Summary of Ch. Lossen's Lectures

Lecture I
 Application of Groebner bases/syzygy computations.

-> Submodule membership problem
   * Groebner bases for submodules
   * Division with remainder
   * Compute a representation of a submodule element in terms of the original
       generators.  (Need to store syzygies.)
   * This solves the lifting problem.  Given three free modules F, G, H with
     maps f : G--->F<----H : g with in(f) contained in im(g), then there exists
     a lift h:G--->H such that the diagram commutes.
   * Radical membership problem.  Decide if g is in radical of <f1,...,fn>
        Rabinowich trick: this is equivalnet to 1 \in < I, 1-tg>

-> Projective closure
   * Compute a GB for I in a global degree order, then the homogenization of the
      GB generates the homogenization of I.

-> Elimination
   * Can use elimination to compute the intersection of two ideals
   * Used to compute quotients
   * And saturation 

-> Geometric meaning 
   * Saturation:  Variety of I : J^\infty is the closure of V(I)-V(J).
       [Caution: if variety is only closed points in the given field, 
                 this will not work.]
   * Elimination: Projection
   * Implicitization

Lecture II:   Constructive module Theory

-> Modules are represented via their presentations
   * How to represent a morphism of modules
   * Computation of Kernels and CoK
-> Modules are represented via their presentations
   * How to represent a morphism of modules
   * Computation of Kernels and CoKernels.

(Much more, FS could not stay awake..)

-> Computation of linear series on a curve.