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# Cohomologies.

Yoshifumi Tsuchimoto

Let be an abelian category. For any object of , the extension group is defined to be the derived functor of the hom'' functor

We note that the functor is a bifunctor''. We may thus consider the right derived functor of and that of . Fortunately, both coincide: The extension group may be calculated by using either an injective resolution of the second variable or a projective resoltuion of the first variable . See [1, Proposition 8.4,Corollary 8.5].

EXAMPLE 06.1   Let us compute the extension groups .
1. We may compute them by using an injective resolution

of .
2. We may compute them by using a free resolution

of .

EXERCISE 06.1   Compute an extension group for modules of your choice. (Please choose a non-trivial example).

In the last lecture we mentioned the notion of injective hulls. Although they are not essential part of our lecture, students may find it interesting to calculate some of the injective hulls of known modules. So we write down some definitions and results related to them.

DEFINITION 06.2   Let be an -module. An -module is called an essential extension of if every non-zero submodule of intersect non-trivially. We denote this as .

Such an essential extension is called maximal if no module properly containing E is an essential extension of M .

LEMMA 06.3   A module is injective if and only if has no proper essential extensions.

LEMMA 06.4   Let be a ring. Let be -modules. We consider a family of modules which satisfy the following properties.
• is an -submodule of which contains .
• is an essential extension of .
Then:
1. The set has a maximal element.
2. If is an injective -module, then any maximal element of is injective.

THEOREM 06.5   For any -module , there exists an injective module which contains whichis minimal among such. The module is unique up to a (non-unique) isomorphism.

DEFINITION 06.6   Such in the above theorem is called the injective hull of .

Injective hulls may then be used to obtain the minimal injective resolution'' of a module.

EXAMPLE 06.7   Let be a positive integer. The injective hull of a -module is equal to . Thus an injective resolution of is given as follows.

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2010-06-03