# Questions tagged [d-modules]

Modules over rings of differential operators.

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### Equivalence of categories of $D$-modules on a singular $X$

Is $D^b(Mod_{qc}(D_X)) \to D^b_{qc}(D_X)$ an equivalence of categories for singular $X$? Where $Mod_{qc}(D_X)$ is the category of quasi-coherent modules over $D_X$ and $D^b_{qc}(D_X)$ is the category ...

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### Confusion on summand of Hochschild homology of D-modules

I've encountered the following confusion.
Start with the category of perfect D-modules on $\mathbb{A} ^{1}$, which I'll denote $D$. We have the object $\mathcal{O}$, which is exceptional in the sense ...

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### Riemann Hilbert Correspondence with fixed stractification

Riemann Hilbert Correspondence states for complex manifold $X$, the bounded derived category of $D$ modules on $X$ with cohomology being regular holonomic is equivalent to the bounded derived category ...

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### Holonomic D-module is locally artinian, what about the other implication?

Let $X$ be a complex manifold, and $M$ a coherent $\mathcal{D}_X$-module. Suppose that for all $x\in X$, the stalk $M_x$ is an artinian $\mathcal{D}_{X,x}$-module. Does it follow that $M$ is holonomic?...

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### Bernstein-Sato polynomial

Let $f$ be a polynomial. It is well-known that there exits a polynomial $b_f(s)$, such that $P\cdot f^{s+1}=b_f(s)f^s$ for some differential operator $P$. The polynomial $b_f(s)$ has been studied very ...

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### Remark 12.8.8 in Arinkin--Gaitsgory

I can not understand Remark 12.8.8 in the preprint "SINGULAR SUPPORT OF COHERENT SHEAVES AND THE GEOMETRIC LANGLANDS CONJECTURE". I am somewhat embarrased by the degree of my confusion, hopefully ...

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### Intuition behind characteristic variety [closed]

Given a $D$-module $M$ on variety $X$, one could define $ch(M)$, the characteristic variety of $M$ as the support of $gr(M)$ inside the cotangent bundle of $X$. What is the intuition behind this ...

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### Differential operator of globally unbounded order on connected complex manifold?

Let $X$ be a connected complex manifold. Consider the module $\mathcal{D}_X$. I recall hearing somewhere that one has to be careful with regards to differential operators that are not globally of ...

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### Interesting (non) examples of singular support

I'm trying to better understand singular support of sheaves on smooth manifolds---to this end: What are examples of conical subsets of $T^*X$ that cannot arise as the singular support of a sheaf on $...

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### Example of tensor category with non-simple unit $J\to \mathbb{1} \to Q$ and suitably extension $Q\to M\to J$

Edit: Thanx very much to Neil Strickland for quickly explaining to us that the following cannot be realized over finite commutative $\mathbb{C}$-algebras, as I had originally asked.
I know that there ...

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### Regular integrable connections and regular holonomic modules

I am trying to prove the following $\mathrm{Conn}^{\mathrm{reg}}(X) = \mathrm{Conn}(X) \cap \mathrm{Mod}_{rh}(\mathcal{D}_X)$.
Here an integrable connection on a smooth algebraic variety $X$ is a $\...

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### Functoriality and proofs

Recently, I began to study $\mathcal{D}$-modules and one of the major problems I encountered is that many natural transformations on the (derived) category of $\mathcal{D}$-modules are defined as the ...

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### Regular holonomic $\mathcal{D}$-modules

An holonomic $\mathcal{D}_X$-module on a smooth algebraic variety over $\mathbb{C}$ is called regular if all its composition factors are minimal extensions of the form $L(Y,N)$, where $N$ is a regular ...

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### Additivity of characteristic cycle of holonomic D-module

Let $\mathcal{M}$ be a holonomic D-module on a complex analytic (or alternatively, algebraic) manifold $X$. One can attach to it (using a good filtration) a characteristic cycle $Ch(\mathcal{M})$ ...

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### Is direct image of simple $D$-module is also simple?

(I have asked the question
The commutativity of minimal extension $\cdots$ and I simplify this question to the next simple question:)
Let $X$ be a rational variety over $\mathbb{C}$, $\phi : \hat{X} \...

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### The commutativity of minimal extension and direct image by blowing-down

Let $X$ be a sooth algebraic variety over $\mathbb{C}$.
Let us assume that there exists the commutative diagram
$\require{AMScd}$
\begin{CD}
U @>{i}>> \hat{X}\\
@| @VV{\phi}V\\
U @>{j}>&...

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251 views

### Kozsul resolution of $\mathcal{O}_X$

Let $i: X \hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. In the book D-modules, perverse sheaves and representation theory the authors say that there exists a locally free ...

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### Is there a Lie group Fourier transform for D-modules?

