Questions tagged [d-modules]

Modules over rings of differential operators.

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26 votes
2 answers
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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 differential operator corresponding ...
Ali Taghavi's user avatar
20 votes
3 answers
4k 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 ...
Jan Weidner's user avatar
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83 votes
2 answers
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Why is differential Galois theory not widely used?

E.R. Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
Henry.L's user avatar
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28 votes
8 answers
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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/...
Saal Hardali's user avatar
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20 votes
2 answers
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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 ...
Ongaro Nyang''s user avatar
18 votes
2 answers
1k 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 ...
user avatar
13 votes
2 answers
1k 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 ...
Ben Webster's user avatar
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12 votes
1 answer
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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 $...
Saal Hardali's user avatar
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8 votes
1 answer
1k views

Tensor product of $\mathcal{D}$-modules and constructible sheaves

The Riemann-Hilbert correspondence, as proved by Kashiwara and Mebkhout, says that for X a smooth algebraic variety over $\mathbb{C}$ there is an equivalence of triangulated categories $D^b_c(X,\...
ChrisLazda's user avatar
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8 votes
1 answer
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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 ...
Jan Weidner's user avatar
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5 votes
1 answer
301 views

Operations on perverse sheaves on disk

The category of perverse sheaves on the disk is isomorphic to the category of diagrams $$E\substack{\substack{c\\\to}\\\substack{v\\\leftarrow}}F$$ With $E,V$ finite dimensional vector spaces, and ...
user2520938's user avatar
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5 votes
1 answer
693 views

What's the relationship between a $E_2$-Hochschild Cohomology module and a D-module?

Let's say for simplicity $A$ is a smooth algebra over a field $k$ ($A$ and $k$ are discrete commutative rings but from now on we are fully derived), and we will consider the $E_2$ algebra $HH^{\bullet}...
davik's user avatar
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4 votes
1 answer
496 views

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 ...
Federico Barbacovi's user avatar
2 votes
1 answer
917 views

Is simple non-holonomic D-module a local concept?

It is well known that we can use the Riemann-Hilbert correspondence to describe holonomic D-modules in terms of a category of perverse sheaves on some variety $X$. And $U|\rightarrow Perv(U)$ is a ...
Shizhuo Zhang's user avatar
2 votes
1 answer
98 views

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}>&...
Y. M.'s user avatar
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2 votes
0 answers
92 views

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 ...
AK12N1's user avatar
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1 vote
1 answer
179 views

Cyclic vector of holomorphic vector bundle with flat connection over compact Riemann surface

I originally posted the question on math.stackexhange, but there doesn't seem to be an answer. I apalogize in advance for cross posting. Let $E\rightarrow X$ be a holomorphic vector bundle over a ...
mtraube's user avatar
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