1 Definition and motivation
Derived functors are systematic constructions in homological algebra that extend ordinary functors by recording the failure of exactness. When a functor does not preserve exact sequences, its derived version extracts additional information from resolutions and packages it into homology or cohomology groups. This makes derived functors a central tool for studying modules, sheaves, chain complexes, and other algebraic objects.
1.1 Exact and non-exact functors
A functor is exact if it preserves short exact sequences. Many functors of interest are only left exact or right exact: they preserve some, but not all, exactness properties. For example, taking invariants, global sections, or tensoring with a fixed module typically loses information. Derived functors measure that loss in a controlled way, producing higher-level invariants that vanish precisely when the original functor behaves exactly.
1.2 Idea of resolving objects
The basic strategy is to replace an object by a resolution built from simpler objects, such as projective or injective ones, on which the functor is easier to evaluate. After applying the functor to the resolution, one computes homology or cohomology of the resulting complex. The outcome is independent of the particular resolution chosen, so the construction gives a well-defined family of invariants.
1.3 Universal properties and characterization
Derived functors can often be characterized by universal properties. In many settings, they are the best possible extension of a functor to a homological context while remaining compatible with exact sequences and acyclic objects. This viewpoint clarifies why they are unique up to canonical isomorphism and why different concrete constructions lead to the same result.
2 Classical constructions
Classical derived functors are divided into right and left derived functors, depending on whether the original functor is left exact or right exact. The appropriate choice of injective or projective resolutions determines the direction in which higher invariants are formed.
2.1 Right derived functors
Right derived functors arise from left exact functors. Since left exactness controls kernels but not cokernels, the missing information appears in higher cohomological degrees. These constructions are especially common in sheaf theory and in the study of Hom-type functors.
2.1.1 Injective resolutions
To compute a right derived functor, one typically embeds the object into an injective resolution. Injective objects are flexible enough that the original functor can be applied term by term. The cohomology of the resulting complex then defines the derived functor groups.
2.1.2 Higher cohomology objects
The higher derived functors of a left exact functor are often interpreted as higher cohomology objects. They generalize familiar cohomology theories by describing obstructions to extending local data, lifting morphisms, or solving equations globally. Their vanishing often signals that the original functor behaves especially well on the object in question.
2.2 Left derived functors
Left derived functors are associated with right exact functors. Since right exactness preserves cokernels but not kernels, the missing structure is recovered by resolving objects using projectives or other suitably flat objects and then taking homology.
2.2.1 Projective resolutions
A projective resolution replaces an object by a chain complex of projective objects mapping onto it. Because projectives admit lifts against surjections, they simplify the evaluation of right exact functors. The homology of the image complex yields the left derived functor groups.
2.2.2 Higher homology objects
The higher groups produced in this way are often viewed as higher homology objects. They measure the extent to which a right exact functor fails to preserve injectivity of maps or kernel information. In algebra, these groups frequently detect torsion, extension phenomena, and nontrivial relations among generators.
2.3 Derived functors in abelian categories
Derived functors are most naturally formulated in abelian categories, where kernels, cokernels, and exact sequences are available. In this setting, one can define derived functors abstractly without relying on a specific category of modules. The abelian framework ensures that homological constructions behave consistently and that standard comparison arguments apply.
3 Basic properties
Derived functors satisfy a collection of structural properties that make them workable in practice. These properties explain why they fit naturally into the broader language of homological algebra.
3.1 Functoriality
Derived functors are functorial in the objects they act on, meaning that morphisms induce corresponding morphisms between derived groups. This compatibility allows calculations to respect maps, commutative diagrams, and induced structures. Functoriality is essential for comparing different resolutions and for transporting information through exact sequences.
3.2 Long exact sequences
A key feature of derived functors is that short exact sequences often produce long exact sequences after derivation. These sequences connect ordinary invariants to higher ones and provide a powerful computational tool. They also encode how local changes in one object affect the entire homological picture.
3.3 Independence of resolution
Although derived functors are computed using resolutions, the result does not depend on the particular resolution chosen, provided the resolutions are appropriate. Comparison theorems show that different choices lead to chain-homotopy equivalent outcomes. This independence is what makes derived functors intrinsic invariants rather than artifacts of a specific calculation.
3.4 Natural transformations
Natural transformations between functors induce corresponding transformations between their derived functors under suitable hypotheses. This compatibility is important when comparing related constructions, such as tensor products and Hom functors, or when passing between geometric and algebraic contexts. Naturality ensures that derived constructions remain coherent across entire categories.
4 Examples
Several standard examples illustrate the utility of derived functors. These constructions appear throughout algebra and geometry because they translate difficult extension problems into computable homological data.
4.1 Ext functors
Ext groups are derived functors of Hom. They classify extensions of modules and more generally measure the ways one object can be built from another through successive extensions. The first Ext group often describes equivalence classes of short exact sequences, while higher Ext groups encode deeper extension data.
4.2 Tor functors
Tor groups are derived functors of the tensor product. They measure the failure of tensoring to be exact and are closely tied to torsion phenomena. In module theory, Tor groups detect how modules interact under base change and frequently appear in computations involving rings, ideals, and flatness.
