# Questions tagged [d-modules]

Modules over rings of differential operators.

260
questions

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### Modern treatment of $q$-differential operators/$\mathcal{D}_q$ modules?

The basic idea of $q$-differential operators: replace
$$\partial\cdot x^n\ =\ nx^{n-1} \hspace{10mm}\rightsquigarrow\hspace{10mm} \partial\cdot x^n \ =\ [n]_q x^{n-1} $$
where $[n]_q=(q^{n}-1)/(q-1)$ ...

3
votes

0
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### What is a twisted D-module?

Let $X/\mathbb{C}$ be an abelian variety, $Y$ be the dual abelian variety, and $P$ be the Poincaré bundle on $X\times Y$. On p.207, Correction to “Sheaves with connection on abelian varieties” (by M. ...

2
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0
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### Compact generators of $\mathcal{D}\text{-Mod}$ via Mayer Vietoris

Let $X$ be a complex variety (or any space for whom a category of sheaves with the six functors is defined), and
$$i\ :\ Z\ \to\ X\ \leftarrow\ U\ :\ j$$
be complementary open and closed embeddings.
...

8
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### What's a holonomic D-module from the point of view of de Rham spaces?

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. We can consider its de Rham space $X_\text{dR}$ as the sheaf on $(\textsf{Sch}/\mathbb{C})_\text{ét}$ defined by $X_\text{dR}(S):= X(S_\text{...

5
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### D-modules generated by derivatives of Delta function

We consider D-modules over the affine line $\mathbb{A}_{\mathbb{C}}^{1}$, i.e. modules of Weyl algebra $A_1(\mathbb{C})=\mathbb{C}[x,\partial]$. There is a set of microfunctions (ref. A Primer of ...

3
votes

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### Confusion about definition of crystals

In the notes by Lurie there seems to be two possible definition for crystals which both makes sense for arbitrary functors $X : \mathrm{CRing}_k \to \mathrm{Set}$. ($k$ here is a field. We probably ...

8
votes

1
answer

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### What are twisted Verma modules? Basic properties?

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbf{C}$, $\lambda$ be a weight in the nonnegative Weyl chamber and $w_1,w_2\in W$. Then the twisted Verma modules are defined e.g. in Andersen and ...

12
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2
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### Can we use Mann's six-functor formalism with D-modules?

In a recent course in Bonn, P. Scholze explains a formalization of a six-functor formalism due to L. Mann. In this axiomatization, three of the functors $f_!,f^*,\otimes$ are "constructed" (...

2
votes

1
answer

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### D modules over nodal curves

Is there a simple description for $D$ modules over $\text{Spec}\left(k\left[x,y \right] / \left(xy\right) \right)$ (say k is algebraically closed of characteristic 0)? Is there a description of the $D$...

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### Local cohomology with coefficients in ideals of parameters

I'm not an expert in local cohomology, but the following problems have come up in my work, and I'd like to get a sense of where things stand.
Let $\mathbb{A}^n=\operatorname{Spec} \mathbb{C}[x_1, \...

4
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### Confusion about D-affineness and jet sheaves on projective line

I have encountered what feels like a very basic confusion relating to the Beilinson Bersnstein localisation theorem.
This theorem in particular states that if $F$ is a $D_{\mathbb{P}^{1}}$ module, ...

2
votes

0
answers

149
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### Smooth pullback of holonomic D-modules is fully faithful

Let $X$ and $Y$ be (not necessarily smooth) algebraic varieties over an algebraically closed field of characteristic $0$, and suppose we have a smooth surjective map $f: X \to Y$ of relative dimension ...

2
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### Generalizations of elliptic chain complexes

I would like to know if it is possible to generalize the notion of elliptic chain complex of differential operators to different contexts, whether geometric or non geometric. I have in mind D-geometry....

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0
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### Good D-module is induced by coherent O-module?

Let $X$ be a compact complex manifold. In Definition 4.24, p.78 of D-modules and Microlocal Calculus (by Kashiwara), a coherent $D_X$-module $F$ is called good if there is a directed family $\{G_i\}...

2
votes

1
answer

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### About the support of a holonomic D-module

Let $X$ be a smooth algebraic variety over $\mathbb{C}$ and let $M^\bullet\in\mathsf{D}^b_\text{h}(\mathcal{D}_X)$ be a complex of D-modules with holonomic cohomologies. We define the support of $M^\...

3
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1
answer

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### Artin vanishing for D-modules (i.e., when is $f_+$ t-exact?)

Let $f:X\to S$ be a morphism between algebraic varieties which are smooth over a field of characteristic zero. We define the (derived) direct image functor $f_+:\mathsf{D}^b(\mathcal{D}_X)\to \mathsf{...

6
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### D-modules on singular varieties; forgetful functors, and t-structures

Let $Z$ be a singular variety over the complex numbers with a closed embedding $i: Z \hookrightarrow X$ into a smooth variety $X$. One can define the derived category $\mathcal{D}(Z)$ of D-modules on $...

