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# bgg -- the ith differential of the complex R(M)

- Usage:
    `bgg(i,M,E)`
- Inputs:
  - i, an integer, the cohomological index
  - M, a module, graded S-module
  - E, a polynomial ring, exterior algebra
- Outputs:
  - a matrix, a matrix representing the ith differential

## Description

This function takes as input an integer i and a finitely generated graded S-module M, and returns the ith map in R(M), which is an adjoint of the multiplication map between M_i and M_{i+1}.

    i1 : S = ZZ/32003[x_0..x_2];

    i2 : E = ZZ/32003[e_0..e_2, SkewCommutative=>true];

    i3 : M = coker matrix {{x_0^2, x_1^2, x_2^2}};

    i4 : bgg(1,M,E)

    o4 = {-2} | e_1 e_0 0   |
         {-2} | e_2 0   e_0 |
         {-2} | 0   e_2 e_1 |

                 3      3
    o4 : Matrix E  <-- E

    i5 : bgg(2,M,E)

    o5 = {-3} | e_2 e_1 e_0 |

                 1      3
    o5 : Matrix E  <-- E

## See also

- symExt -- the first differential of the complex R(M)

## Ways to use bgg:

- `bgg(ZZ,Module,PolynomialRing)`

## For the programmer

The object bgg is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/BGG.m2:722:0.