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Dmodules :: DHom

DHom -- D-homomorphisms between holonomic D-modules

Synopsis

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, 1 differential variables
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.

See also

Ways to use DHom :

For the programmer

The object DHom is a method function with options.