D.15.10 gradedModules_lib

Library:

gradedModules.lib

Purpose:

Operations with graded modules/matrices/resolutions

Authors:

Oleksandr Motsak <U@D>, where U=motsak, D=mathematik.uni-kl.de
Hanieh Keneshlou <hkeneshlou@yahoo.com>

Overview:

The library contains several procedures for constructing and manipulating graded modules/matrices/resolutions. Basics about graded objects can be found in [DL].
Throughout this library graded objects are graded maps, that is, matrices with polynomials, together with grading weights for source and destination. Graded modules are implicitly given as coker of a graded map. Note that in special cases we may also consider submodules in S^r generated by columns of a graded polynomial matrix (or a graded map).

Note:

set assumeLevel to positive integer value in order to auto-check all assumptions. We denote the current basering by S.

References:

[DL] Decker, W., Lossen, Ch.: Computing in Algebraic Geometry, Springer, 2006

Procedures:

D.15.10.1 grobj		construct a graded object (map) given by matrix M
D.15.10.2 grtest		check whether A is a valid graded object
D.15.10.3 grdeg		compute graded degrees of columns of the map M
D.15.10.4 grview		view the graded structure of map M
D.15.10.5 grshift		shift graded module coker(M) by +d
D.15.10.6 grzero		presentation of S(0)^1
D.15.10.7 grtwist		presentation of S(d)^r
D.15.10.8 grtwists		presentation of S(v[1])+...+S(v[size(v)])
D.15.10.9 grsum		direct sum of two graded modules coker(M) + coker(N)
D.15.10.10 grpower		direct p-th power of graded module coker(M)
D.15.10.11 grtranspose		un-ordered graded transpose of map M
D.15.10.12 grgens		try to compute submodule generators of coker(M)
D.15.10.13 grpres		presentation of submodule generated by columns of F
D.15.10.14 grorder		reorder cols/rows of M for correct graded-block-structure
D.15.10.15 grtranspose1		reordered graded transpose of map M
D.15.10.16 TestGRRes		compute/order/transpose a graded resolution of ideal I
D.15.10.17 KeneshlouMatrixPresentation		build some presentation with intvec v
D.15.10.18 grsyz		syzygy of Im(M)
D.15.10.19 grres		resolution of Im(M) of length l... minimal?
D.15.10.20 grlift		graded lift, gens!
D.15.10.21 grprod		composition of graded maps (product of matrices?)
D.15.10.22 grgroebner		Groebner Basis of Im(M) as a graded object
D.15.10.23 grconcat		sum of maps into the same target module
D.15.10.24 grrndmat		generate random matrix compatible with src and dst gradings
D.15.10.25 grrndmap		generate random 0-deg homomorphism src(S) -> src(D)
D.15.10.26 grrndmap2		generate random 0-deg homomorphism dst(S) -> dst(D)
D.15.10.27 grlifting		RND! chain lifting
D.15.10.28 grlifting2		RND! chain lifting
D.15.10.29 mappingcone
D.15.10.30 grlifting3		RND! chain lifting? probably wrong one
D.15.10.31 mappingcone3
D.15.10.32 grrange		get the row-weightings
D.15.10.33 grneg		graded object given by -A
D.15.10.34 matrixpres		matrix presentation of direct sum of Omega^{a[i]}(i)