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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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