Given a D-module $\mathcal{M}$ on $\mathbb{C}^n$, its Fourier transform $\widehat{\mathcal{M}}$ is equal to $\mathcal{M}$ as a set, but its module structure over $\mathbb{C}[x_1,...,x_n,\partial_1,...,...

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### Canonical map in the direct image of $\mathscr{D}_X$

Let $f : X \to Y$ be a proper holomorphic map between holomorphic manifolds. We work with $\mathscr{D}$-modules. Consider the transfer bi-modules $\mathscr{D}_{Y\leftarrow X}.$ Can one find a ...

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### Isomorphism classes of rings of differential operators

Let $X$ be your "favourite" kind of space, and let $\mathcal{D}_X$ be the (sheaf of) ring(s) of differential operators on $X$. What does the ring $\mathcal{D}_X$ tell us about $X$?
I know this might ...

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288 views

### Characteristic variety of a D-module

Let $X$ be an algebraic variety, $M \in Mod(\mathcal{D}_X)$. I am studying the characteristic variety associated to this module, and I am trying to understand why all the different definitions ...

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### Differential operators and quasi-finite morphisms

Let $ X, Y $ be smooth affine varieties over $ \mathbb C $. Let $ T : X \rightarrow Y $ be a dominant quasi-finite morphism and let $ T^\# : \mathbb C[Y] \rightarrow \mathbb C[X] $ be the resulting ...

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### A canonical isomorphism in derived categories of D-modules

I am learning D-modules recently, and my question might be technical. It arises from Lemma 2.6.13 in Hotta-Takeuchi-Tanisaki's book, which states that there exists a canonical isomorphism
$$
R\mathcal ...

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### Derived category of $\mathcal{D}_X$ modules

Define $D^{\text{#}}_{\bullet}(Mod(\mathcal{D}_X))$ to be the full subcategory of the derived category $D^{\text{#}}(Mod(\mathcal{D}_X))$ of complexes of $\mathcal{D}_X$-modules whose cohomology ...

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### How to exchange left and right $\mathcal{D}_X$-modules when $X$ is not smooth?

When $X$ is a smooth scheme (over something of characteristic $0$), one can exchange left and right $\mathcal{D}_X$-modules ($\mathcal{D}_X$ means the sheaf of differential operators) by tensor with (...

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### Transfer modules and Weyl algebra

Let $V$ be a $\mathbb{C}$-vectorial space of dimension $n$ and $V^*$ the complex dual space.
I would like to understand the following isomorphism $$D_{V^* \leftarrow V \times V^*} \overset{L}\otimes_{...

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### Beilinson-Bernstein localization, equivariant modules

I have a question regarding the equivariance in the Beilinson-Bernstein localization. Let $G$ be an simply connected algebraic group over a discrete valuation ring $R$ of mixed characteristic $(0,p)$ ...

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### Holonomic modules and Holonomic functions

Let
$$f_{d}(h):=\sum_{k=1}^{d}(-1)^k\binom{d-1}{k-1}\prod_{i=1}^{d}G((i-k)h) . $$
I have proved that $ F(x):=\sum_{d=1}f_{d}\frac{x^{d}}{d!}\in \mathbb{C}(h)[[x]]$ is holonomic and arrive at a ...

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### About an algebraic construction of a sheaf of formal microdifferential operators

While reading these notes by Victor Ginzburg on $D$-modules I found a certain construction of Microlocailzation in the algebraic setting which unfortunately doesn't seem to be elaborated on a lot in ...

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### $\mathscr{D}$-module external tensor product and analytification

Let $X$ and $Y$ be smooth complex varieties and $M\in D^b_{\mathrm{hol}}(\mathscr{D}_X)$ resp. $N\in D^b_{\mathrm{hol}}(\mathscr{D}_Y)$ holonomic $\mathscr{D}$-modules (resp. complexes with holonomic ...

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### The minimal injective $R$-resolution of a $D$-module

The most general version of the question I want to ask is:
Let $R$ be a regular (commutative, Noetherian) ring containing a field $k$ of characteristic $0$, let $D = D(R,k)$ be the ring of $k$-...

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### Riemann-Hilbert correspondence for non-flat connections

First of all, let me warn that my knowledge of the correspondence is rather superficial, and I apologize for any technical inaccuracies below.
Setting
Let $X$ be a smooth complex algebraic variety, ...

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### Prime spectrum of the derived category of holonomic $\mathcal{D}$-modules?

Let $X$ be a smooth algebraic (/projective if it simplifies things considerably) variety over $\mathbb{C}$ and consider the derived category $\mathcal{C}=D_h^b(\mathcal{D}_X)$ of bounded complexes of $...