4.3 Sheaf cohomology
Sheaf cohomology is a geometric example of a derived functor, obtained from the global section functor on sheaves. It measures the failure of global sections to capture all local data on a space. In geometry, these cohomology groups provide invariants that reflect the shape, connectivity, and algebraic structure of the underlying space.
4.4 Derived functors in algebraic topology
In algebraic topology, derived functors appear in cohomology theories built from chain complexes and coefficient systems. They help express invariants such as cohomology with local coefficients, spectral sequence terms, and obstructions to extending maps. This perspective connects topological problems to algebraic calculations.
5 Derived categories viewpoint
The derived category formalism packages chain complexes and homological information into a setting where quasi-isomorphisms become invertible. This viewpoint gives a more flexible and conceptual framework for derived functors.
5.1 Chain complexes and homotopy categories
Chain complexes provide the raw material for derived constructions, while the homotopy category identifies maps that differ by chain homotopy. Passing from complexes to the derived category further localizes with respect to quasi-isomorphisms. This process isolates the homological content of a complex and suppresses irrelevant chain-level details.
5.2 Total derived functors
Total derived functors extend ordinary functors to the derived category. Rather than applying a functor directly to objects, one applies it to a suitable replacement complex and then interprets the result in the derived setting. This approach unifies right and left derived functors and makes composition and comparison more systematic.
5.3 Derived equivalences
Derived equivalences are equivalences between derived categories that preserve homological structure at a deeper level than ordinary equivalences of categories. They often reveal that two seemingly different mathematical objects share the same derived invariants. Such equivalences are central in modern representation theory and algebraic geometry.
6 Computation techniques
Although derived functors are conceptually abstract, they are often computed using explicit tools. These methods convert the definitions into workable algebraic formulas and comparisons.
6.1 Resolutions and spectral sequences
Resolutions remain the fundamental computational device, but spectral sequences often make complicated calculations manageable by filtering complexes and extracting successive approximations. They are especially useful when a direct computation of a derived functor is difficult. Spectral sequences can relate unknown derived groups to more accessible intermediate data.
6.2 Čech methods
Čech methods compute sheaf-theoretic derived functors by using open covers and associated combinatorial complexes. They are particularly effective for spaces with suitable coverings, where local sections can be assembled into global information. Čech complexes often give concrete representatives for cohomology classes.
6.3 Dimension shifting
Dimension shifting is a technique that relates higher derived functors to lower ones by using exact sequences and resolutions strategically. It can reduce a difficult higher-degree problem to a more familiar base case. This method is frequently used in proofs and in computations involving Ext, Tor, and cohomology.
6.4 Grothendieck spectral sequence
The Grothendieck spectral sequence addresses the derived functor of a composite functor. Under suitable hypotheses, it relates the derived functors of each component to those of the composite. This is one of the most important tools for navigating complex compositions in algebra and geometry.
7 Applications
Derived functors occur throughout modern mathematics because they convert structural questions into computable invariants. Their range of applications reflects their ability to detect hidden layers of information.
7.1 Homological algebra
In homological algebra, derived functors are foundational. They organize extension theory, exactness properties, and the study of resolutions into a unified framework. Many central results in the subject are formulated in terms of derived constructions.
7.2 Algebraic geometry
In algebraic geometry, derived functors appear in the study of sheaves, cohomology of varieties, and the behavior of morphisms between geometric objects. They help measure how local geometric information assembles into global structure. Higher direct images and cohomology groups are especially important in this area.
7.3 Representation theory
Representation theory uses derived functors to study modules over algebras and representations of groups and Lie-type structures. Ext and Tor reveal extension classes, projective dimensions, and relationships between modules. Derived methods also provide refined invariants that distinguish representations with similar ordinary character data.
7.4 Topology
In topology, derived functors support the construction and analysis of cohomology theories, fibrations, and local coefficient systems. They help describe obstructions, classify extensions, and organize the algebraic information attached to spaces. Their use in spectral sequences makes them especially effective in computations involving filtered or fibered spaces.
8 Variants and generalizations
The derived functor idea extends beyond the classical abelian setting. Modern formulations broaden its scope to encompass more general categories, categorical adjunctions, and higher structures.
8.1 Derived functors in non-abelian settings
In non-abelian contexts, the standard theory of kernels and exact sequences is not always available. Nevertheless, derived-like constructions can still be defined using homotopical or categorical replacements. These variants preserve the guiding idea of measuring deviation from exactness in a generalized setting.
8.2 Left and right Kan extensions
Kan extensions provide a categorical framework that often parallels derived functors. Under suitable hypotheses, derived functors can be viewed as homological versions of Kan extensions. This connection clarifies how universal mapping properties underlie many constructions in category theory.
8.3 Sheaves and derived categories of sheaves
The theory of sheaves benefits greatly from derived categories, where sheaf cohomology and higher direct image functors fit into a coherent formalism. Derived categories of sheaves allow one to treat local-to-global phenomena with greater flexibility. They are especially useful when pushing forward, pulling back, or comparing sheaf complexes along a morphism.
8.4 Higher and infinity-categorical derived functors
Higher category theory and infinity-categories provide a modern setting for derived constructions that retain homotopical information at all levels. In these frameworks, derived functors are often encoded by homotopy-coherent universal properties rather than by strict resolutions alone. This viewpoint connects classical homological algebra with contemporary approaches in geometry and topology.