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

## Synopsis

• Usage:
F = cosyzygyRes(len, M)
• Inputs:
• len, an integer, how long a chain of cosyzygies
• M, , Should be a CM module over a Gorenstein ring
• Outputs:
• F, , 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 : ChainComplex

## Ways to use cosyzygyRes :

• cosyzygyRes(Module)
• cosyzygyRes(ZZ,Module)

## For the programmer

The object cosyzygyRes is .