An overview of computational methods, available in algebra and algebraic geometry is presented in the form of two lectures. The basic methods, based on Gröbner bases are presented in the first lecture. In the second lecture, an algorithm is presented to find the radical of an ideal. Some possible questions that lead to computational techniques are:
- Find the kernel of a ring map, the integral closure of a domain, the blowup ring, the normal cone;
- Find the intersection of a set of ideals, the ideal quotient of two ideals, the ideal quotient of two ideals, the radical of an ideal, the associated primes of an ideal, the primary decomposition of an ideal, the Hilbert function of an homogeneous ideal, the codimension of an ideal;
- Find the finite free resolution of a module, the annihilator of a module, \(Hom(M,N),M^*,M^{**}\), the homology, Ext, Tor;
- Find the module corresponding to the normal, canonical or tangent sheaf, the image of a map corresponding to some sections of a given line bundle, the cohomology of a sheaf on projective space, secant loci, tangent developable, singular locus;
- Find the tangent cone of a local ring, the inverse system of a zero- dimensional ideal, a minimal set of generators for an ideal.
All except integral closures and primary decompositions have been implemented in Macaulay. These exceptions will be implemented in the future.