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# isSyzygy -- Tests if a module is a d-th syzygy

- Usage:
    `b = isSyzygy(M,d)`
- Inputs:
  - M, a module, over a polynomial ring.
  - d, an integer, positive
- Outputs:
  - b, a Boolean value, returns true if M is a d-th syzygy, and false otherwise.

## Description

This algorithm is based upon the methods described in the book of Evans and Griffith (Syzygies. London Mathematical Society Lecture Note Series, 106. Cambridge University Press, Cambridge, 1985.)

    i1 : kk=ZZ/32003

    o1 = kk

    o1 : QuotientRing

    i2 : S=kk[a..d]

    o2 = S

    o2 : PolynomialRing

    i3 : F=res (ideal vars S)^2

          1      10      20      15      4
    o3 = S  <-- S   <-- S   <-- S   <-- S  <-- 0

         0      1       2       3       4      5

    o3 : ChainComplex

NOTE: We are viewing a syzygy module as a cokernel of an appropriate map.

    i4 : isSyzygy(coker F.dd_3,3)  -- the cokernel defined by the 3rd map is not a 3rd syzygy

    o4 = false

    i5 : isSyzygy(coker F.dd_4,3)  -- the cokernel defined by the 4th map is a 3rd syzygy

    o5 = true

This function is called within evansGriffith.

## Ways to use isSyzygy:

- `isSyzygy(Module,ZZ)`

## For the programmer

The object isSyzygy is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Bruns.m2:463:0.