# Projective and injective modules

Examples, counterexamples and relations between free, projective, flat and torsion free modules.

Projective modules over local rings are free.

In the ring the ideals and  are projective non-free modules.
In fact, therefore the ideals are maximal and we have a short exact sequence . But , and R is a projective module, so the sequence splits and I and J are summand of a free module, hence projective.

In the ring where k is a field, the ideal is torsion free but not flat. Consider the short exact sequence . Then tensor with I to get . The element is in the ker but it is not zero. For that we use the Lemma 6.4 in Eisenbud “Commutative algebra with a view toward algebraic geometry”, which states that ,
and and . Since k[x,y] is a UFD, we have and so and similarly . Plugging into the first ones which yields which means that 1 is a linear combination of elements in I which is impossible.

References: Rotman “An introduction to homological algebra”, http://en.wikipedia.org/wiki/File:Module_properties_in_commutative_algebra.svghttp://blog.jpolak.org/?p=363, http://stacks.math.columbia.edu/tag/058Z, Ex 3.25 from http://math.uga.edu/~pete/MATH8020C3.pdf

# More localization

The localization is an exact functor.

We have as A-modules and -modules.

Local properties: an A-module M is zero if and only if its localizations at prime ideals are, if and only if the localizations at the maximal ideals are.

In all ideals are extension of ideals of A. An ideal I of A extends to the whole ring if and only if it meets S.  Prime ideals of the localization are in 1-1 correspondence to prime ideals of A which do not meet S. The nilradical of the localization is the localization of the nilradical.

References: Atiyah-Macdonald chap 3.

# Map between rings

Let 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 both as R-modules and as R/I-modules.

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

Show that is not isomorphic to as vector spaces over the reals .

Definition of multiplicative system and of the ring of fractions.