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### Who are the compact generators in the derived category of $\mathcal{D}_X$-modules?

Let $X$ be a smooth affine variety over $\mathbb{C}$ and let $\mathcal{D}_X$ be its algebra of differential operators.
Consider $\mathcal{C}=\mathcal{D}_X$-$\text{mod}$, the stable $\infty$ category ...

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### Regular holonomic D-modules as generalisation of regular singular points

I'm trying to understand why the definition of a regular holonomic D-module is a good generalisation of the usual definition of a regular singular point for a differential equation. More precisely, ...

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### Do almost commutative flat degenerations induce equality in K-theory? (Or: Is the characteristic variety actually a support of a class in $K$-theory?)

I intentionally phrased the title to match a different question which is almost identical to the one i'm asking. However similar, the answer there, which uses commutative algebraic geometry, is not ...

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### What is a Green's function in the language of $\mathcal{D}$-modules?

Let $P$ be an analytic linear differential operator defined on some open interval $X=(a,b)$ and $\mathcal{M}=\mathcal{D}_X / \mathcal{D}_X \bullet P$ the corresponding $\mathcal{D}$-module. I'm trying ...

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### Relation between the b-function of a holonomic module and the b-function of its corresponding holonomic dual

Let $D=k[x_1,\ldots, x_n, \partial_1, \ldots, \partial_n]$ be the Weyl algebra with $k$ a field of characteristic zero.
Given a holonomic D-module $M$ we know that its holonomic dual $M^{'}=\text{Ext}...

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### Algorithm for holonomic sequence

A sequence $(a_n)_{n\geq 0}$ of complex numbers is called polynomially recursive (P-recursive) or holonomic if there exists a number $r$ and rational functions $P_1(n), \ldots, P_r(n)$ such that $a_n =...

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### Is $\mathbb{P}^1$ the only smooth projective curve with a locally split tangent lie algebroid?

Let $C$ be a smooth projective curve over an algebraically closed field $k$.
The tangent lie algeborid $\mathcal{T}_C$ of $C$ is just sheaf of vector fields on $C$ equipped with the usual lie ...

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### Deligne's Canonical Extension in Algebraic Varieties?

Suppose $C/\mathbb{Q}$ is an algebraic curve (not necessarily complete) defined over $\mathbb{Q}$, and $p$ is a $\mathbb{Q}$ valued point of $C$. Suppose there is a algebraic fibration
\begin{equation}...

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### $q$-crystals - is there such a thing?

There are several important facts that I first heard about here on MO. One of the most enlightening of these is that $\mathscr D$-modules on a scheme $X$ may be viewed as sheaves on the groupoid of ...

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### A global geometric formulation of the fundamental theorem of Picard Vessiot theory?

Let $X$ be a smooth curve over an algebraically closed field of characteristic 0. The category of quasi-coherent $D$-modules $\mathcal{D}_X$-$Mod$ is a symmetric monoidal abelian category. We can ...

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### Is there a useful theory of D-modules on smooth (non-analytic) manifolds?

D-modules are related to flat connections on vector bundles, end hence local systems. The theory of D-modules (and related notions such as crystals etc.) seems to be very popular in complex analytic ...

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### Are simple Poisson $A$-modules finitely generated as $A$-modules?

Let $A$ be a Poisson $\mathbb{C}$-algebra: $A$ has a the structure of a complex commutative algebra and at the same time it carries the structure of a Lie algebra, with Lie bracket $\{\cdot, \cdot\}$. ...

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### Bernstein's theorem

In the book "An introduction to the theory of local zeta functions" prof. Igusa presents Bernstein's theorem as follows:
Let $K_0$ be a field, and write $K=K_0(s)$. Let $f\in K_0[x_1,\dots,x_n]\...

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### Example of non-holonomic D-module and explicit computation of characteristic variety

I'm currently trying to have a better understanding of the concepts of characteristic variety and holonomic $D$-modules (let us assume that they are coherent) on a holomorphic manifold $X$. I know ...

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### Applications of Microlocal Analysis?

What examples are there of striking applications of the ideas of Microlocal Analysis?
Ideally i'm looking for specific results in any of the relevant fields (PDE, algebraic/differential geometry/...

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### How can I construct D-modules over projective space using explicit differential equations?

Over $\mathbb{A}^n$, it is easy to construct D-modules by writing down an explicit linear system of PDE's and then writing a presentation of the associated D-module
$$
\mathcal{D}^n \xrightarrow{} \...

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### What kind of algebraic object is $\mathcal{D}_X$? (algebra of diifferential operators). What's special about modules over it?

Let $R$ be a regular ring over a field of char 0. Let $X=Spec R$ and $D=\mathcal{D}_X$
the algebra of differential operators over it.
The overall vague question is what kind of algebraic object is $...