4
votes

0
answers

81
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### Holonomic distributions in the analytic setting

We are interested in a reference to the notion of Holonomic distributions in the real analytic setting. Namely, distributions that generate a Holonomic D-module under the action of the algebra of ...

1
vote

1
answer

206
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### Every rank 1 local system $L$ satisfying $m^*L=L\boxtimes L$ comes from the Lang torsor? The same holds for D-modules?

Let $G$ be a commutative connected algebraic group over a field $k$, with group operation $m:G\times G\to G$. If $k=\mathbb{F}_q$, we may use a character $\varphi:G(k)\to\overline{\mathbb{Q}}_\ell^\...

5
votes

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139
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### Concrete computation of pullback of the $D$-module $\mathbb{C}[t,t^{-1}]$

I have some problems in calculating some example explicitly. Consider
$$ \{0\} \overset{i}{\rightarrow} \mathbb{C} \overset{j}{\leftarrow} \mathbb{C}^*.$$
Then $Rj_+\mathbb{C}[t,t^{-1}] = \mathbb{C}[t,...

3
votes

1
answer

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### Explicit computation of D-modules pullback

Consider the $D_{\mathbb{A}^1}$-module $M:=D_{\mathbb{A}^1}/(x)$, and the map $f:z\mapsto z^k$. I want to know $f^*(M)$. I believe it only has a single non-zero cohomology, namely in degree $0$, which ...

2
votes

1
answer

155
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### Confusion about Wakimoto's chiral differential operators on $\mathbf{P}^1$

It is a classical result of Wakimoto that the sheaf of chiral differential operators $D_{ch}$ on $\mathbf{P}^1$ has global sections
$$D_{ch}(\mathbf{P}^1)\ \simeq\ L_{-2}(\mathfrak{sl}_2)$$
the simple ...

5
votes

1
answer

245
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### Two identities involving Ext functors in the context of D-modules

I have several questions regarding proposition 2.3 in "Cherednik and Hecke algebras of varieties with a finite group action", by Pavel Etingof. Let $X$ be a complex affine algebraic variety ...

5
votes

1
answer

257
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### Holonomic = annihilated by some differential operator

Let $X$ be a smooth variety over a characteristic zero field $k$. It seems to be well-known that
"A coherent $\mathcal{D}_X$-module is holonomic if and only if 'every element is annihilated by a ...

4
votes

0
answers

128
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### D-module theoretic Chern characters and an index-type theorem

Let $X$ be a smooth projective variety over $\mathbf{C}$. Consider the category $\mathbf{D}_{X}^{\text{perf}}$ of perfect complexes of left $D$-modules on $X$. It is well known that there is an ...

4
votes

1
answer

300
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### Compatibility between the functors of $\mathcal{O}_X$-modules and $\mathcal{D}_X$-modules

Let $f:X\to Y$ be a morphism between smooth algebraic varieties over $\mathbb{C}$.
We have natural functors $f^!:\mathsf{D}_{\text{qc}}(\mathcal{D}_Y)\to \mathsf{D}_{\text{qc}}(\mathcal{D}_X)$, $f_*:\...

3
votes

1
answer

285
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### Koszul complex of a $\mathcal{D}$-module

I am trying to understand the construction of push forward in the derived category of $\mathcal{D}$-modules using the Koszul complex. My question will be about the latter notion.
Let $X$ be a smooth ...

7
votes

0
answers

193
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### A reference for Bernstein's approach to KL conjectures

The exposition in the famous J. Bernstein's lecture notes on D-modules is rather sketchy.
Fortunately, there are many other sources for the information from this text (like Hotta's excellent "D-...

2
votes

1
answer

263
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### Open/closed embeddings and the de Rham space

Let $U\to X$ be an open immersion of schemes and denote by $D$ the (say reduced) complement. Then by applying the de Rham functor, we get morphisms
$$U_{dR}\to X_{dR}\leftarrow D_{dR}$$
of the ...

3
votes

0
answers

238
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### An algebraic proof: A line bundle on a curve with a connection must be of degree 0

Let me state it using the language of $D$-modules. Let $X$ be a smooth projective curve and $\mathcal{L}$ a line bundle on it. Assume that $\mathcal{L}$ has a left action of $\mathcal{D}_X$. Then show ...

3
votes

1
answer

223
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### do hyperfunction solutions always exist?

I have two questions---the second question (which is what I'm really interested in) is a generalization of the first, but I think the first may be more likely to get an answer. I'll be happy with an ...

4
votes

1
answer

494
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### Six functor formalism for quasi-coherent $D$-modules

Let $X$ be a smooth scheme over a field $k$ and let $\mathsf{D}_{\text{qc}}(\mathcal{D}_X)$ be the full subcategory of $\mathsf{D}(\mathcal{D}_X\mathsf{-Mod})$ composed of the complexes of $\mathcal{D}...

8
votes

1
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391
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### D-modules as ind-coherent sheaves over positive characteristics?

There is an interpretation of D-modules over "sufficiently nice" prestacks $X$ (read: various finiteness conditions apply, perhaps even smoothness) by Gaitsgory and Rozenbylum (see chapter I....

