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\textbf{Universit\`a di Trieste}\\
\textbf{Corso di Studi in Matematica}\\
Laurea Magistrale\\
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{\large \textbf{Algebra Computazionale}}\\
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\textbf{a.\ a.\ 2012--13}\\
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\textbf{docente: prof.\ Alessandro Logar}
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\textbf{Ideals, modules, term order} --- 
Noetherian rings and modules. Hilbert basis theorem. Terms on 
$P=K[x_1, \dots, x_n]$ and on $P^r$. Term orders. Macaulay Basis Theorem.
Algorithm of division of polynomials. 
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\textbf{Graded rings} --- Graded rings, graded modules. Homomorphisms of 
graded rings. Shift.

\textbf{Gr\"obner basis, I} --- Definition of a $\sigma$- Gr\"obner basis.
Characterizations of Gr\"obner bases.
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\textbf{Syzygies} --- Definition of syzygies. Syzygies of monomial modules 
and their computation.
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\textbf{Gr\"obner bases, II} --- Lifting of syzygies. Normal form of
an element w.r.t.\ a Gr\"obner basis. Buchberger algorithm. Schreyer theorem
for computing syzygies. Sygygise Hilbert theorem.
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\textbf{Properties of graded modules} --- Minimal sistem of generators of 
a graded module. Nakayama lemma (for homogeneous modules). Homogeneous 
Gr\"obner bases. FFR of a graded module over $P$. Betti numbers.

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

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\textbf{Bibliography}
\begin{enumerate}
\item Martin Kreuzer, Lorenzo Robbiano. \textsl{Computational Commutative 
algebra~1} Springer-Verlag.
\item Martin Kreuzer, Lorenzo Robbiano. \textsl{Computational Commutative 
algebra~2}, Springer-Verlag.
\item David Eisenbud. \textsl{Commutative Algebra vith a view toward Algebraic 
Geometry} Springer-Verlag.
\item Bernd Sturmfels. \textsl{Computational invariant theory}, 
Springer-Verlag.
\item M.F.\ Atiyah, I.G.\ Macdonald, \textsl{Commutative algebra}.
\end{enumerate}

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