evansGriffith -- Reduces the rank of a syzygy

Usage:

N = evansGriffith(M,d)
Inputs:
- M, a matrix, over a polynomial ring whose cokernel is an d-th syzygy.
- d, an integer, positive
Outputs:
- N, a matrix, with the same source and kernel as M, but such that coker N is a dth syzygy of rank d.

Description

The routine reduces the target of M by elementary moves (see elementary) involving just d+1 variables. The outcome is probabilistic, but if the routine fails, it gives an error message.

See 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..e]

o2 = S

o2 : PolynomialRing

i3 : i=ideal(a^2,b^3,c^4, d^5)

             2   3   4   5
o3 = ideal (a , b , c , d )

o3 : Ideal of S

i4 : F=res i

      1      4      6      4      1
o4 = S  <-- S  <-- S  <-- S  <-- S  <-- 0
                                         
     0      1      2      3      4      5

o4 : ChainComplex

i5 : f=F.dd_3

o5 = {5} | c4  d5  0   0   |
     {6} | -b3 0   d5  0   |
     {7} | a2  0   0   d5  |
     {7} | 0   -b3 -c4 0   |
     {8} | 0   a2  0   -c4 |
     {9} | 0   0   a2  b3  |

             6      4
o5 : Matrix S  <-- S

i6 : EG = evansGriffith(f,2)  -- notice that we have a matrix with one less row, as described in elementary, and the target module rank is one less.

o6 = {5} | c4  d5  0                                                     
     {6} | -b3 0   d5                                                    
     {7} | 0   -b3 -8251a4+5071a3b-9480a2b2+12365a3c+8231a2bc+5026a2c2-c4
     {7} | a2  0   2653a4-6203a3b-11950a2b2-13508a3c+5864a2bc+10259a2c2  
     {8} | 0   a2  -7501a3+9534a2b-7216a2c                               
     ------------------------------------------------------------------------
     0                                                       |
     0                                                       |
     -8251a2b3+5071ab4-9480b5+12365ab3c+8231b4c+5026b3c2     |
     2653a2b3-6203ab4-11950b5-13508ab3c+5864b4c+10259b3c2+d5 |
     -7501ab3+9534b4-7216b3c-c4                              |

             5      4
o6 : Matrix S  <-- S

i7 : isSyzygy(coker EG,2)

o7 = true

This is called within bruns.

Ways to use evansGriffith:

evansGriffith(Matrix,ZZ)

For the programmer

The object evansGriffith is a method function.

The source of this document is in Bruns.m2:425:0.