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41 views

Why does this identity hold for D-modules?

Consider $\mathbb{C}^n$ with coordinates $x_1,\ldots,x_n$ and let $\partial_1,\ldots,\partial_n$ be the corresponding vector fields. Then the canonical free rank 1 D-module should be $$ \mathbb{C}[x_1,...
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38 views

Bernstein polynomial of $x_1^2+\cdots+x_n^2$

According to S. C. Coutinho’s A Primer of Algebraic $D$-modules (see p. 95), the Bernstein polynomial of $p=x_1^2+x_2^2+\cdots+x_n^2\in K[x_1,\dots,x_n]$ (with $K$ an arbitrary field of characteristic ...
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Characteristic Variety of the Principal Symbol solves PDE system?

In the study of partial differential equations, it is often considered enough to analyze the principal symbols and their characteristic variety (see, https://en.wikipedia.org/wiki/...
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D-module push forward definition

I know it is quite difficult to define the push forward of a $D$-module. I read there that, for $f$ a map $X\longrightarrow Y$, if $M$ is a left or right $D_{X}$-module, in general, $f_{∗}M$ is not ...
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40 views

When is a D-module holonomic?

Suppose that I have a small family of partial differential equations such that $y(x)$ must be a solution to $$ P_i(x,D) y(x)=0,\ \ \ \ \ \ i=1,...,m $$ for all $i=1,...,m$, where I am using multi-...
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1answer
198 views

Defining a sheaf of differential operators

I have two questions related to the coordinate-free definition of $\mathcal{D}$-modules provided in Ginzburg's notes (see pages 24-25): Exercise 2.1.13 says that Diff($M,M$) is almost-commutative, i....
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61 views

A question on Weyl algebra $A_1$

This question is from A Primer of Algebraic D-Modules by S.C. Coutinho, Ex 2.4.10. In the following $A_1$ means the Weyl algebra generated by $x$ and $d$. Let $f: A_1^2\to A_1$ be the map defined ...
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Tensor product of $A_n$ modules/ localisation at ring of differentials

I'm working through Coutinho's "A Primer of Algebraic D-Modules" and I've gotten stuck on the following question: Let $p \in K[x_1, \ldots ,x_n]$ be non-zero, and let $A_n$ be the Weyl Algebra. Show ...
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43 views

Space of polynomial solutions of $P(f) = 0$ for $P \in A_1 (\mathbb{C})$ has finite dimension

I've been trying to work through Coutinho's Primer on Algebraic D-Modules and I'm having trouble with proving the following exercise: Let $P \in A_1 (\mathbb{C})$, the first Weyl Algebra over $\...
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1answer
152 views

holonomic D-modules

I am trying to develop an intuition about holonomic D-modules and find the literature formidable (I study physics). My question is, given a linear differential operator in n-variables, $x=(x_1,...,...
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1answer
224 views

A problem in elementary calculus

Let $P(x), Q(x)$ be two polynomials with real coefficients and set $F(x) = \frac{P(x)}{Q(x)}$. Consider a table which has the function $e^{\int_0^x F \, dx}$ on it. The table has the set of rules that ...
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153 views

Is it really unknown that every endomorphism of the Weyl algebra $A_1$ is an isomorphism?

Here $A_1 := K\{x\cdot-, \frac{d}{dx}\} \subset \operatorname{End}_K(K[x])$ for some characteristic-zero field $K$. I found this claim in Coutinho's "A Primer of Algebraic D-Modules." If this is ...
6
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1answer
154 views

Perverse sheaves (or D-modules) on vector spaces, constructible with respect to a hyperplane arrangement

Let $V$ be a finite dimensional complex vector space and let $\mathcal{A}$ be a finite collection of hyperplanes in $V$. Stratify $V$ by the intersections of elements of $\mathcal{A}$, and consider ...
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0answers
256 views

Recommendation textbooks on D-module

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

Holonomic ideals and D-finite power series

I would like to understand the connection between the term D-finite power series (in n variables) and the term of a holonomic module over the Weyl algebra A_n. A power series $f \in K[[x_1,...,x_n]]$ ...
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399 views

questions about coroot

I am reading the lecture notes of geometric representation theory: http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf. I have a question on coroot. In general, if we have a root $\alpha$, then the ...
3
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1answer
200 views

What theorem of Serre's is being referenced?

I am reading Joseph Bernstein's notes on D-modules which are available online here In section 1 (page 1 of the pdf) Bernstein writes "By Serre's theorem this condition is local." I was wondering to ...