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# koszul(ZZ,Matrix) -- a differential in a Koszul complex

- Function: koszul
- Usage:
    `g = koszul(i,f)`
- Inputs:
  - i, an integer
  - f, a matrix, a 1 by n matrix
- Outputs:
  - g, a matrix, the i-th differential in the Koszul complex of the matrix f

## Description

    i1 : R = QQ[x_1..x_4];

    i2 : f = matrix{{x_1..x_4}}

    o2 = | x_1 x_2 x_3 x_4 |

                 1      4
    o2 : Matrix R  <-- R

To see the second differential in the Koszul complex of the matrix f look at:

    i3 : koszul(2,f)

    o3 = {1} | -x_2 -x_3 0    -x_4 0    0    |
         {1} | x_1  0    -x_3 0    -x_4 0    |
         {1} | 0    x_1  x_2  0    0    -x_4 |
         {1} | 0    0    0    x_1  x_2  x_3  |

                 4      6
    o3 : Matrix R  <-- R

## Ways to use this method:

- koszul(ZZ,Matrix) -- a differential in a Koszul complex

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/functions/koszul-doc.m2:51:0.