6
votes

0
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### Definition of the tensor product of $D$-modules

Let $\mathsf{D}^b_h(X)$ be the full subcategory of $\mathsf{D}^b(D_X\textsf{-Mod})$ (the bounded derived category of left $D_X$-modules) consisting of the complexes with holonomic cohomology. The ...

6
votes

0
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246
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### The geometric "hands-on" vs. algebraic approach to nearby cycles

Feel free to skip to the question below; the following is just context and discussion:
An interesting, but seemingly less used result in the theory of nearby cycles of constructible sheaves is (...

3
votes

0
answers

128
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### "higher" micro-support

Recall that for a sheaf $F$ on an analytic manifold $X$ the micro-support consists of those $\omega\in T^*X$ for which there exists a $C^1$ function $f$ defined around $\pi(\omega)$ with $f(\pi(\omega)...

7
votes

2
answers

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### Why is the ring of Grothendieck differential operators bad when $X$ is singular?

$\DeclareMathOperator\Diff{Diff}$Suppose for simplicity that $X$ is affine, it is then possible to define $\Diff(X)$ — the ring of Grothendieck differential operators. When $X$ is smooth, then
...

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0
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154
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### When does a $D$-module think it’s a pullback along a smooth morphism?

Let $X$ and $Y$ be two algebraic varieties, and let $f: X \to Y$ be a morphism. Suppose $A$ is a holonomic $D$-module on $Y$. In this situation we can pull $A$ back to $X$ using either the $!$ or $*$ ...

7
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answers

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### Tensor-hom adjunction for $\mathcal{D}$-modules

Let $X$ be a smooth equidimensional scheme over $\mathbb{C}$. Given two left $\mathcal{D}_X$-modules $M$ and $N$, we endow their tensor product (as $\mathcal{O}_X$-modules) $M\otimes_{\mathcal{O}_X} N$...

9
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295
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### Understanding the mixed Hodge structure on D-modules on $\mathbb{C}$

Consider the category of regular D-modules on $\mathbb{C}$ (let's say as a complex manifold). It's a well-known fact that if you consider the subcategory of these where the singular support is the ...

3
votes

1
answer

267
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### Iterating specialization of sheaves?

This is a question about the operation of taking the specialization of sheaves along a subspace. I'll recall the settings in which I've encountered a notion of specialization of sheaves:
The real, ...

2
votes

0
answers

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### On the use of the fundamental exact sequence of K\"ahler differentials in a paper of Lyubeznik

Let $k$ be a field, $R := k[x_1, \cdots , x_n]$ the polynomial ring in $n$ indeterminates over $k$ and $f$ a nonzero element of $R$. The following paper of Lyubeznik which I have been recently reading,...

2
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0
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### Socle of a quotient of the ring of differential operators of a polynomial ring

I have been reading the following paper:
https://www.sciencedirect.com/science/article/pii/S002240491000263X
Proposition 2.4(ii) shows that if $\mathfrak D$ is a ring of $k$-linear differential ...

3
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0
answers

281
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### Cohomological dimension for stacks

If $X$ is a scheme (maybe with conditions), I'm pretty sure that the ($\ell$-adic/de Rham) rational cohomology $H^*(X,\mathcal{F}$) of an $\ell$-adic sheaf/holonomic $D$-module $\mathcal{F}$ vanishes ...

5
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### Geometric interpretation of $\mathbb{C}^{\times}$-gerbes

Let $X$ be a (nice enough, e.g. smooth etc.) variety over the complex numbers, and let $\mathcal{G}$ be a gerbe on $X$. Then $\mathcal{G}$ is classified by a cohomology class in $\alpha \in H^2(X, \...

4
votes

1
answer

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### Symmetric powers of curves and completion along the diagonal

Given a smooth curve $C$, denote by $\text{Sym}^d(C)$ its $d$-th symmetric power. Let $\Delta$ be the diagonal subvariety which is defined as the codimension $1$ subvariety that at least two of the ...

8
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### $K$-theory of $D$-modules

I have to admit I don't know much about topics appearing in this question, I just see very rough connections between these objects:
According to this page 23, a different $t$-structure on $D^b(\text{...

5
votes

1
answer

189
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### Is analytification of regular holonomic D modules a fully faithful functor?

It seems to me that by the algebraic Riemann Hilbert functor(which factors through the analytification map) and also the analytic Riemann Hilbert functor that the (derived) category of (algebraic) ...

3
votes

1
answer

272
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### Is the abelian category of pure Hodge modules semi-simple?

I am a beginner in the subject, and at the moment I am trying to understand basic properties of the main objects of the M. Saito's theory of the mixed Hodge modules in general.The question in the ...

3
votes

1
answer

186
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### Is it possible that $\int |f^2(z)|^{t+1}(P\bar{P})\phi(z) dz=0$ for all compactly supported $\phi$?

When proving that the log-canonical threshold is minus the largest root of the Bernstein-Sato polynomial one considers the integral
$$\int |f^2(z)|^{t+1}(P\bar{P}(\phi(z))) dz,$$
where $P$ is a linear ...