### 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)

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