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Questions tagged [d-modules]

This tag is for questions relating to -Modules.

The origin of -modules is in the work of the Japanese school of Mikio Sato in the mid-twentieth century on algebraic analysis. The aim of this program was to understand systems of linear partial differential equations on manifolds, and their generalizations, using the techniques of algebraic geometry and sheaf theory.

A -module on an (algebraic or complex analytic) variety X is the notion generalising the one of finite rank vector bundles with flat connection.

The theory of algebraic -modules provides a bridge from algebra to analysis and topology. It can be regarded as "mildly non-commutative algebra" in the sense that it is obtained by extending the methods of commutative algebra to non-commutative rings (or rather sheaves of rings) of algebraic differential operators on complex varieties. As such it gives a substitute in algebraic geometry for the theory of linear partial differential equations.

For more details see

https://en.wikipedia.org/wiki/D-module

https://www.encyclopediaofmath.org/index.php/D-module

72 questions
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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 true for arbitrary $n$ it implies the Jacobian…
Daniel McLaury
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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 the category of perverse sheaves on $V$ (or, if you…
Stephen
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Is it true that $M\boxtimes N = p_1^* M\otimes_{\mathcal{O}_{X\times X}}p_2^* N$ for D-modules?

Let $X$ be a smooth algebraic variety over $\mathbb{C}$ and consider the projections $p_1,p_2:X\times X\to X$. If $M$ and $N$ are left $\mathcal{D}_X$-modules, their exterior tensor product is defined as $$M\boxtimes N := \mathcal{D}_{X\times…
Gabriel
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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, $x=(x_1,...,x_n)$ (using multi-index notation), $…
alphanzo
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There aren't *left* $D$-modules on a singular variety?

In these notes one can read on page $5$: "it may seem surprising, but one cannot define left $D$-modules for non-smooth schemes". Could someone elaborate on it? If I'm not mistaken, to define $D$-modules on a singular scheme $Z$, we embed $Z$ into a…
student
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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 field of charateristic $0$ and $K$ a closed subgroup of $G$ with corresponding lie algebras $\mathfrak{g},…
C.Niculescu
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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 However, I am a beginner in this theory and I…
trequartista
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The relative de Rham complex

Here (at the bottom of page 13) it's stated that for a smooth map $f:X\to S$, the relative Spencer complex $$\Omega_{X/S}^\bullet \otimes_{f^{-1}\mathcal{O}_S} D_X$$ is a resolution of the transfer module $$D_{S\leftarrow X} \ = \ f^{-1}(D_S…
Meow
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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 that the $A_n$ module $K[X,p^{-1}] \otimes _{K[X]}…
CameronJWhitehead
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It's true that $\mathrm{D}_{X\times Y}(M^\bullet\boxtimes N^\bullet)=\mathrm{D}_X(M^\bullet)\boxtimes \mathrm{D}_Y(N^\bullet)$ for D-modules?

Let $X,Y$ be smooth algebraic varieties over $\mathbb{C}$, $M^\bullet\in\mathsf{D}^b_{\text{coh}}(\mathcal{D}_X)$, $N^\bullet\in\mathsf{D}^b_{\text{coh}}(\mathcal{D}_Y)$, and denote by $\mathrm{D}_X$ the Verdier duality functor over $X$. I wonder if…
Gabriel
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Does Verdier Duality Fix Simple Perverse Sheaves?

Suppose you have a connected reductive algebraic group $G$ acting on a "nice" variety $X$, so that the orbits decompose $X$ into a Whitney stratification. Consider the category of Perverse Sheaves $\textbf{Per}(X)$ resulting from this…
James Steele
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Idea behind the definition of direct image of $\mathscr{D}$-modules

Let $f:X\to Y$ be a morphism of smooth algebraic varieties over $\mathbb{C}$. Since $\mathcal{D}_{X\to Y}$ is a $(\mathcal{D}_X,f^{-1}\mathcal{D}_Y)$-bimodule, given a right $\mathcal{D}_X$-module $M$, we have a natural right $\mathcal{D}_Y$-module…
Gabriel
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Derivations of the algebra $K[x,y]/(y^2-x^3)$

Let $K$ be a field of characteristic zero, and let $S=K[x,y]/(y^2-x^3)$. It is easy to see that $S\cong R:=K[t^2,t^3]$ via the isomorphism induced by the ring homomorphism $K[x,y]\to R$ given by $x\mapsto t^2, y\mapsto t^2$. It is known that for any…
Albert
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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 $f \in K[[x_1,...,x_n]]$ is called $D$-finite if all partial…
user7475
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Algebraic cohomology $R\Gamma_{[Z]}(\cdot)$ for locally closed analytic set $Z$

I really searched for this quite a while and didn't find an answer - I hope I didn't miss anything obvious. Let $X$ be a complex manifold ($\mathscr O_X$ the sheaf of holomorphic functions on $X$) and $Z\subset X$ a closed analytic subset, $I_Z$…
user178979
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