Derived FunctorsEdit
Derived functors occupy a central place in modern algebra and geometry by codifying how and why certain constructions fail to be exact. They turn an idea about a functor into a family of invariants that illuminate structure across algebra, topology, and geometry. Prototypical examples such as Ext and Tor arise by applying the general philosophy to the functors that appear most often in algebra: Hom and ⊗. The development of derived functors links the explicit mechanics of modules and sheaves to global, functorial information about spaces and groups, providing a unifying language that is at once computational and conceptually revealing.
Historically, the subject grew out of a need to organize and systematize cohomology theories. Pioneers in the field showed how to extract higher information from objects by resolving them with simpler, well-understood building blocks, such as projective or injective objects. The resulting invariants behave well with respect to exact sequences and long exact sequences, and they interact nicely with other fundamental constructions. Beyond pure algebra, derived functors connect to the language of category theory and the modern framework of derived category, which packages all higher information into a single homological setting. In practice, they give concrete computational tools for areas like group cohomology, sheaf cohomology, and algebraic topology.
This article presents the subject from a viewpoint that emphasizes rigor, clarity, and practical applicability. It covers the basic ideas, standard constructions, and the main families of derived functors, while also addressing how the topic has been debated within the mathematical community. The emphasis is on how derived functors organize and reveal obstructions to exactness, rather than on any particular school of mathematical thought.
The basic idea
Let A be an abelian category with enough projectives or injectives, and let F be a functor from A to another abelian category B. If F is exact, there is little to say: F preserves exact sequences. When F is not exact, one can ask how far it is from preserving exactness. Derived functors measure precisely that failure.
Right derived functors: If F is left exact, its higher right derived functors R^nF (for n ≥ 1) capture the obstructions to F preserving surjections and extending exact sequences. A standard way to compute them is to replace an object A ∈ A by a projective resolution P_* → A, apply F to the resolution, and take the homology of the resulting chain complex. The resulting groups R^nF(A) are functorial in A and fit into long exact sequences derived from short exact sequences in A.
Left derived functors: Dually, if F is right exact, its higher left derived functors L_nF measure the obstruction to F preserving injective properties. One computes them by replacing A with an injective resolution A → I^*, applying F, and taking homology.
For concrete algebra, the most familiar instances are:
Ext^n_R(M, N): the n-th right derived functor of Hom_R(M, −) with M fixed, or equivalently of Hom_R(−, N) in the second variable. Ext^n captures n-fold extensions of M by N and classifies ways to fit N into a chain of extensions ending with M. It is computed by taking a projective resolution P_* of M and applying Hom_R(P_*, N), then taking cohomology.
Tor_n^R(M, N): the n-th left derived functor of ⊗R with M and N as inputs. Tor^R_n(M, N) measures how far tensoring with N fails to be exact on the left, and it is computed by taking a projective resolution P* of M, forming P_* ⊗_R N, and taking homology.
These constructions are functorial in both variables and come with natural long exact sequences arising from short exact sequences. They arise in many guises across mathematics and are compatible with the standard tools of homological algebra, such as connecting homomorphisms and the six-term exact sequences that appear in cohomology theories.
A unifying perspective views derived functors as a way to extend F to a functor on the derived category, capturing all higher information at once. In this framework, F is replaced by its total derived functor, and the various R^nF or L_nF are realized as parts of a single object in the derived category. This modern language streamlines many arguments and clarifies how different functors interact.
Examples in action include cohomology theories:
Sheaf cohomology H^n(X, F) is the n-th right derived functor of the global sections functor Γ(X, −) on sheaves over a space X. This connects geometric information about X with algebraic data in the category of sheaves.
Group cohomology H^n(G, M) arises as the n-th derived functor of taking G-invariants, with M a G-module. It encodes obstructions to lifting properties from fixed points to more general G-actions.
Topological invariants, via singular or simplicial cohomology, can be interpreted through derived functors in appropriate algebraic models, linking geometric intuition to algebraic computation.
Construction and technical foundations
The standard approach to defining derived functors relies on resolutions in abelian categories. The key technical tools are:
Resolutions: A projective resolution P_* → A is an exact chain complex with projective P_i, starting from P_0 → A → 0 and continuing to the left. An injective resolution A → I^* is a cochain complex with injective I^i that captures A in degree 0.
Enough projectives/injectives: The category A must have enough projectives to build projective resolutions for all objects (for right derived functors), or enough injectives to build injective resolutions (for left derived functors). In many familiar settings, such as modules over a ring or sheaves of abelian groups on a topological space, these conditions are satisfied.
