and perverse sheaves

Given a smooth complex algebraic variety, the Riemann-Hilbert-correspondence tells us, that the category of perverse sheaves is equivalent to the category of regular, holonomic D-modules.
However not ...

**5**

votes

**1**answer

478 views

### Reconstruction from category of D-modules on variety

Arinkin has a theorem which says that an abelian variety can be reconstructed from its derived category of coherent D-modules.
D.Orlov conjectured that this theorem is true for any variety.
My ...

**5**

votes

**1**answer

431 views

### What kind of algebra has geometric realization as in “Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups”

In Chriss and Ginzburg's book "Representation Theory and Complex Geometry" as well as the paper "Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups", the group algebra ...

**5**

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**1**answer

244 views

### D-modules on affine space that are regular at infinity

If I have a D-module $M$ on $\mathbb{A}^n$ (which is essentially the same thing as a module over the Weyl algebra), then I can push this D-module forward to $\mathbb{P}^n$ to get a D-module $j_*M$ on ...

**5**

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**1**answer

457 views

### Localizability of differential operators a la Grothendieck

Hello,
Maybe this question is trivial, so sorry
Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1).
Then we can define the module of differential operators $D^{\leq n} ...

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**0**answers

86 views

### Characteristic Cycles and Nearby Cycles

Let $Y$ be a smooth algebraic variety over $\mathbb{C}$, let $X = Y \times \mathbb{C}$ and let $f: X \to \mathbb{C}$ be the projection. Let $M$ be a (not necessarily regular) holonomic $D_X$-module ...

**5**

votes

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174 views

### Computing the constant D-module on an intersection

Let $M$ be a smooth variety, say over the complex numbers, and let $i:W \hookrightarrow M, j: Z \hookrightarrow M$ be smooth subvarieties. Let $i_{+},j_{+}$ denote (derived) pushforward of D-modules, ...

**5**

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315 views

### Blow ups and Characteristic varieties

Let $f:Y\to X$ be a morphism of smooth varieties (complex analytic or algebraic as you wish). We have
$$
T^* Y \overset{f_d}{\longleftarrow} Y\times_X T^*X \overset{f_\pi}{\longrightarrow} T^* X
...

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437 views

### V-filtration of D-modules associated to a monomial

Hi
In Mixed Hodge modules Saito computes the Verdier specialisation of a D-modules with respect to a monomial $g = x_1^{m_1}\ldots x_n^{m_n}$. This is a very nice result as I find such explicit ...

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votes

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596 views

### How is this observation related to Koszul duality?

Let $X$ be a smooth variety, $\mathcal D$ the sheaf of algebraic differentail operators, $\Omega$ the algebraic deRham complex and $\mathcal M$ a quasi coherent $\mathcal O_X$-module.
Now there is a ...

**4**

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**1**answer

480 views

### Analogues of D-modules and constructible sheaves

For a smooth complex variety, one can consider the category of say holonomic $\mathcal D$ modules on it. It is equipped with the deRham functor, which turns a $\cal D$-module into a constructible ...