PolyExt -- Ext groups between a holonomic module and a polynomial ring

Synopsis

Usage:
PolyExt M, PolyExt I; RatExt(i,M), RatExt(i,I)
Inputs:
- M, a module, over the Weyl algebra D
- I, an ideal, which represents the module M = D/I
- i, an integer, nonnegative
Optional inputs:
- Strategy => ...,
Outputs:
- a hash table or a module, the Extⁱ group(s) between holonomic M and the polynomial ring

Description

The Ext groups between a D-module M and the polynomial ring are the derived functors of Hom, and are finite-dimensional vector spaces over the ground field when M is holonomic.

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

i2 : M = W^1/ideal(x^2*D^2)

o2 = cokernel | x2D2 |

                            1
o2 : W-module, quotient of W

i3 : PolyExt(M)

                      2
o3 = HashTable{0 => QQ }
                      2
               1 => QQ

o3 : HashTable

Caveat

Does not yet compute explicit representations of Ext groups such as Yoneda representation.

Ways to use `PolyExt` :

PolyExt(Ideal)
PolyExt(Module)
PolyExt(ZZ,Ideal)
PolyExt(ZZ,Module)

PolyExt -- Ext groups between a holonomic module and a polynomial ring

Synopsis

Description

Caveat

See also

Ways to use PolyExt :

Ways to use `PolyExt` :