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D.2.12 schreyer_lib
- Library:
- schreyer.lib
- Purpose:
- Schreyer resolution computations and helpers for derham.lib
- Author:
- Oleksandr Motsak <U@D>, where U={motsak}, D={mathematik.uni-kl.de}
- Overview:
- The library contains several procedures for computing a/part of Schreyer
resoltion (cf. [SFO]), and some helpers for derham.lib (which requires
resolutions over the homogenized Weyl algebra) for that purpose.
The input for any resolution computation is a set of vectors M in form of a
module over some basering R. The helpers works both in the commutative and
non-commutative setting (cf. [MO]), that is the ring R may be non-commutative,
in which case the ring ordering over it must be global. They produce/work with
partial Schreyer resolutions of (R^rank(M))/M in form of a specially constructed
ring (endowed with a special ring ordering that will be extended in the course
of a resolution computation) containing the following objects:
RES: the list of modules contains the images of maps (also called syzygy
modules) substituting the computed beginning of a Schreyer resolution, that is,
each syzygy module is given by a Groebner basis with respect to the
corresponding Schreyer ordering. RES starts with a zero map given by rank(M)
zero generators indicating that the image of the first differential map is
zero. The second map RES[2] is given by M, which indicates that the resolution
of (R^rank(M))/M is being computed.
MRES: the module is a direct sum of modules from RES and thus comprises all
computed differentials. Syzygies are shifted so that gen(i) is mapped to MRES[i]
under the differential map.
Here, we call a free resolution a Schreyer resolution if each syzygy
module is given by a Groebner basis with respect to the corresponding Schreyer
ordering. A Schreyer resolution can be much bigger than a minimal resolution of
the same module, but may be easier to construct. The Schreyer ordering
succesively extends the starting module ordering on M (defined in Singular by
the basering R) and is extended to higher syzygies using the following
definition:
a < b if and only if (d(a)<d(b)) OR ( (d(a)=d(b) AND (comp(a)<comp(b)) ),
where d(a) is the image of an under the differential (given by MRES), and
comp(a) is the module component, for any module terms a and b from the same
higher syzygy module.
- Note:
- Since most comutations require the module syzextra.so, please be make sure
to build it into Singular on Windows.
- References:
[BMSS] Burcin, E., Motsak, O., Schreyer, F.-O., Steenpass, A.:
Refined algorithms to compute syzygies, 2015 (to appear).
[SFO] Schreyer, F.O.: Die Berechnung von Syzygien mit dem verallgemeinerten
Weierstrassschen Divisionssatz, Master's thesis, Univ. Hamburg, 1980.
[MO] Motsak, O.: Non-commutative Computer Algebra with applications:
Graded commutative algebra and related structures in Singular with applications,
Ph.D. thesis, TU Kaiserslautern, 2010.
Procedures:
See also:
lres;
res;
sres;
syz.
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