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# cosyzygyRes -- cosyzygy chain of a Cohen-Macaulay module over a Gorenstein ring

- Usage:
    `F = cosyzygyRes(len, M)`
- Inputs:
  - len, an integer, how long a chain of cosyzygies
  - M, a module, Should be a CM module over a Gorenstein ring
- Outputs:
  - F, a complex, last map is presentation of M

## Description

the script returns the dual of the complex F obtained by resolving the cokernel of the transpose of the presentation of M for len steps. Thus M is the len-th syzygy of the module resolved by F. When the first argument len is omitted, the value defaults to len = 2.

    i1 : S = ZZ/101[a,b,c];

    i2 : R = S/ideal"a3,b3,c3";

    i3 : M = module ideal vars R;

    i4 : betti presentation M

                0 1
    o4 = total: 3 6
             1: 3 3
             2: . 3

    o4 : BettiTally

    i5 : betti (F = cosyzygyRes(3,M))

                0 1 2 3 4
    o5 = total: 3 1 1 3 6
            -7: 3 1 . . .
            -6: . . . . .
            -5: . . . . .
            -4: . . . . .
            -3: . . . . .
            -2: . . 1 3 3
            -1: . . . . 3

    o5 : BettiTally

    i6 : cosyzygyRes M

          1      1      3      6
    o6 = R  <-- R  <-- R  <-- R

         0      1      2      3

    o6 : Complex

## Ways to use cosyzygyRes:

- `cosyzygyRes(Module)`
- `cosyzygyRes(ZZ,Module)`

## For the programmer

The object cosyzygyRes is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/CompleteIntersectionResolutions.m2:4319:0.