Duality between Ext and Tor functors

Introduction

In homological algebra, Ext and Tor functors play a central role in understanding the structure of module categories. These functors provide a bridge between the categorical and algebraic properties of modules, enabling the study of derived functors, exact sequences, and the behavior of modules under various operations. The duality between Ext and Tor is a profound concept that reflects the interplay between different homological constructions. While Ext and Tor are often treated as distinct tools, their relationship is deeply intertwined, particularly in the context of derived categories and dualizable objects. This article explores the foundational definitions, the nature of their duality, and the implications of this relationship in homological algebra.

Definitions of Ext and Tor Functors

The Ext and Tor functors are derived functors constructed from the Hom and tensor functors, respectively. These functors are essential in homological algebra for measuring the extent to which certain properties are preserved under exact sequences.

The Ext functor, denoted $\text{Ext}^n(-, -)$, is defined as the right-derived functor of the Hom functor $\text{Hom}(-, -)$, applied to the category of modules over a ring $R$. Specifically, for a pair of $R$-modules $A$ and $B$, $\text{Ext}^n(A, B)$ measures the failure of the Hom functor to be exact in the derived sequence. This is computed using projective resolutions of $B$, where the derived Hom functor is obtained by applying Hom to the resolution and then taking the nth term in the long exact sequence.

Similarly, the Tor functor, denoted $\text{Tor}^n(-, -)$, is the right-derived functor of the tensor functor $\otimes$, applied to the category of modules over $R$. For a pair of $R$-modules $A$ and $B$, $\text{Tor}^n(A, B)$ quantifies the failure of the tensor functor to be exact in the derived sequence. This is computed using injective resolutions of $A$, where the derived tensor functor is obtained by applying $\otimes$ to the resolution and then taking the nth term in the long exact sequence.

The duality between Ext and Tor arises when considering the dualities of the underlying categories. For example, in the case of modules over a ring $R$, the dualities between Ext and Tor are often established through the use of projective and injective resolutions, particularly when $R$ is a field or a commutative ring with certain properties.

The Duality Between Ext and Tor

The duality between Ext and Tor is a fundamental property of homological algebra, reflecting the symmetry between the two functors in certain contexts. While Ext and Tor are not isomorphic in general, they are dual to each other under specific conditions. This duality is often expressed through the isomorphism $\text{Ext}^n(A, B) \cong \text{Tor}^n(B, A)$ for certain classes of modules, particularly when $A$ and $B$ are projective or injective.

In the case of modules over a ring $R$, the duality between Ext and Tor is particularly evident when considering the dualities of projective and injective modules. For instance, if $A$ is a projective $R$-module, then $\text{Ext}^n(A, B) = 0$ for all $n \geq 1$, and similarly, if $B$ is an injective $R$-module, $\text{Tor}^n(A, B) = 0$ for all $n \geq 1$. This symmetry highlights the deep connection between the two functors.

The dualities between Ext and Tor are often formalized through the use of derived functors and the properties of the derived categories. For example, in the derived category $\mathbf{D}(R)$, the Ext and Tor functors are dual to each other under the action of the derived Hom and derived tensor functors. This duality is crucial in the study of derived functors and their applications in homological algebra.

Properties and Examples

The properties of Ext and Tor functors are closely tied to the structure of the modules and the ring $R$ over which they are defined. For instance, in the case of modules over a commutative ring $R$, the Ext and Tor functors satisfy certain exactness properties, such as the long exact sequences and the vanishing properties under projective or injective modules.

Consider the case of modules over a field $k$, where the duality between Ext and Tor becomes more explicit. For example, if $A$ and $B$ are finitely generated $k$-modules, then $\text{Ext}^n(A, B)$ and $\text{Tor}^n(A, B)$ are both zero for all $n \geq 1$, reflecting the fact that $k$-modules are projective and injective in certain cases. However, this duality is not always straightforward, as the behavior of Ext and Tor can vary significantly depending on the specific modules and rings involved.

Another example is the use of Ext and Tor in the study of module categories. For instance, the Ext functor can be used to classify