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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 ...
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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, ...
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1answer
51 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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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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113 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, ...
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222 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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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 ...
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1answer
139 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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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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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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286 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 ...
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1answer
195 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 ...