(i) Prove that the covariant functor E=Extℤ^1(G, □) is right exact for every abelian group G, an

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(i) Prove that the covariant functor E=Extℤ^1(G, □) is right...

(i) Prove that the covariant functor $E=\operatorname{Ext}_{\mathbb{Z}}^{1}(G, \square)$ is right exact for every abelian group $G$, and compute its left derived functors $L_{n} E$. (ii) Prove that the contravariant functor $F=\operatorname{Ext}_{\mathbb{Z}}^{1}(\square, G)$ is right exact for every abelian group $G$, and compute its left derived functors $L_{n} F$. (See the footnote on page 370 .)

(i) Prove that the covariant functor $E=\operatorname{Ext}_{\mathbb{Z}}^{1}(G, \square)$ is right exact for every abelian group $G$, and compute its left derived functors $L_{n} E$.
(ii) Prove that the contravariant functor $F=\operatorname{Ext}_{\mathbb{Z}}^{1}(\square, G)$ is right exact for every abelian group $G$, and compute its left derived functors $L_{n} F$. (See the footnote on page 370 .)

Key Concepts

Right Exactness and Derived Functors

Right exact functors are ones that preserve the exactness at the right end of a short exact sequence (specifically, they maintain cokernels). However, many such functors are not exact and their deviation from exactness is measured by their left derived functors. These derived functors quantify the failure of a functor to be exact and are instrumental in developing a deeper understanding of its homological behavior.

Covariant and Contravariant Functors

In category theory, covariant functors preserve the direction of morphisms between objects, while contravariant functors reverse them. This difference is significant in homological algebra because the construction of derived functors and the application of resolutions may differ based on the variance. The Ext functor can be seen in both covariant and contravariant forms, which affects its exactness properties and the method of computing its derived functors.

Projective Resolution

A projective resolution is an exact sequence made up of projective objects that terminates at a given module or object in an abelian category. These resolutions are used to compute derived functors like Ext by replacing objects with projective ones so that the functor becomes easier to analyze. This construction is crucial in understanding the computation of left derived functors for right exact functors such as the ones in the problem.

Ext Functor

The Ext functor is a central notion in homological algebra that measures the classes of extensions between objects in an abelian category. It is defined as the derived functor of the Hom functor, capturing obstructions to splitting short exact sequences and classifying extensions. For example, Ext¹ takes an object and a module to the group of equivalence classes of extensions, and its derived functors provide higher extension groups.

Injective Resolution

An injective resolution is the dual concept to a projective resolution; it is an exact sequence consisting of injective objects beginning from a given object. Injective resolutions are typically employed to compute right derived functors when dealing with left exact functors. Understanding both projective and injective resolutions is important when analyzing the behavior of functors like Ext, especially for contravariant cases.

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