# Course Identification

Introduction to the theory of algebraic D-modules
20214151

## Lecturers and Teaching Assistants

Prof. Dmitry Gourevitch
N/A

## Course Schedule and Location

2021
First Semester
Tuesday, 10:15 - 13:00
27/10/2020

## Field of Study, Course Type and Credit Points

Mathematics and Computer Science: Lecture; Elective; Regular; 3.00 points

N/A

## Prerequisites

Basic algebraic geometry.

20

English

## Attendance and participation

Expected and Recommended

Pass / Fail

100%

Other

N/A
N/A
-
N/A

9

## Syllabus

1. Algebraic properties of modules over the Weyl algebra D=Dn.
(i) Bernstein fi ltration of the Weyl algebra. Filtrations of D-modules. Good
filtrations.
(ii) Noetherian properties.
(iii) Associated varieties, Hilbert polynomials. Dimension and degree of a D-
module.
(iv) Basic inequality.
(v) Holonomic D-modules and their properties.

2. Relation with analysis.
(i) D-modules and systems of differential equations. Solutions of D-modules. Left
and right D-modules.
(ii) Digression on the theory of generalized functions and distributions.
(iii) Regularization of distributions xs and Qs using differerential equations.

3. Standard functors for D-modules.
(i) Inverse image functor
(ii) Relation between left and right D-modules
(iii) Direct image functor
(iv) Fourier transform

4. Properties of functors on categories of D-modules.

5. Stability of the holonomic property.

6. Further applications to analysis: regularization of di fferent type of integrals.

6. Homological techniques in study of D-modules.

7. Geometric picture of the the algebra D of differential operators.
(i) Geometric fi ltration on the algebra D.
(ii) Associated varieties for D-modules. Noetherian properties.
(iii) Comparison of two approaches.
(iv) Equivalence of two approaches to holonomic modules.

7. D-modules on smooth affine algebraic varieties.
(i) Short digression into affine algebraic varieties. The sheaf OX and the category M(OX)
of O-modules on X. Localization of OX-modules.
(ii) Recall of basic properties of smooth varieties.
(iii) Basic de finition and the structure of the algebra D = DX of differential operators on a smooth affine algebraic variety X.
(iv) Category M(D) of D-modules on a smooth affine algebraic variety X. Localization of D-modules.
(v) Relation between left and right D-modules.

8. Basic functors between D-modules.
(i) Basic functors and their properties.
(ii) Kashiwara's lemma
(iii) Reduction to the case of affine space (Weyl algebra).

9. Study of D-modules using the geometric filtration.
(i) Associated module. Noetherian properties.
(ii) Singular support of a D-module. Relation with Kashiwara lemma.
(iii) Proof of the basic inequality.

## Learning Outcomes

Upon successful completion of this course students should:

1. Demonstrate knowledge of the advanced language of algebraic D-modules and thus get access to the wide literature that uses it.
2. Use the algebraic technique of D-modules to solve the complicated analytic problems concerning invariant distributions.