Algebra 4 – May 29th – Projective and injective modules

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 R=\mathbb{Z}[\sqrt{-5}] the ideals I=(3, 1+\sqrt{-5}) and J=(3, 1-\sqrt{-5}) are projective non-free modules.
In fact, R/I\cong R/J\cong \mathbb{Z}_3 therefore the ideals are maximal and we have a short exact sequence 0\to I\cap J\to I\oplus J\to R\to 0. But I\cap J=IJ=(3), 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 R=k[x,y] where k is a field, the ideal I=(x,y) is torsion free but not flat. Consider the short exact sequence 0\to I\to R \to R/I\to 0. Then tensor with I to get I\otimes I\to I\to I/I^2. The element x\otimes y + y\otimes(-x) 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 a_{11}m_1+ a_{12}m_2=x, a_{21}m_1+ a_{22}m_2=y
and a_{11}y+ a_{21}(-x)=0 and a_{12}y+ a_{22}(-x)=0. Since k[x,y] is a UFD, we have a_{11}y=a_{21}x and (b_{11}x)y=(b_{21}y)x so b_{11}=b_{21} and similarly b_{12}=b_{22}. Plugging into the first ones b_{11}m_1x+ b_{12}m_2x=x which yields b_{11}m_1+ b_{12}m_2=1 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

Algebra 4 – May 22th – More localization

More localization

The localization is an exact functor.

We have S^{-1}A\otimes_A M\cong S^{-1}M as A-modules and S^{-1}A-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 S^{-1}A 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.

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.