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5.1.127 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 some partial
computations 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
==>
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
==>
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