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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:

D.15.17.1 Sres  compute Schreyer resolution of module M of maximal length len
D.15.17.2 Ssyz  compute Schreyer resolution of module M of length 1
D.15.17.3 Scontinue  extend currently active resolution by (at most) len syszygies


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