DHom -- D-homomorphisms between holonomic D-modules

Synopsis

Usage:
DHom(M,N), DHom(M,N,w), DHom(I,J)
Inputs:
- M, a module, over the Weyl algebra D
- N, a module, over the Weyl algebra D
- I, an ideal, which represents the module M = D/I
- J, an ideal, which represents the module N = D/J
- w, a list, a positive weight vector
Optional inputs:
- Strategy => ...,
Outputs:
- a hash table, a basis of D-homomorphisms between holonomic D-modules M and N

Description

The set of D-homomorphisms between two holonomic modules M and N is a finite-dimensional vector space over the ground field. Since a homomorphism is defined by where it sends a set of generators, the output of this command is a list of matrices whose columns correspond to the images of the generators of M. Here the generators of M are determined from its presentation by generators and relations.

The procedure calls Drestriction, which uses w if specified.

The algorithm used appears in the paper 'Computing homomorphisms between holonomic D-modules' by Tsai-Walther(2000). The method is to combine isomorphisms of Bjork and Kashiwara with the restriction algorithm.

i1 : W = QQ[x, D, WeylAlgebra=>{x=>D}]

o1 = W

o1 : PolynomialRing

i2 : M = W^1/ideal(D-1)

o2 = cokernel | D-1 |

                            1
o2 : W-module, quotient of W

i3 : N = W^1/ideal((D-1)^2)

o3 = cokernel | D2-2D+1 |

                            1
o3 : W-module, quotient of W

i4 : DHom(M,N)

o4 = {| -xD+x+1 |, | -D+1 |}

o4 : List

Caveat

Input modules M, N, D/I and D/J should be holonomic.

Ways to use `DHom` :

DHom(Ideal,Ideal)
DHom(Module,Module)
DHom(Module,Module,List)

DHom -- D-homomorphisms between holonomic D-modules

Synopsis

Description

Caveat

See also

Ways to use DHom :

Ways to use `DHom` :