Algebra 4 – May 16th – Localization

Localization

The map $A\to S^{-1}A$ is neither injective nor surjective in general. The localization at a prime ideal is a local ring. Localization of a module.

References: Atiyah-Macdonald chap 3.

Algebra 4 – May 15th – Map between rings

Map between rings

Let $f:A\to B$ be a morphism of rings. We define the extended ideal and the contracted ideal. Basic properties. Then to every B-module we associate an A-module via the restriction of coefficients, and to every A-module a B-module using the tensor product.

Let R be a ring, I an ideal and M a R-module. Show that $R/I\otimes_R M\cong M/IM$ both as R-modules and as R/I-modules.

If M is a free A-module, then $M\otimes_A B$ is a free B-module of same rank.

Show that $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$ is not isomorphic to $\mathbb{C}$ as vector spaces over the reals .

Definition of multiplicative system and of the ring of fractions.

References: Atiyah-Macdonald chap 1&2, link.

Prossime lezioni di Algebra 4

Ci sarà un piccolo cambiamento nelle lezioni di settimana prossima: venerdì 16 dalle 9:30 alle 10:30 in Aula 9 terrà lezione il Prof. Mazza per esercitazioni invece che la Prof.ssa Mantovani. Il resto della settimana resta inalterato.

Algebra 4 – April 24th – More tensor product of modules

More tensor product of modules

Let R be a ring and I and J ideals. Then $R/I\otimes R/J\cong R/(I+J)$ . Tensor product of free modules is a free module. Example of non-simple tensor.

Tensor product and sequences, the tensor product is right exact but not left exact. Tensor product and direct sums and direct products.

References: Atiyah-Macdonald chap 2, link.

Algebra 4 – April 10th – Tensor product of modules

Tensor product of modules

Definition of tensor product and basic properties. Show that $2\otimes 1$ is zero in $\mathbb{Z} \otimes \mathbb{Z}_2$ but not in $2 \mathbb{Z} \otimes \mathbb{Z}_2$ .

References: Atiyah-Macdonald chap 2, link.