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D.15.14 gradedModules_lib

Operations with graded modules/matrices/resolutions
Oleksandr Motsak <U@D>, where U=motsak, D=mathematik.uni-kl.de
Hanieh Keneshlou <hkeneshlou@yahoo.com>

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).

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

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


D.15.14.1 grobj  construct a graded object (map) given by matrix M
D.15.14.2 grtest  check whether A is a valid graded object
D.15.14.3 grdeg  compute graded degrees of columns of the map M
D.15.14.4 grview  view the graded structure of map M
D.15.14.5 grshift  shift graded module coker(M) by +d
D.15.14.6 grzero  presentation of S(0)^1
D.15.14.7 grtwist  presentation of S(d)^r
D.15.14.8 grtwists  presentation of S(v[1])+...+S(v[size(v)])
D.15.14.9 grsum  direct sum of two graded modules coker(M) + coker(N)
D.15.14.10 grpower  direct p-th power of graded module coker(M)
D.15.14.11 grtranspose  un-ordered graded transpose of map M
D.15.14.12 grgens  try to compute submodule generators of coker(M)
D.15.14.13 grpres  presentation of submodule generated by columns of F
D.15.14.14 grorder  reorder cols/rows of M for correct graded-block-structure
D.15.14.15 grtranspose1  reordered graded transpose of map M
D.15.14.16 TestGRRes  compute/order/transpose a graded resolution of ideal I
D.15.14.17 KeneshlouMatrixPresentation  build some presentation with intvec v
D.15.14.18 grsyz  syzygy of Im(M)
D.15.14.19 grres  resolution of Im(M) of length l... minimal?
D.15.14.20 grlift  graded lift, gens!
D.15.14.21 grprod  composition of graded maps (product of matrices?)
D.15.14.22 grgroebner  Groebner Basis of Im(M) as a graded object
D.15.14.23 grconcat  sum of maps into the same target module
D.15.14.24 grrndmat  generate random matrix compatible with src and dst gradings
D.15.14.25 grrndmap  generate random 0-deg homomorphism src(S) -> src(D)
D.15.14.26 grrndmap2  generate random 0-deg homomorphism dst(S) -> dst(D)
D.15.14.27 grlifting  RND! chain lifting
D.15.14.28 grlifting2  RND! chain lifting
D.15.14.29 mappingcone  
D.15.14.30 grlifting3  RND! chain lifting? probably wrong one
D.15.14.31 mappingcone3  
D.15.14.32 grrange  get the row-weightings
D.15.14.33 grneg  graded object given by -A
D.15.14.34 matrixpres  matrix presentation of direct sum of Omega^{a[i]}(i)