5.1.109 res

Procedure from library standard.lib (see standard_lib).

Syntax:

res ( ideal_expression, int_expression [, any_expression ])
res ( module_expression, int_expression [, any_expression ])

Type:

resolution

Purpose:

computes a (possibly minimal) free resolution of an ideal or module using a heuristically chosen method.
The second (int) argument (say, k) specifies the length of the resolution. If it is not positive then k is assumed to be the number of variables of the basering.
If a third argument is given, the returned resolution is minimized.

Depending on the input, the returned resolution is computed using the following methods:

quotient rings:

nres (classical method using syzygies) , see nres.

homogeneous ideals and k=0:

lres (La’Scala’s method), see lres.

not minimized resolution and (homogeneous input with k not 0, or local rings):

sres (Schreyer’s method), see sres.

all other inputs:

mres (classical method), see mres.

Note:

Accessing single elements of a resolution may require that some partial computations have to be finished and may therefore take some time.

See also betti; hres; ideal; lres; minres; module; mres; nres; resolution; sres.

Example:

  ring r=0,(x,y,z),dp;
ideal i=xz,yz,x3-y3;
def l=res(i,0); // homogeneous ideal: uses lres
l;
→  1      3      2      
→ r <--  r <--  r
→ 
→ 0      1      2      
→ resolution not minimized yet
→ 
print(betti(l), "betti"); // input to betti may be of type resolution
→            0     1     2
→ ------------------------
→     0:     1     -     -
→     1:     -     2     1
→     2:     -     1     1
→ ------------------------
→ total:     1     3     2
l[2];         // element access may take some time
→ _[1]=-x*gen(1)+y*gen(2)
→ _[2]=-x2*gen(2)+y2*gen(1)+z*gen(3)
i=i,x+1;
l=res(i,0);   // inhomogeneous ideal: uses mres
l;
→  1      3      3      1      
→ r <--  r <--  r <--  r
→ 
→ 0      1      2      3      
→ resolution not minimized yet
→ 
ring rs=0,(x,y,z),ds;
ideal i=imap(r,i);
def l=res(i,0); // local ring not minimized: uses sres
l;
→   1       1       
→ rs <--  rs
→ 
→ 0       1       
→ resolution not minimized yet
→ 
res(i,0,0);     // local ring and minimized: uses mres
→   1       1       
→ rs <--  rs
→ 
→ 0       1       
→