nres matrix computation [max depth]

Compute a minimal free resolution of the module M presented by matrix . The columns of the matrix of the result minimally generate the syzygy module of M. A standard basis is computed for each of these syzygy modules.

If a maximum depth, d, is specified, then only the first d matrices of the resolution are computed. The default for d is the number of variables in the base ring of matrix . If the base ring is a polynomial ring, and not a quotient ring, then Hilbert's syzygy theorem says that the length of the resolution is bounded by the number of variables.

For example,

%  type j
; -bc+ad -bc+ad b3-a2c c3-bd2 -ac2+b2d 

%  nres j w
; 2.3...4...5...
; computation complete after degree 5

%  pres w
; 
; ----------------------------------

; bc-ad b3-a2c c3-bd2 ac2-b2d 
; 
; ----------------------------------

; ac bd c2 b2 
; -d 0  0  -c 
; 0  -a -b 0  
; -b c  d  -a 
; 
; ----------------------------------

; -c 
; -b 
; a  
; d  
; 
; ----------------------------------

The nres command

Note that the columns of w.1 minimally generate the submodule corresponding to matrix , the columns of w.2 minimally generate the syzygies on these columns, the columns of w.3 minimally generate the syzygies on w.2, etc.

The computation proceeds degree by (slanted) degree. For more details, see section 7.4.