Categoria: didattica algebra 4 2015-2016

Localizzazione di moduli, proprietà locali. Moduli proiettivi, prime proprietà e varie implicazioni.

Note in PDF.

Lemma di Nakayama e localizzazione di anelli e moduli.

Note in PDF.

Prodotti tensore: definizione e prime proprietò.

Note in PDF.

Updated PDF of notes

In these lectures we recall the proofs that every vector space has a basis and two vector spaces of one vector space have the same number of elements. Then we show that maps from a free module are determined by a basis, and show that two basis of a free module are equipotent (if the ring is commutative with unity) and thus define the rank. We show that free modules lift epimorphisms and if a quotient of a module is free, then it’s also a direct summand. We define the torsion of a module. If R is a PID, every finitely generated torsion free module is free. If R PID every fintely generated module splits as a sum of its torsion submodule and a free module. If R is PID, every submodule of a free module is itself free and has a lower rank.

Reference: Rotman “Advanced modern algebra” and Barbieri-Viale “Cos’è un numero” PDF of notes (password protected)