DExt -- Ext groups between holonomic modules

Synopsis

Usage:

DExt(M,N), DExt(M,N,w)
Inputs:
- M, a module, over the Weyl algebra D
- N, a module, over the Weyl algebra D
- w, a list, a positive weight vector
Optional inputs:
- Info => ..., default value 1,
- Output => ..., default value HomologyModules,
- Special => ..., default value None,
- Strategy => ..., default value Schreyer,
Outputs:
- a hash table, the Ext groups between holonomic D-modulesM and N

Description

The Ext groups between D-modules M and N are the derived functors of Hom, and are finite-dimensional vector spaces over the ground field when M and N are holonomic.

The procedure calls Drestriction, which uses w if specified.

The algorithm used appears in the paper 'Polynomial and rational solutions of holonomic systems' by Oaku-Takayama-Tsai (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 variable(s)

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

o2 = cokernel | xD-x |

                            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 : DExt(M,N)

                      2
o4 = HashTable{0 => QQ }
                      2
               1 => QQ

o4 : HashTable

Caveat

Input modules M, N should be holonomic.Does not yet compute explicit representations of Ext groups such as Yoneda representation.

Ways to use DExt:

DExt(Module,Module)
DExt(Module,Module,List)

For the programmer