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D.15.17 schreyer_lib
- Library:
- schreyer.lib
- Purpose:
- Helpers for computing a Schreyer resolution in
derham.lib
- Author:
- Oleksandr Motsak <U@D>, where U={motsak}, D={mathematik.uni-kl.de}
- Overview:
The library contains helper procedures for computing a Schreyer resoltion (cf. [SFO]),
originally meant to be used by derham.lib (which requires resolutions over the homogenized Weyl algebra).
The library works both in the commutative and non-commutative setting (cf. [MO]).
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 input for the resolution computations is a set of vectors M in form of a module over some basering R .
The ring R may be non-commutative, in which case the ring ordering should be global.
These procedures produce/work with partial Schreyer resolutions of (R^rank(M))/M in form of
a ring (endowed with a special ring ordering that will be extended in the course of a resolution computation)
containing a list of modules RES and a module MRES :
The list of modules RES 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.
The list 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.
The module MRES 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.
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 a 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.
- References:
- [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
- Note:
- requires the dynamic or built-in module
syzextra
Procedures:
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