Changeset b1645e in git
 Timestamp:
 Dec 19, 2013, 5:08:20 PM (11 years ago)
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 (u'fiekerDuVal', '117eb8c30fc9e991c4decca4832b1d19036c4c65')(u'spielwiese', 'b52fc4b2495505785981d640dcf7eb3e456778ef')
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 789d6fd59d9c7d6f129c8226b0d7ad732069ada8
 gitauthor:
 Oleksandr Motsak <motsak@mathematik.unikl.de>20131219 17:08:20+01:00
 gitcommitter:
 Oleksandr Motsak <motsak@mathematik.unikl.de>20131219 17:09:18+01:00
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Singular/LIB/schreyer.lib
r789d6f rb1645e 6 6 AUTHOR: Oleksandr Motsak <U@D>, where U={motsak}, D={mathematik.unikl.de} 7 7 KEYWORDS: Schreyer ordering; Schreyer resolution; syzygy 8 OVERVIEW: 9 @* The library contains helper procedures for computing a Schreyer resultion (cf. [SFO]) 10 for @code{derham.lib} also in noncommutative setting (cf. [MO]). 11 We call a free resulution a Schreyer resolution if any higher syzygy is a Goebner base of a previous one 12 with respect to the corresponding Schreyer ordering. 13 Schreyer resolution can be much bigger than a minimal resolution of the same module one but can be easier to construct. 14 @* Input for resolution computations is a set of vectors @code{M} in form of a module over some basering @code{R}. 15 The ring @code{R} may be noncommutative, in which case the ring ordering should be global. 16 @* These procedures produce/work with partial Schreyer resolutions of @code{(R^rank(M))/M} in form of 17 a ring (endowed with a special ring ordering that will be extended in the course of a resolution computation) 18 containing a list of modules @code{RES} and a module @code{MRES}: 19 @* The list of modules @code{RES} contains the images of maps (also called syzygies) substituting the 20 computed beginning of a Schreyer resolution, that is, each syzygy module is a Groebner Basis 21 of the previous one with respect to corresponding Schreyer induced ordering. 22 @* The list @code{RES} starts with a zero map given by @code{rank(M)} zero generators indicating that the image of 23 the first differential map is zero. The second map @code{RES[2]} is given by @code{M}, which indicates that 24 the resolution is of @code{(R^rank(M))/M} is being computed. 25 @* The module @code{MRES} is a direct sum of modules from @code{RES} and thus comprises all computed differentials. 26 @* Syzygies are shifted so that @code{gen(i)} is mapped to @code{MRES[i]} under the differential map. 27 @* Schreyer ordering extends the starting module ordering on @code{M} (defined in Singular by the basering @code{R}) 28 and is extended to higher syzygies using the following definition: 29 @* a < b if and only if (d(a) < d(b)) OR ( (d(a) = d(b) AND (comp(a) < comp(b)) ), 30 @* where @code{d(a)} is the image of a under the differential (given by @code{MRES}), 31 and @code{comp(a)} is the mod. component, for any module terms @code{a} and @code{b} from the same free module. 32 REFERENCES: 33 [SFO] Schreyer, F.O.: Die Berechnung von Syzygien mit dem verallgemeinerten Weierstrassschen Divisionssatz, 34 Master's thesis, Univ. Hamburg, 1980. 35 [MO] Motsak, O.: Noncommutative Computer Algebra with applications: Graded commutative algebra and related 36 structures in Singular with applications, Ph.D. thesis, TU Kaiserslautern, 2010 37 38 NOTE: requires the dynamic or builtin module @code{syzextra} 8 39 9 40 PROCEDURES: 10 Sres(M,len) compute Schreyer resolution of module M of maximal length len 11 Ssyz(M) compute Schreyer resolution of module M of length 1 12 Scontinue(len) extend currently active resolution by (at most) len syszygies 13 14 NOTE: requires the dynamic or builtin module: syzextra 41 Sres(M,len) compute Schreyer resolution of module M of maximal length len 42 Ssyz(M) compute Schreyer resolution of module M of length 1 43 Scontinue(len) extend currently active resolution by (at most) len syszygies 15 44 "; 16 45 … … 332 361 RETURN: nothing, instead it changes the currently active resolution 333 362 PURPOSE: extends the currently active resolution by at most len syzygies 334 NOTE:must be used within a ring returned by Sres or Ssyz363 ASSUME: must be used within a ring returned by Sres or Ssyz 335 364 EXAMPLE: example Scontinue; shows an example 336 365 " … … 362 391 "USAGE: Ssyz(module M) 363 392 RETURN: ring, containing a Schreyer resolution 364 PURPOSE: computes a Schreyer resolution of M of length 1 365 NOTE: the output is explained inSres393 PURPOSE: computes a Schreyer resolution of M of length 1 (see the library overview) 394 SEE ALSO: Sres 366 395 EXAMPLE: example Ssyz; shows an example 367 396 " … … 394 423 "USAGE: Sres(module M, int len) 395 424 RETURN: ring, containing a Schreyer resolution 396 PURPOSE: computes a Schreyer resolution of (basering^rank(M))/M with at most len syzygy modules 397 NOTE: input is a set of vectors M over a basering. The ring basering may be noncommutative. 398 @* If given len is zero then nvars(basering) + 1 is used instead. 399 @* Schreyer resolution is represented by a list of modules RES and a module MRES 400 belonging to a specially constructed ring, which is endowed with a Schreyer ordering. 401 @* The list of modules RES contains the images of maps (also called syzygies) subsituting the 402 computed beginning of a Schreyer free resolution of (baseRing^rank(M))/M. 403 @* The leading zero map RES[1] with rank(M) zero generators indicates that the image of 404 the first differential map is zero. The second map RES[2] is given by M, which indicates that 405 the resolution is of (baseRing^rank(M))/M is being computed. 406 @* The module MRES is a direct sum of modules from RES and comprises all computed differential maps. 407 @* Syzygies are shifted so that gen(i) is mapped to MRES[i] under the differential. 408 @* Schreyer ordering extends an arbitrary starting module ordeing (defined by basering) 409 and is extended to higher syzygt modules using the following definition: 410 @* a < b if and only if (d(a) < d(b)) OR ( (d(a) = d(b) AND (comp(a) < comp(b)) ), 411 @* where d(a) is the image of a under the differential (given by MRES), 412 and comp(a) is the mod. component, for any module terms a and b. 413 @* Syzygies are given by Groebner bases with respect to corresponding Schreyer orderings. 425 PURPOSE: computes a Schreyer resolution of M of length at most len (see the library overview) 426 NOTE: If given len is zero then nvars(basering) + 1 is used instead. 427 SEE ALSO: Ssyz 414 428 EXAMPLE: example Sres; shows an example 415 429 "
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