Università di Trieste Corso di Studi in Matematica Laurea Magistrale

Algebra Computazionale

a. a. 2012–13

docente: prof. Alessandro Logar

Ideals, modules, term order — Noetherian rings and modules. Hilbert basis theorem. Terms on P = K[x_{1}, …, x_{n}] and on P^{r}. Term orders. Macaulay Basis Theorem. Algorithm of division of polynomials.

Graded rings — Graded rings, graded modules. Homomorphisms of graded rings. Shift.

Gröbner basis, I — Definition of a σ- Gröbner basis. Characterizations of Gröbner bases.

Syzygies — Definition of syzygies. Syzygies of monomial modules and their computation.

Gröbner bases, II — Lifting of syzygies. Normal form of an element w.r.t. a Gröbner basis. Buchberger algorithm. Schreyer theorem for computing syzygies. Sygygise Hilbert theorem.

Properties of graded modules — Minimal sistem of generators of a graded module. Nakayama lemma (for homogeneous modules). Homogeneous Gröbner bases. FFR of a graded module over P. Betti numbers.

Hilbert function and HIlbert series for a graded module — Definition and properties of the Hilbert function. Computation of the HF via Macaulay basis theorem. Definition and properties of the Hilbert series.

Invariant polynomials — Symmetrical polynomials. Action of GL(n, ℂ) on P. Action of finite subgroups Γ of GL(n, ℂ) on P. Reynolds operator. Hilbert finiteness theorem. Molien theorem for computing the Hilbert series of P^{Γ}. Computation of the fundamental invariants of P^{Γ}.

Bibliography

1. Martin Kreuzer, Lorenzo Robbiano. Computational Commutative algebra 1 Springer-Verlag.

2. Martin Kreuzer, Lorenzo Robbiano. Computational Commutative algebra 2, Springer-Verlag.

3. David Eisenbud. Commutative Algebra vith a view toward Algebraic Geometry Springer-Verlag.

4. Bernd Sturmfels. Computational invariant theory, Springer-Verlag.

5. M.F. Atiyah, I.G. Macdonald, Commutative algebra.