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# DExt -- Ext groups between holonomic modules

- 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.

## See also

- DHom -- D-homomorphisms between holonomic D-modules
- Drestriction -- restriction modules of a D-module

## Ways to use DExt:

- `DExt(Module,Module)`
- `DExt(Module,Module,List)`

## For the programmer

The object DExt is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/BernsteinSato/DOC/DHom.m2:116:0.