Definition, Construction, and Functorial Properties of Derived Functors

Introduction

Derived functors are central to homological algebra, providing a systematic way to extend ordinary functors to the derived category of chain complexes. They arise when a functor fails to preserve exactness, necessitating the construction of higher extensions to capture essential information about the original functor's behavior. In the derived category, derived functors are defined using resolutions, such as injective or projective resolutions, and are pivotal in computing derived limits, derived colimits, and other homotopy-theoretic constructs. This article explores the foundational concepts of derived functors, their construction through resolutions, and their functorial properties, emphasizing their role in modern algebraic topology and sheaf theory.

Definition

A derived functor is a functor defined on the derived category of chain complexes, which extends a given functor to the category of chain complexes while accounting for loss of exactness. Given a functor $ F: \mathcal{A} \rightarrow \mathcal{B} $ between abelian categories, the derived functor $ \mathbf{R}F $ is constructed by resolving the domain object $ A \in \mathcal{A} $ with a complex $ \mathbf{C}^{\bullet} $ in $ \mathcal{A} $, and then applying $ F $ to $ \mathbf{C}^{\bullet} $. The derived functor $ \mathbf{R}F $ is defined as the derived functor of $ F $ with respect to the resolution $ \mathbf{C}^{\bullet} $.

The left derived functor $ \mathbf{R}^n F $ is computed as the nth derived functor of $ F $ with respect to the resolution, while the right derived functor $ \mathbf{R}^n F $ is defined similarly using a different resolution. These derived functors are essential in homological algebra, as they allow the study of functors that are not exact, such as the derived functors of the Hom functor or the derived functors of the tensor product.

Construction

The construction of derived functors relies on the use of resolutions, which are complexes of objects in the domain category that approximate the original objects. For example, the derived functor of a functor $ F: \mathcal{A} \rightarrow \mathcal{B} $ is often constructed using an injective resolution of the domain object, as injective resolutions are well-suited for computing derived functors due to their universal property.

In detail, the left derived functor $ \mathbf{R}^n F $ is defined as the nth derived functor of $ F $ with respect to a resolution $ \mathbf{C}^{\bullet} $. This is computed by taking the nth homology of the complex $ F(\mathbf{C}^{\bullet}) $, which is isomorphic to the derived functor $ \mathbf{R}^n F $. Similarly, the right derived functor $ \mathbf{R}^n F $ is defined using a projective resolution of the domain object, as projective resolutions are effective for computing derived functors in the context of derived categories.

The derived functors are not unique, as different resolutions can yield different derived functors. However, they are isomorphic in certain cases, such as when the original functor is exact. For instance, the derived functors of the Hom functor $ \mathbf{R}^n \operatorname{Hom} $ are isomorphic to the Hom functor of the complexes $ \mathbf{C}^{\bullet} $ and $ \mathbf{D}^{\bullet} $, provided the resolutions are compatible. This ensures that derived functors capture the essential homotopy-theoretic information of the original functor.

Functorial Properties

Derived functors exhibit strong functorial properties, including compatibility with other functors and natural transformations. For instance, if $ \mathbf{G} $ is another functor from the derived category, then the derived functors $ \mathbf{R}^n \mathbf{G} $ are compatible with the derived functors $ \mathbf{R}^m \mathbf{F} $, ensuring that the derived functors of composite functors are isomorphic to the derived functors of the individual functors.

Moreover, derived functors preserve exactness in certain contexts. For example, if $ F $ is an exact functor, then the derived functors $ \mathbf{R}^n F $ are exact for all $ n \geq 0 $. However, when $ F $ is not exact, the derived functors may not preserve exactness, necessitating the use of resolutions to capture the full homotopy-theoretic behavior.

Natural transformations between derived functors are also well-defined. If $ \Phi: \mathbf{R}^n F \rightarrow \mathbf{R}^m G $ is a natural transformation, then it must satisfy certain compatibility conditions with the resolutions used to compute the derived functors. These properties ensure that derived functors are consistent across different resolutions and functors, making them indispensable tools