4.1 Injective resolutions

Injective resolutions are a key tool in homological algebra and sheaf theory. They allow us to replace objects in abelian categories with complexes of injective objects, enabling the computation of derived functors and cohomology.

These resolutions are particularly useful in categories where not every object is projective, like sheaves on topological spaces. They provide a way to study an object's properties by examining its injective resolution, which is unique up to homotopy equivalence.

Definition of injective resolutions

• Injective resolutions are a fundamental tool in homological algebra and sheaf theory that allow for the computation of derived functors and cohomology
• They provide a way to replace an object in an abelian category with a complex of injective objects, which can be used to study the original object's properties
• Injective resolutions are particularly useful in categories where not every object is projective, such as the category of sheaves on a topological space

Objects in an abelian category

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• An abelian category is an additive category with kernels, cokernels, and a compatibility condition between the two
• Objects in an abelian category include modules over a ring, sheaves on a topological space, and abelian groups
• Morphisms in an abelian category are group homomorphisms that respect the additive structure and commute with the kernels and cokernels

Injective objects

• An object $I$ in an abelian category is called injective if for every monomorphism $f: A \to B$ and every morphism $g: A \to I$, there exists a morphism $h: B \to I$ such that $h \circ f = g$
• Injective objects are dual to projective objects and play a crucial role in the construction of injective resolutions
• Examples of injective objects include divisible abelian groups and injective modules over a ring

Chain complexes

• A chain complex is a sequence of objects and morphisms in an abelian category, with the composition of consecutive morphisms being zero
• Chain complexes are used to study homological invariants such as cohomology and homology
• Injective resolutions are specific types of chain complexes where each object is injective, and the complex is exact

Exactness of injective resolutions

• An injective resolution of an object $A$ in an abelian category is a chain complex $I^\bullet$ of injective objects together with a morphism $A \to I^0$ such that the augmented complex $0 \to A \to I^0 \to I^1 \to \cdots$ is exact
• Exactness means that at each position, the kernel of the outgoing morphism is equal to the image of the incoming morphism
• The exactness of injective resolutions is crucial for their applications in derived functors and cohomology

Existence of injective resolutions

• The existence of injective resolutions is a fundamental result in homological algebra and sheaf theory
• In categories with enough injectives, every object admits an injective resolution
• The proof of existence relies on transfinite induction and the construction of injective hulls

Grothendieck's theorem

• Grothendieck's theorem states that in a Grothendieck abelian category (GAC), every object admits an injective resolution
• GACs are abelian categories with a generator and where direct limits are exact
• Examples of GACs include module categories and categories of sheaves on a topological space

Proof in categories with enough injectives

• In an abelian category with enough injectives, the existence of injective resolutions can be proved by inductively constructing the resolution
• The proof involves embedding the object into an injective object, forming the quotient, and repeating the process
• The resulting complex is an injective resolution of the original object

Construction of injective hulls

• An injective hull of an object $A$ is a minimal injective object containing $A$ as a subobject
• Injective hulls are unique up to isomorphism and can be used to construct injective resolutions
• In module categories, injective hulls can be constructed using the Baer criterion and Zorn's lemma

Properties of injective resolutions

• Injective resolutions possess several important properties that make them valuable tools in homological algebra and sheaf theory
• These properties include homotopy invariance, uniqueness up to homotopy equivalence, and functoriality
• Understanding these properties is crucial for applying injective resolutions to derive functors and compute cohomology

Homotopy invariance

• Injective resolutions are homotopy invariant, meaning that if two injective resolutions of an object are homotopy equivalent, they yield the same derived functors and cohomology
• Homotopy equivalence between chain complexes is a weaker notion than isomorphism, allowing for more flexibility in constructing resolutions
• Homotopy invariance is a key property that enables the well-definedness of derived functors

Uniqueness up to homotopy equivalence

• Any two injective resolutions of an object are homotopy equivalent
• This uniqueness property ensures that the choice of injective resolution does not affect the resulting derived functors or cohomology
• The proof of uniqueness involves constructing chain maps between the resolutions and showing that they are homotopy equivalences