Independence of the resolution: The derived functors R^nF(A) (or L_nF(A)) are independent, up to canonical isomorphism, of the specific choice of resolution. This is what makes the construction intrinsic to the object A and the functor F, not to the particular resolution used.
Exact sequences and functoriality: A short exact sequence in A yields a long exact sequence in the corresponding derived functors, which is the cornerstone of computational techniques. This structure provides a way to propagate local information to global conclusions.
Generalized frameworks: In modern practice, many mathematicians organize these ideas via the language of derived categorys and, in some contexts, via model category structures. While these frameworks are not strictly necessary for every computation, they provide a robust, unified viewpoint and facilitate more systematic proofs.
Important derived functors
Ext and Tor are the two flagship families, but the same philosophy applies to more general F:
Ext^n_R(M, N): as described, Ext arises from the right derived functors of Hom and has a natural interpretation as equivalence classes of n-fold extensions. It is a fundamental tool in algebra, with applications ranging from module theory to algebraic geometry and representation theory.
Tor_n^R(M, N): the left derived functors of tensor product, capturing how tensoring with N fails to be exact. Tor plays a central role in flatness questions and intersection theory, among other areas.
Other derived functors RF^n and LF^n: For a general functor F between abelian categories, the higher derived functors RF^n or LF^n measure the failure of F to preserve exactness in the appropriate direction. They appear in various contexts, including cohomology of sheaves, Homological stability questions, and spectral sequence computations.
Spectral sequences: A powerful computational device that arises naturally when composing functors or filtering chain complexes. Spectral sequences relate different layers of derived functors and often converge to the invariants of interest. The Grothendieck spectral sequence, among others, connects compositions of derived functors with the derived functors of the composition, providing a bridge between disparate constructions.
Applications and computations:
In algebra, deriving Hom and tensor products yields explicit invariants like Ext and Tor, which classify extensions and interactions of modules over a ring.
In geometry, the derived functor formalism underpins sheaf cohomology and the cohomology of schemes, connecting local data to global invariants that reflect geometric structure.
In topology, derived functors translate chain-level constructions into homological invariants, linking algebraic objects to topological properties.
Construction in practice and foundations
Two standard practical routes are used to build and compute derived functors:
Projective-resolution route for right derived functors: When F is left exact, one takes a projective resolution P_* → A, applies F, and computes the cohomology of F(P_*). This yields R^nF(A).
Injective-resolution route for left derived functors: When F is right exact, one takes an injective resolution A → I^, applies F, and computes the homology of F(I^). This yields L_nF(A).
The robustness of the theory rests on the fact that different resolutions give canonically isomorphic results, ensuring that the derived functors are invariants of the original input object and the chosen functor, not of the chosen resolution.
In modern contexts, the derived category viewpoint reframes these constructions as Hom-objects in a category of complexes up to quasi-isomorphism. This perspective highlights universal properties and clarifies the relationships among different functors.
Controversies and debates
Within the mathematical community, derived functors sit at the crossroads of tradition, abstraction, and computation. Key points of discussion include:
Foundations and size issues: The use of universes and large-cardinal assumptions in some parts of the Grothendieck tradition has sparked debate about the most economical and philosophically satisfying foundations. Proponents argue these tools simplify statements and proofs, while critics worry about unnecessary ontological baggage.
Abstraction vs. concrete calculation: A perennial tension exists between highly abstract frameworks (such as derived categories and model categories) and hands-on, concrete computations. Proponents of abstraction emphasize unification and power to prove broad theorems; critics caution that excessive machinery can obscure explicit, checkable results, especially for students learning the subject.
Pedagogy and access: As with many advanced topics, there is disagreement about how early in a mathematical curriculum to introduce derived functors and their modern language. Some instructors favor concrete, example-driven development; others promote the categorical viewpoint from the outset, arguing that it mirrors the structure of contemporary research.
The role of theory in applications: Derived functors have shown wide applicability, from topology to geometry to number theory. In discussions about education and research priorities, questions arise about how to balance deep theoretical development with problem-driven applications that have clear computational payoffs.
From a practical standpoint, the consensus is that the theory provides a coherent and productive framework for understanding and computing invariants across a broad spectrum of mathematics. Critics of over-generalization argue for maintaining a strong connection to concrete computations and explicit examples, while supporters stress that the long-term payoff of the abstract formalism includes greater clarity and the discovery of new connections.
The discussion around foundational choices is not about excluding useful results but about balancing rigor, tractability, and interpretability. In the end, the derived functor formalism has proven its value by delivering precise invariants and providing powerful tools for a wide array of mathematical disciplines, even as the field continues to refine its foundational underpinnings.