Left and Right Exact Functors

Introduction

In homological algebra, exact functors play a pivotal role in understanding the structure of modules and their relationships. A functor $ F: \mathcal{A} \to \mathcal{B} $ is said to be exact if it preserves short exact sequences. The distinction between left exact and right exact functors arises from their behavior with respect to the first and second terms of a short exact sequence, respectively. These concepts are foundational in the study of derived functors, projective and injective resolutions, and the formation of derived categories. This article explores the definitions, properties, and applications of left and right exact functors, emphasizing their significance in modern algebraic topology and homological theory.

Definition of Exact Functors

A functor $ F: \mathcal{A} \to \mathcal{B} $ is exact if it preserves short exact sequences. Specifically, for any short exact sequence in $ \mathcal{A} $:
$$ 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0, $$
the corresponding sequence in $ \mathcal{B} $:
$$ 0 \to F(A) \xrightarrow{F(f)} F(B) \xrightarrow{F(g)} F(C) \to 0, $$
is also exact. Exact functors are essential in preserving the structural properties of modules and their derived functors, enabling the analysis of cohomological phenomena.

Left Exactness

A functor $ F $ is left exact if it preserves the exactness of sequences up to the first term. That is, for any short exact sequence:
$$ 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0, $$
the sequence:
$$ 0 \to F(A) \xrightarrow{F(f)} F(B) \xrightarrow{F(g)} F(C) \to 0, $$
is exact. This property ensures that the image of the first term is preserved, and the kernel of the first map coincides with the kernel of the second map. Left exactness is crucial in the study of derived functors, as it guarantees that the functor $ F $ can be used to construct derived functors from the original functor.

Right Exactness

Conversely, a functor $ F $ is right exact if it preserves the exactness of sequences up to the second term. For any short exact sequence:
$$ 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0, $$
the sequence:
$$ 0 \to F(A) \xrightarrow{F(f)} F(B) \xrightarrow{F(g)} F(C) \to 0, $$
is exact. This property ensures that the image of the second term is preserved, and the cokernel of the second map coincides with the cokernel of the first map. Right exactness is vital in the study of injective resolutions and the formation of derived functors, as it allows for the construction of resolutions that capture higher cohomological information.

Properties of Left and Right Exact Functors

The relationship between left and right exactness is dual. The dual of a left exact functor is a right exact functor, and vice versa. This duality is a cornerstone of homological algebra, enabling the translation of properties between functors and their duals. Additionally, the preservation of exactness in left or right exact functors ensures that the functors are compatible with the structure of the underlying categories.

For example, the functor $ F = \text{Hom}(-, -) $ is left exact, as it preserves the exactness of sequences up to the first term. Similarly, the functor $ F = \text{Tor}(-, -) $ is right exact, as it preserves the exactness of sequences up to the second term. These examples illustrate the practical implications of left and right exactness in specific contexts.

Applications in Homological Algebra

Left and right exact functors are indispensable in the study of derived functors, which are used to compute cohomology groups of a complex. The derived functor $ R^i F $ of a left exact functor $ F $ is defined as the right derived functor, capturing the failure of $ F $ to be exact. Similarly, the derived functor $ R^i F $ of a right exact functor $ F $ is defined as the left derived functor. These derived functors are central to the study of sheaf cohomology, derived categories, and the resolution of complexes.