Changeset 33694c in git for Singular/LIB/schreyer.lib

Timestamp:

Dec 19, 2013, 2:34:22 PM (10 years ago)

Author:

Oleksandr Motsak <motsak@…>

Branches:

(u'spielwiese', '8e0ad00ce244dfd0756200662572aef8402f13d5')

Children:

1966e4f1761815810664e4021de885a45ad57a25

Parents:

ec896331e6b7d2d8e1e5007a0322eb8424cd90cd

git-author:

Oleksandr Motsak <motsak@mathematik.uni-kl.de>2013-12-19 14:34:22+01:00

git-committer:

Oleksandr Motsak <motsak@mathematik.uni-kl.de>2013-12-19 14:35:31+01:00

Message:

Update the documentation for schreyer.lib

File:

• Singular/LIB/schreyer.lib (modified) (4 diffs)

Singular/LIB/schreyer.lib

@@ -3 +3 @@
 category="General purpose";
 info="
-LIBRARY: schreyer.lib     Helpers for working with the Schreyer induced ordering
+LIBRARY: schreyer.lib     Helpers for computing a Schreyer resolution in derham.lib 
 AUTHOR:  Oleksandr Motsak <U@D>, where U={motsak}, D={mathematik.uni-kl.de}
 
 PROCEDURES:
- Sres(M,l)      Schreyer resolution of module M of maximal length l
- Ssyz(M)        Schreyer resolution of module M of length 1
- Scontinue(l)   continue the resolution computation by most l steps
-
-KEYWORDS:  syzygy; Schreyer induced ordering; Schreyer free resolution
-NOTE:  requires the dynamic module: syzextra
+ Sres(M,len)      compute Schreyer resolution of module M of maximal length len
+ Ssyz(M)          compute Schreyer resolution of module M of length 1
+ Scontinue(len)   extend currently active resolution by (at most) len syszygies
+
+KEYWORDS: Schreyer ordering; Schreyer resolution; syzygy
+NOTE:  requires the dynamic or built-in module: syzextra
 ";
 
@@ -329 +329 @@
 
 proc Scontinue(int l)
-"USAGE:  Scontinue(l)
-RETURN:  nothing, instead it changes RES and MRES variables in the current ring
-PURPOSE: computes further (at most l) syzygies 
-NOTE:    must be used within a ring returned by Sres or Ssyz. RES and MRES are
-         explained in Sres
+"USAGE:  Scontinue(int len)
+RETURN:  nothing, instead it changes the currently active resolution
+PURPOSE: extends the currently active resolution by at most len syzygies 
+NOTE:    must be used within a ring returned by Sres or Ssyz
 EXAMPLE: example Scontinue; shows an example
 "
@@ -361 +360 @@
 
 proc Ssyz(module M)
-"USAGE:  Ssyz(M)
-RETURN:  ring, containing a list of modules RES and a module MRES
-PURPOSE: computes the first syzygy module of M (wrt some Schreyer ordering)
-NOTE:    The output is explained in Sres
+"USAGE:  Ssyz(module M)
+RETURN:  ring, containing a Schreyer resolution
+PURPOSE: computes a Schreyer resolution of M of length 1
+NOTE:    the output is explained in Sres
 EXAMPLE: example Ssyz; shows an example
 "
@@ -393 +392 @@
 
 proc Sres(module M, int l)
-"USAGE:  Sres(M, l)
-RETURN:  ring, containing a list of modules RES and a module MRES
-PURPOSE: computes (at most l) syzygy modules of M wrt the classical Schreyer
-         induced ordering with gen(i) > gen(j) if i > j, provided both gens
-         are from the same syzygy level.
-NOTE:    RES contains the images of maps subsituting the beginning of the
-         Schreyer free resolution of baseRing^r/M, while MRES is a sum of
-         these images in a big free sum, containing all the syzygy modules.
-         The syzygy modules are shifted so that gen(i) correspons to MRES[i].
-         The leading zero module RES[0] indicates the fact that coker of the
-         first map is zero. The number of zeroes inducates the rank of input.
-NOTE:    If l == 0 then l is set to be nvars(basering) + 1
+"USAGE:  Sres(module M, int len)
+RETURN:  ring, containing a Schreyer resolution
+PURPOSE: computes a Schreyer resolution of (basering^rank(M))/M with at most len syzygy modules,
+         computed with respect to a Schreyer (induced) ordering.
+NOTE:    Input is a set of vectors M over a basering. basering may be non-commutative. 
+NOTE:    Schreyer resolution is represented by a list of modules RES and a module MRES 
+         belonging to a specially constructed ring, which is endowed with a Schreyer ordering.
+         The list of modules RES contains the images of maps (also called syzygies) subsituting the
+         computed beginning of a Schreyer free resolution of (baseRing^rank(M))/M.
+         The leading zero map RES[1] with rank(M) zero generators indicates that the image of 
+         the first differential map is zero. The second map RES[2] is given by M, which indicates that 
+         the resolution is of (baseRing^rank(M))/M is being computed.
+         The module MRES is a direct sum of modules from RES and comprises all computed differential maps.
+         Syzygies are shifted so that gen(i) is mapped to MRES[i] under the differential.
+         Syzygies are given by Groebner bases with respect to corresponding Schreyer orderings.
+NOTE:    Schreyer ordering extends an arbitrary starting module ordeing (defined by basering) 
+         and is extended to higher syzygt modules 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 mod. component, for any module terms a and b.
+NOTE:    If len == 0 then len is set to be nvars(basering) + 1
 EXAMPLE: example Sres; shows an example
 "