Give an example of an R-module B for which L0 HomR(B, □) is not naturally isomorphic to HomR(B,

step 1 In this problem, we're given a set where A is equal to A, B, T. The OHA contains the elements, A

Give an example of an R-module B for which L0 HomR(B, □) is...

Key Concepts

Natural Isomorphism

A natural isomorphism is an isomorphism between two functors that is consistent across the entire category; that is, it respects the morphisms between objects. The idea of a natural isomorphism is crucial in category theory because it ensures that the relationship between functors is canonical and does not depend on arbitrary choices. In the context of derived functors, the failure of a natural isomorphism between L? Hom_R(B, -) and Hom_R(B, -) signals an intrinsic limitation tied to the algebraic properties of the module B.

Projective Module

A projective module is one with the property that the Hom functor Hom_R(P, -) is exact. When a module B is not projective, the failure of this exactness necessitates using derived functors (like L?) to capture additional homological information. Thus, non-projective modules provide a context where the natural isomorphism between a functor and its zeroth derived functor fails.

Left Derived Functor

Left derived functors are tools in homological algebra used to systematically extend a functor that is only left exact so that its behavior on projective resolutions gives a measure of how far the functor is from being exact. The zeroth left derived functor (here noted as L?) usually recovers the original functor if the functor were exact; differences indicate where exactness fails.

Hom Functor

The Hom functor, denoted Hom_R(B, -) when considering a fixed module B, assigns to each R-module M the abelian group of R-linear maps from B to M. This functor plays a key role in many areas of algebra, and its properties such as (left/right) exactness depend critically on the nature of the module B.

R-Module

An R?module is a structure that generalizes the notion of vector spaces, but over a ring instead of a field. This concept is central in algebra and homological algebra, as many constructions (like Hom, tensor products, and derived functors) are defined on the category of R-modules.

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