Construction of Tor Functor (Projective Resolution Method) and Properties

Introduction

In homological algebra, the Tor functor plays a pivotal role in understanding the structure of modules over rings, particularly in the context of derived functors and derived categories. The Tor functor, denoted as $\text{Tor}_n(M, N)$, measures the failure of the Hom functor to preserve exact sequences, providing insight into the interactions between modules. Its construction relies on the use of projective resolutions, a foundational technique in homological algebra. This article explores the rigorous construction of the Tor functor via projective resolutions and delves into its key properties, emphasizing its significance in algebraic geometry, representation theory, and the study of derived categories.

Construction of Tor Functor (Projective Resolution Method)

The Tor functor is defined as the derived functor of the Hom functor, $\text{Hom}_R(-, -)$, applied to the category of chain complexes. To compute $\text{Tor}_n(M, N)$, one typically constructs a projective resolution of the module $M$ and evaluates the Hom functor on the resulting complex. Specifically, let $P^\bullet$ be a projective resolution of $M$, and consider the complex $\text{Hom}_R(P^\bullet, N)$. The derived functor $\text{Tor}_n(M, N)$ is then obtained as the $n$-th homology of $\text{Hom}_R(P^\bullet, N)$.

The projective resolution method ensures that the computation of $\text{Tor}_n(M, N)$ is well-defined, even when $M$ is not projective. This approach leverages the fact that projective resolutions are exact, allowing the use of the Hom functor to capture the essential homological information. For example, if $M$ is a finitely generated module over a Noetherian ring, the projective resolution can be constructed as a finite chain complex of free modules, simplifying the computation of derived functors.

The construction of $\text{Tor}_n(M, N)$ via projective resolutions is closely tied to the concept of derived functors. In this context, the Tor functor is a derived functor of the Hom functor, and its computation relies on the acyclicity of the projective resolution. This method provides a systematic way to analyze the homological properties of modules, particularly in the study of injective and projective modules, and their duals.

Properties of Tor Functor

The Tor functor exhibits several key properties that make it a central object in homological algebra.

Exactness and Functoriality

The Tor functor is exact in the category of chain complexes when applied to the projective resolution of the module $M$. This property ensures that the derived functors are well-defined and do not depend on the specific projective resolution chosen. Furthermore, the Tor functor is functorial in both arguments, meaning that $\text{Tor}_n(M, N)$ is naturally isomorphic to $\text{Tor}_n(N, M)$ under certain conditions, such as when $M$ and $N$ are projective.

Dual Relationship with Ext Functor

The Tor functor is dual to the Ext functor in the sense that $\text{Tor}_n(M, N)$ is isomorphic to $\text{Ext}_n^*(N, M)$ in the category of chain complexes. This duality reflects the symmetry between homological functors and their derived counterparts, providing a powerful tool for analyzing the structure of modules.

Long Exact Sequence in Cohomology

A fundamental property of the Tor functor is its compatibility with short exact sequences. For a short exact sequence $0 \to N \to P \to M \to 0$, the derived functors $\text{Tor}_n(N, P)$, $\text{Tor}_n(N, M)$, and $\text{Tor}_n(N, P)$ form a long exact sequence in cohomology. This sequence allows for the computation of $\text{Tor}_n(M, N)$ by analyzing the cohomology of the projective resolution, offering a systematic approach to homological computations.

Computation via Projective Resolutions

The projective resolution method simplifies the computation of $\text{Tor}_n(M, N)$ by reducing it to the evaluation of the Hom functor on the projective resolution of $