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A.3.4 Free resolution
In SINGULAR a free resolution of a module or ideal has its own type:
resolution . It is a structure that stores all information related to
free resolutions. This allows partial computations of resolutions via
the command res . After applying res , only a pre-format of the
resolution is computed which allows to determine invariants like
Betti-numbers or homological dimension. To see the differentials
of the complex, a resolution must be converted into the type list which
yields a list of modules: the k-th module in this
list is the first syzygy-module (module of relations) of the (k-1)st module.
There are the following commands to compute a resolution:
res
- res
computes a free resolution of an ideal or module using a heuristically
chosen method.
This is the preferred method to compute free resolutions of ideals or
modules.
lres
- lres
computes a free resolution of an ideal or module with LaScala's
method. The input needs to be homogeneous.
mres
- mres
computes a minimal free resolution of an ideal or module with the syzygy
method.
sres
- sres
computes a free resolution of an ideal or module with Schreyer's
method. The input has to be a standard basis.
nres
- nres
computes a free resolution of an ideal or module with the standard basis
method.
minres
- minres
minimizes a free resolution of an ideal or module.
syz
- syz
computes the first syzygy module.
res(i,r) , lres(i,r) , sres(i,r) , mres(i,r) ,
nres(i,r) compute the first r modules of the resolution
of i, resp. the full resolution if r=0 and the basering is not a qring.
See the manual for a precise description of these commands.
Note: The command betti does not require a minimal
resolution for the minimal Betti numbers.
Now let us take a look at an example which uses resolutions: The Hilbert-Burch
theorem says that the ideal i of a reduced curve in
has a free resolution of length 2 and that i is given by the 2x2 minors
of the 2nd matrix in the resolution.
We test this for two transversal cusps in
Afterwards we compute the resolution of the ideal j of the tangent developable
of the rational normal curve in
from above.
Finally we demonstrate the use of the type resolution in connection with
the lres command.
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