Functoriality of injective resolutions

• Injective resolutions are functorial, meaning that a morphism between objects induces a morphism between their injective resolutions
• Functoriality allows for the study of derived functors as functors between derived categories
• The induced morphism between injective resolutions is unique up to homotopy, ensuring the well-definedness of the derived functor

Applications of injective resolutions

• Injective resolutions have numerous applications in homological algebra and sheaf theory
• They are essential tools for computing derived functors, cohomology, and understanding the global properties of sheaves
• Injective resolutions provide a powerful framework for studying the homological invariants of objects in abelian categories

Derived functors

• Derived functors are a way to extend a left or right exact functor to a functor between derived categories
• Injective resolutions are used to define right derived functors, such as the Ext and sheaf cohomology functors
• The derived functor is obtained by applying the original functor to an injective resolution and taking the cohomology of the resulting complex

Computation of cohomology

• Injective resolutions are a key tool for computing the cohomology of objects in abelian categories
• To compute the cohomology of an object, one first constructs an injective resolution and then applies the global section functor to obtain a complex of abelian groups
• The cohomology groups are the homology groups of this complex, which measure the deviation from exactness

Injective resolutions in sheaf cohomology

• In the category of sheaves on a topological space, injective resolutions are used to define sheaf cohomology
• Sheaf cohomology is a powerful invariant that captures global information about the sheaf and the underlying space
• Injective resolutions provide a canonical way to compute sheaf cohomology and study its properties, such as the existence of long exact sequences

Examples of injective resolutions

• Concrete examples of injective resolutions help to illustrate the concepts and techniques involved in their construction and application
• Injective resolutions can be constructed in various abelian categories, such as module categories and categories of sheaves
• Examining specific examples deepens the understanding of injective resolutions and their role in homological algebra and sheaf theory

In module categories

• In the category of modules over a ring, injective resolutions can be constructed using injective modules
• For a module $M$, an injective resolution is a complex $0 \to M \to I^0 \to I^1 \to \cdots$ where each $I^i$ is an injective module
• Examples of injective modules include the divisible abelian groups $\mathbb{Q}$ and $\mathbb{Q}/\mathbb{Z}$, and the injective hull of a module

In categories of sheaves

• In the category of sheaves on a topological space, injective resolutions are constructed using injective sheaves
• An injective sheaf is a sheaf that satisfies the sheaf-theoretic analogue of the injectivity property for modules
• Examples of injective sheaves include the sheaf of locally constant functions with values in an injective module and the sheaf of rational functions on an algebraic variety

Concrete constructions of injective resolutions

• Explicit constructions of injective resolutions often involve the use of injective hulls and direct sums
• For a module $M$, an injective resolution can be constructed by embedding $M$ into its injective hull $I^0$, forming the quotient $M^1 = I^0/M$, and repeating the process with $M^1$
• In the category of sheaves, injective resolutions can be constructed using the Godement resolution, which involves the sheafification of the direct sum of skyscraper sheaves

Comparison with projective resolutions

• Injective resolutions are often studied alongside their dual counterparts, projective resolutions
• While injective resolutions are used to compute right derived functors, projective resolutions are used for left derived functors
• Understanding the similarities and differences between injective and projective resolutions is important for their effective application in homological algebra and sheaf theory

Dual notions

• Injective and projective objects are dual notions in an abelian category
• An object $P$ is projective if for every epimorphism $f: B \to C$ and every morphism $g: P \to C$, there exists a morphism $h: P \to B$ such that $f \circ h = g$
• Projective resolutions are chain complexes of projective objects with a morphism from the original object to the complex, satisfying a universal property

Advantages vs disadvantages

• Injective resolutions have the advantage of always existing in categories with enough injectives, such as Grothendieck abelian categories
• Projective resolutions, on the other hand, may not exist in some categories, such as the category of sheaves on a general topological space
• However, projective resolutions can be easier to construct explicitly in certain cases, such as for modules over a ring

When to use injective vs projective resolutions

• The choice between using injective or projective resolutions depends on the specific problem and the functors involved
• Injective resolutions are typically used for right derived functors, such as the Ext and sheaf cohomology functors
• Projective resolutions are used for left derived functors, such as the Tor functor in module categories
• In some cases, both injective and projective resolutions can be used to compute the same invariants, providing alternative approaches to the problem