# DExt -- Ext groups between holonomic modules

## Synopsis

• Usage:
DExt(M,N), DExt(M,N,w)
• Inputs:
• M, , over the Weyl algebra D
• N, , over the Weyl algebra D
• w, a list, a positive weight vector
• Outputs:
• , 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``` ```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 reprentations 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)