Glossary

5.3 Characterizations and examples

3 min read•august 7, 2024

Projective and injective modules are key players in homological algebra. They're like the superheroes of module theory, swooping in to solve problems and make life easier for algebraists.

This section dives into characterizing these modules and provides examples. We'll see how projective and injective modules relate to other concepts like flat modules, vector spaces, and abelian groups.

Projective and Injective Modules

Properties and Constructions of Projective and Injective Modules

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    • Smallest that maps onto a given module
    • Unique up to isomorphism
    • Can be used to measure the complexity of a module
    • Smallest containing a given module as a submodule
    • Unique up to isomorphism
    • Dual notion to the projective cover
    • Can be constructed using the

Homological Dimensions

    • Shortest length of a projective resolution of a module
    • Measures how far a module is from being projective
    • Projective modules have projective dimension 0
    • Used to define the global dimension of a ring
    • Shortest length of an injective resolution of a module
    • Measures how far a module is from being injective
    • Injective modules have injective dimension 0
    • Used to define the global dimension of a ring

Special Module Types

Flat and Cotorsion Modules

    • Module MM such that the functor −⊗RM- \otimes_R M is exact
    • Generalizes the notion of a flat ring homomorphism
    • Projective modules are flat, but the converse is not always true
    • Example: Over a principal ideal domain, a module is flat if and only if it is torsion-free
    • Module CC such that ExtR1(F,C)=0\text{Ext}^1_R(F, C) = 0 for all flat modules FF
    • Dual notion to flat modules
    • Injective modules are cotorsion, but the converse is not always true

Vector Spaces

    • Module over a field
    • Has a basis, allowing for coordinate representation of elements
    • Dimension of a vector space is the cardinality of its basis
    • Example: Rn\mathbb{R}^n is a vector space over R\mathbb{R} with the standard basis {e1,…,en}\{e_1, \ldots, e_n\}

Ring Properties

Finiteness Conditions on Rings

    • Ring in which every ideal is finitely generated
    • Equivalent to the ascending chain condition on ideals
    • Examples: Fields, principal ideal domains, and polynomial rings over Noetherian rings
    • Ring in which every descending chain of ideals stabilizes
    • Implies the ring is Noetherian
    • Examples: Fields and finite-dimensional algebras over fields

Abelian Groups

Examples and Properties

  • examples
    • Z\mathbb{Z} under addition
    • Q\mathbb{Q}, R\mathbb{R}, and C\mathbb{C} under addition
    • Z/nZ\mathbb{Z}/n\mathbb{Z} (cyclic groups) under addition
    • The multiplicative group of nonzero elements of a field
  • Abelian groups are Z\mathbb{Z}-modules
    • The ring action is given by integer multiplication
    • Allows for the application of module-theoretic concepts to abelian groups
  • are direct sums of cyclic groups
    • Classification up to isomorphism

Key Terms to Review (16)

