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

Synopsis

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.