Projective resolution

Introduction

A projective resolution is a fundamental concept in homological algebra, a branch of mathematics that studies homology in a general algebraic setting. Projective resolutions are used to compute derived functors, such as Ext and Tor, which are central to many areas of algebra and topology. This article delves into the detailed structure, properties, and applications of projective resolutions.

Definition and Basic Properties

A projective resolution of a module \( M \) over a ring \( R \) is an exact sequence of \( R \)-modules: \[ \cdots \to P_2 \to P_1 \to P_0 \to M \to 0 \] where each \( P_i \) is a projective module. The sequence is exact, meaning that the image of each map is equal to the kernel of the next. The modules \( P_i \) are chosen to be projective to facilitate the lifting of homomorphisms, which is a key property in homological algebra.

Projective modules are direct summands of free modules, and they have the property that any surjective homomorphism onto them splits. This property ensures that projective resolutions can be constructed for any module, although the length and complexity of the resolution can vary.

Construction of Projective Resolutions

The construction of a projective resolution typically involves the following steps: 1. **Start with the module \( M \)**: Identify a surjective homomorphism from a projective module \( P_0 \) to \( M \). 2. **Lift to the next level**: Find a projective module \( P_1 \) and a surjective homomorphism from \( P_1 \) to the kernel of the previous map. 3. **Iterate**: Continue this process, constructing a sequence of projective modules and homomorphisms.

For example, if \( M \) is a finitely generated module over a Noetherian ring, one can often use free modules (which are projective) to construct the resolution.

Applications in Homological Algebra

Projective resolutions are used to define and compute derived functors, which measure the failure of certain functors to be exact. Two of the most important derived functors are: - **Ext**: Measures extensions of modules. - **Tor**: Measures the torsion product of modules.

These functors have applications in various areas of mathematics, including algebraic topology, where they are used to study homotopy groups and cohomology.

Examples and Computations

Consider the module \( \mathbb{Z}/2\mathbb{Z} \) over \( \mathbb{Z} \). A projective resolution of \( \mathbb{Z}/2\mathbb{Z} \) is given by: \[ \cdots \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \xrightarrow{2} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0 \] Here, each \( \mathbb{Z} \) is a free (and hence projective) module, and the maps are multiplication by 2.

Properties of Projective Resolutions

1. 1. Length of a Projective Resolution

The length of a projective resolution of a module \( M \) is the smallest integer \( n \) such that \( P_i = 0 \) for all \( i > n \). If no such \( n \) exists, the module is said to have infinite projective dimension.

1. 1. Projective Dimension

The projective dimension of a module \( M \) is the length of its shortest projective resolution. Modules with finite projective dimension are of particular interest in homological algebra and commutative algebra.