Macaulay2 » Documentation
Packages » BernsteinSato :: rationalFunctionExt
next | previous | forward | backward | up | index | toc

# rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)

- Usage:
    `rationalFunctionExt M, rationalFunctionExt I; rationalFunctionExt(M,f), rationalFunctionExt(I,f);`
    `rationalFunctionExt(i,M), rationalFunctionExt(i,I); rationalFunctionExt(i,M,f), rationalFunctionExt(i,I,f)`
- Inputs:
  - M, a module, over the Weyl algebra D
  - I, an ideal, which represents the module M = D/I
  - f, a ring element, a polynomial
  - i, an integer, nonnegative
- Optional inputs:
  - Strategy => ..., default value Schreyer,
- Outputs:
  - a hash table or a module, the Ext^i group(s) between holonomic M and the polynomial ring localized at the singular locus of M (or at f if specified)

## Description

The Ext groups between 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 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+5)

    o2 = cokernel | xD+5 |

                                1
    o2 : W-module, quotient of W

    i3 : rationalFunctionExt M

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

    o3 : HashTable

## Caveat

Input modules M or D/I should be holonomic.

## See also

- Dresolution -- resolution of a D-module
- Dintegration -- integration modules of a D-module

## Ways to use rationalFunctionExt:

- `rationalFunctionExt(Ideal)`
- `rationalFunctionExt(Ideal,RingElement)`
- `rationalFunctionExt(Module)`
- `rationalFunctionExt(Module,RingElement)`
- `rationalFunctionExt(ZZ,Ideal)`
- `rationalFunctionExt(ZZ,Ideal,RingElement)`
- `rationalFunctionExt(ZZ,Module)`
- `rationalFunctionExt(ZZ,Module,RingElement)`

## For the programmer

The object rationalFunctionExt 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.m2:224:0.