🧬Homological Algebra Unit 5 – Projective and Injective Modules

Key Concepts and Definitions

• Modules generalize the notion of vector spaces by allowing scalars from a ring instead of a field
• Homomorphisms between modules preserve the module structure and are analogous to linear transformations
• Exact sequences are sequences of modules and homomorphisms where the image of each homomorphism equals the kernel of the next
• Projective modules are direct summands of free modules and satisfy a certain lifting property
• Injective modules are analogous to projective modules and satisfy a certain extension property
• Flat modules are a generalization of free modules and are characterized by the tensor product preserving exact sequences
• Resolutions are exact sequences that measure how far a module is from being projective or injective
  • Projective resolutions start with the module and have projective modules to the left
  • Injective resolutions start with the module and have injective modules to the right

Projective Modules: The Basics

• A module $P$ is projective if for every surjective homomorphism $g: B \to C$ and every homomorphism $f: P \to C$, there exists a homomorphism $h: P \to B$ such that $g \circ h = f$
• Equivalently, a module $P$ is projective if the functor $\text{Hom}(P, -)$ is exact
• Every free module is projective, as the lifting property holds by the universal property of free modules
• Every direct summand of a projective module is projective
  • If $P = A \oplus B$ and $P$ is projective, then $A$ and $B$ are also projective
• The direct sum of projective modules is projective
• Over a PID (principal ideal domain), a module is projective if and only if it is free
• Projective modules are flat, but the converse is not true in general

Injective Modules: Core Ideas

• A module $I$ is injective if for every injective homomorphism $f: A \to B$ and every homomorphism $g: A \to I$, there exists a homomorphism $h: B \to I$ such that $h \circ f = g$
• Equivalently, a module $I$ is injective if the functor $\text{Hom}(-, I)$ is exact
• Every divisible abelian group is an injective $\mathbb{Z}$-module
  • Examples include $\mathbb{Q}$ and $\mathbb{Q}/\mathbb{Z}$
• The direct product of injective modules is injective
• Over a Noetherian ring, a module is injective if and only if it is a direct summand of every module containing it
• Injective modules over a PID are divisible, but the converse is not true in general
• Baer's criterion states that a module $M$ over a ring $R$ is injective if and only if every homomorphism from an ideal $I$ of $R$ to $M$ can be extended to a homomorphism from $R$ to $M$

Characterizations and Properties

• A module $P$ is projective if and only if every short exact sequence $0 \to A \to B \to P \to 0$ splits
• A module $I$ is injective if and only if every short exact sequence $0 \to I \to A \to B \to 0$ splits
• Projective modules are flat, and over a perfect ring (e.g., a PID), the converse is also true
• Injective modules are divisible, and over a Dedekind domain, the converse is also true
• A ring $R$ is semisimple if and only if every $R$-module is projective (equivalently, injective)
  • Examples of semisimple rings include fields and finite direct products of matrix rings over fields
• A ring $R$ is quasi-Frobenius if and only if projective $R$-modules and injective $R$-modules coincide
• Over a Noetherian ring, a module is injective if and only if it is a direct sum of indecomposable injective modules

Connections to Other Algebraic Structures

• Projective modules are closely related to flat modules, which are characterized by the tensor product preserving exact sequences
• Injective modules are related to divisible modules, particularly in the context of abelian groups and modules over PIDs
• Projective and injective resolutions are used to define derived functors, such as Ext and Tor, which measure the failure of exactness of certain functors
• The projective dimension and injective dimension of a module are invariants that quantify how far the module is from being projective or injective, respectively
  • A module has projective dimension $n$ if it has a projective resolution of length $n$ and no shorter
  • Similarly, a module has injective dimension $n$ if it has an injective resolution of length $n$ and no shorter
• The global dimension of a ring is the supremum of the projective dimensions of all modules, and it measures the homological complexity of the ring
• Gorenstein rings are characterized by having finite self-injective dimension as both a left and right module over itself

Applications in Homological Algebra

• Projective and injective resolutions are used to compute derived functors, such as Ext and Tor
  • $\text{Ext}^n(A, B)$ is the $n$-th cohomology of the complex $\text{Hom}(P_\bullet, B)$, where $P_\bullet$ is a projective resolution of $A$
  • $\text{Tor}_n(A, B)$ is the $n$-th homology of the complex $A \otimes Q_\bullet$, where $Q_\bullet$ is a projective resolution of $B$
• The projective and injective dimensions of modules are used to define the global dimension of a ring
• Projective and injective modules play a crucial role in the theory of homological dimensions and the characterization of various classes of rings
  • For example, a ring is quasi-Frobenius if and only if projective and injective modules coincide
• The existence of enough projectives or injectives in a category is a key property that allows for the construction of resolutions and the computation of derived functors
• In the category of sheaves over a topological space, injective resolutions are used to define sheaf cohomology, which has applications in algebraic geometry and complex analysis

Examples and Counterexamples

• Over the ring of integers $\mathbb{Z}$, every projective module is free (and thus injective), but there exist injective modules that are not projective, such as $\mathbb{Q}$
• Over a field $k$, every module is both projective and injective, as the category of $k$-modules is semisimple
• The $\mathbb{Z}$-module $\mathbb{Q}/\mathbb{Z}$ is injective but not projective, as it is divisible but not free
• The ring of continuous functions on the unit interval, $C([0, 1])$, is an example of a ring with infinite global dimension, as there exist modules with arbitrarily large projective dimensions
• The ring of upper triangular matrices over a field is an example of a quasi-Frobenius ring that is not semisimple, as projective and injective modules coincide, but not every module is projective
• The $\mathbb{Z}$-module $\mathbb{Z}/2\mathbb{Z}$ is not flat, as tensoring with it does not preserve the injectivity of the map $2: \mathbb{Z} \to \mathbb{Z}$

Advanced Topics and Open Questions

• The finitistic dimension of a ring is the supremum of the projective dimensions of modules with finite projective dimension, and it is an open question whether this is always equal to the global dimension
• The Gorenstein symmetry conjecture states that the injective dimension of a ring as a left module over itself equals its injective dimension as a right module over itself
• The Auslander-Reiten conjecture states that if a ring has a module of finite projective dimension and a module of finite injective dimension, then the ring must be quasi-Frobenius
• The flat cover conjecture asserts that every module has a flat cover, which is a minimal flat module mapping onto it
• The Whitehead problem asks whether every abelian group with a free resolution of finite length is necessarily free
• Generalizations of projective and injective modules, such as Gorenstein projective and Gorenstein injective modules, have been studied in the context of Gorenstein homological algebra
• The study of projective and injective modules in various categories, such as the category of sheaves or the category of representations of a quiver, leads to interesting connections with other areas of mathematics, such as algebraic geometry and representation theory