Abelian group: An abelian group is a set equipped with an operation that combines any two elements to form a third element, satisfying four key properties: closure, associativity, identity, and invertibility, along with commutativity. The commutative property distinguishes abelian groups from general groups, meaning the order of operation does not affect the outcome. This concept is fundamental in understanding structures in algebra, particularly when discussing characterizations and examples as well as foundational axioms in topology.
Artinian Ring: An Artinian ring is a ring in which every descending chain of ideals eventually stabilizes. This means that if you keep taking ideals in a sequence where each one is contained in the previous one, you will eventually reach a point where you can't go any further. This property connects closely with other important features like the structure of modules over the ring, and it plays a significant role in understanding both finite-dimensional representations and various aspects of homological algebra.
Cotorsion module: A cotorsion module is a type of module that satisfies a specific property related to the concept of injective modules and torsion modules. Specifically, a module is cotorsion if every short exact sequence of the form $0 \to A \to B \to M \to 0$, where $M$ is a cotorsion module, splits whenever $A$ is an injective module. This property connects cotorsion modules with the behavior of extensions and how they relate to other modules, making them an important class in homological algebra.
Finitely generated abelian groups: Finitely generated abelian groups are a special class of abelian groups that can be generated by a finite set of elements. This means that every element of the group can be expressed as a finite linear combination of these generators, using coefficients from the integers. These groups play a crucial role in understanding the structure of abelian groups and can often be characterized in terms of their elements and relationships.
Flat Module: A flat module is a type of module over a ring that preserves the exactness of sequences when tensored with any other module. This means that if you have an exact sequence of modules, tensoring it with a flat module will keep it exact. Flat modules are essential in understanding projective modules, resolutions, and have numerous applications in both algebra and topology.
Fundamental theorem of finitely generated abelian groups: The fundamental theorem of finitely generated abelian groups states that every finitely generated abelian group can be expressed as a direct sum of cyclic groups. This theorem provides a clear structure to these groups, showing that they can be decomposed into a combination of finite and infinite cyclic components, making it easier to understand their properties and relationships.
Homological dimensions: Homological dimensions refer to a set of numerical invariants that provide information about the complexity of modules and their relationships to projective, injective, or flat resolutions. These dimensions, such as projective dimension, injective dimension, and flat dimension, help us understand how far a given module is from being projective or injective. They play a crucial role in determining properties of functors, particularly when discussing the Tor functor and its applications.
Injective Dimension: Injective dimension is a measure of how far a module is from being injective, specifically defined as the shortest length of an injective resolution of that module. It plays a vital role in homological algebra, connecting the notions of injective modules, projective resolutions, and derived functors, while also influencing computations involving Tor and Ext.
Injective Envelope: An injective envelope is a minimal injective module that contains a given module as a submodule. It can be thought of as the smallest injective module that 'surrounds' or 'covers' the original module, allowing for an embedding that respects the module's structure. This concept is crucial when working with injective modules and helps in understanding their role within the broader context of module theory.
Injective hull: An injective hull is the smallest injective module that contains a given module as an essential submodule. It can be thought of as a way to 'enlarge' a module so that it becomes injective, which is useful in various algebraic contexts. The concept plays an important role in understanding injective modules, providing essential properties and characterizations that relate to other structures, such as local cohomology and Cohen-Macaulay rings.
Injective Module: An injective module is a type of module that has the property that any homomorphism from a submodule can be extended to the entire module. This means that if you have a short exact sequence where one of the modules is injective, it allows for certain extensions and lifting properties that are crucial in homological algebra. The concept connects deeply with projective modules and plays a significant role in constructing projective and injective resolutions, understanding exact sequences, and utilizing the Ext functor effectively.
Noetherian Ring: A Noetherian ring is a type of ring in which every ascending chain of ideals stabilizes, meaning that any set of ideals does not have an infinite strictly increasing sequence. This property ensures that every ideal in the ring is finitely generated, which is a crucial aspect when studying algebraic structures and modules. The concept connects deeply with important notions in commutative algebra, such as prime ideals and maximal ideals, and plays a vital role in the understanding of algebraic geometry and algebraic topology.
Projective Cover: A projective cover is a special kind of projective module that serves as a minimal projective approximation of an object in a category, particularly in the context of module theory. It is a surjective morphism from a projective module onto the object, such that any other morphism from a projective module to that object factors uniquely through this cover. This concept connects deeply with the characterization of modules and their decompositions.
Projective Dimension: Projective dimension is a homological invariant that measures the complexity of a module by determining the length of the shortest projective resolution of that module. It connects deeply with various concepts in homological algebra, such as resolutions, derived functors, and cohomology theories, showcasing how modules relate to projective modules and their role in understanding other homological properties.
Projective Module: A projective module is a type of module that satisfies a lifting property with respect to homomorphisms, meaning that for any surjective homomorphism, any module homomorphism from the projective module can be lifted to the original module. This concept is crucial for understanding direct sums and the behavior of modules under exact sequences, particularly how projective modules can be used to construct resolutions and relate to the Ext functor.
Vector Space: A vector space is a mathematical structure formed by a collection of vectors, which can be added together and multiplied by scalars to satisfy specific axioms. These axioms ensure that vector addition is commutative and associative, and that there is a zero vector acting as an additive identity. The concept of vector spaces is crucial in various branches of mathematics and physics, allowing for the formulation of linear combinations and transformations.
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