Fact-checked by Grok 2 weeks ago

Derived functor

In homological algebra, derived functors provide a way to extend additive functors between abelian categories by associating to each such functor a sequence of derived functors that quantify the functor's deviation from exactness.^[1]^[2] Left derived functors L_i F are constructed using projective resolutions of the argument, while right derived functors R^i F employ injective resolutions, with the zeroth derived functor coinciding with the original.^[1]^[3] The construction of derived functors relies on the existence of enough projectives or injectives in the domain category, ensuring that resolutions can approximate objects sufficiently for homology computations.^[2] For a left exact functor F, the right derived functor R^i F(A) is defined as the i-th cohomology group of the complex obtained by applying F to an injective resolution of A, and this value is independent of the choice of resolution.^[1] Similarly, for right exact functors, left derived functors arise from projective resolutions via homology.^[3] If F is exact, all higher derived functors vanish for i > 0.^[2] Derived functors satisfy key homological properties, including the formation of long exact sequences from short exact sequences in the domain category.^[1]^[2] Prominent examples include the Ext functors, given by R^i \Hom_R(-, N), which measure extensions in module categories, and the Tor functors, L_i (M \otimes_R -), which detect torsion in tensor products.^[1] These constructions underpin much of homological algebra, facilitating computations in algebraic topology, commutative algebra, and beyond.^[3]

Background Concepts

Abelian categories

An abelian category is an additive category \mathcal{A} in which every morphism admits a kernel and a cokernel, and the canonical morphism from the coimage of any morphism to its image is an isomorphism.^[4] As an additive category, \mathcal{A} possesses a zero object, which serves both as initial and terminal object, and admits finite biproducts, meaning finite direct sums and products coincide and are denoted by \oplus.^[5] These properties ensure that \mathcal{A} is enriched over the category of abelian groups, so the Hom-sets \operatorname{Hom}_{\mathcal{A}}(A, B) form abelian groups for objects A, B \in \mathcal{A}, with composition distributing over addition.^[4] In an abelian category, subobjects of an object A are represented by kernels of morphisms out of A, while quotient objects are given by cokernels of morphisms into A; monomorphisms coincide with kernels and epimorphisms with cokernels.^[5] This structure allows for a precise notion of subobjects and quotients, where a subobject corresponds to an equivalence class of monomorphisms into A with the same image, and quotients are formed by identifying elements via the relation defined by the kernel.^[6] Abelian categories form the foundational setting for homological algebra, enabling the study of exact sequences and derived functors in a general categorical framework that abstracts properties of modules and sheaves.^[6] In such categories, exact sequences are defined as those where the image of each morphism equals the kernel of the subsequent one.^[5] Prominent examples include the category \mathbf{Ab} of abelian groups, where objects are abelian groups and morphisms are group homomorphisms, which is abelian with kernels as normal subgroups and cokernels as quotient groups.^[4] Similarly, for a ring R, the category \mathbf{Mod}_R of left R-modules with R-linear maps is abelian, generalizing the structure of vector spaces or abelian groups.^[5] Another key example is the category of sheaves of abelian groups on a topological space X, denoted \mathbf{Sh}(\mathbf{Ab})_X, where kernels and cokernels are computed sheaf-theoretically, preserving the abelian structure.^[7]

Exact functors and chain complexes

In an abelian category \mathcal{A}, a sequence of morphisms

\cdots \to A_{n-1} \xrightarrow{f_{n-1}} A_n \xrightarrow{f_n} A_{n+1} \to \cdots

is called exact at A_n if the image of f_{n-1} equals the kernel of f_n, that is, \operatorname{im}(f_{n-1}) = \ker(f_n) as subobjects of A_n.^[8] A longer sequence is exact if it is exact at every position.^[8] A short exact sequence is a sequence of the form $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0that is [exact](/page/Exact_sequence) atA, B, and C; this implies fis the kernel ofgandgis the cokernel off$.^[9] An additive functor F: \mathcal{A} \to \mathcal{B} between abelian categories is left exact if it preserves finite limits, equivalently, if whenever $0 \to A \to B \to C

is [exact](/page/Exact_sequence), then &#36;0 \to F(A) \to F(B) \to F(C)

is exact (preserving kernels).^[8] It is right exact if it preserves finite colimits, equivalently, if A \to B \to C \to 0 exact implies F(A) \to F(B) \to F(C) \to 0 exact (preserving cokernels).^[8] The functor F is exact if it is both left and right exact, meaning it preserves all short exact sequences.^[9] A chain complex C_\bullet in an abelian category is a sequence of objects and morphisms (C_n, d_n) for n \in \mathbb{Z}, where each d_n: C_n \to C_{n-1} is a morphism satisfying d_{n-1} \circ d_n = 0 for all n (the differentials compose to zero).^[8] The homology groups of C_\bullet are defined as

H_n(C_\bullet) = \frac{\ker(d_n)}{\operatorname{im}(d_{n+1})}

for each n, measuring the failure of exactness at C_n.^[8] A cochain complex C^\bullet is analogous but with differentials d^n: C^n \to C^{n+1} increasing the index, again satisfying d^{n+1} \circ d^n = 0.^[9] Its cohomology groups are H^n(C^\bullet) = \ker(d^n)/\operatorname{im}(d^{n-1}).^[9] Homological indexing for chain complexes uses decreasing indices (e.g., \cdots \to C_1 \to C_0 \to C_{-1} \to \cdots), with homology H_n in degree n, while cohomological indexing for cochain complexes uses increasing indices (e.g., \cdots \to C^{-1} \to C^0 \to C^1 \to \cdots), with cohomology H^n in degree n.^[8]

Motivation

Limitations of ordinary functors

Ordinary functors in homological algebra, particularly those that are exact, preserve the exactness of short exact sequences, but many fundamental functors are only left exact or right exact, failing to capture the full homological structure of chain complexes. This inadequacy becomes evident when applying such functors to exact sequences, where they do not preserve exactness throughout, leading to a loss of information about higher-order relations in the objects involved.^[10] A classic example is the covariant Hom functor \operatorname{Hom}_R(A, -), which is left exact but not right exact. Consider the short exact sequence of R-modules $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0. Applying \operatorname{Hom}_R(\mathbb{Z}/2\mathbb{Z}, -) yields the sequence $0 \to \operatorname{Hom}_R(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}) \to \operatorname{Hom}_R(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}) \to \operatorname{Hom}_R(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}/2\mathbb{Z}), which simplifies to $0 \to 0 \to 0 \to \mathbb{Z}/2\mathbb{Z}. Here, \operatorname{Hom}_R(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}) = 0 but \operatorname{Hom}_R(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}/2\mathbb{Z}) = \mathbb{Z}/2\mathbb{Z} \neq 0, so exactness fails at the right end. This demonstrates that left exact functors like Hom cannot fully reflect the cokernel structure in exact sequences.^[8]^[10] Similarly, the covariant tensor functor - \otimes_R M is right exact but not left exact. For the same short exact sequence $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0 and M = \mathbb{Z}/2\mathbb{Z}, tensoring gives \mathbb{Z}/2\mathbb{Z} \otimes_R \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \otimes_R \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \otimes_R \mathbb{Z}/2\mathbb{Z} \to 0, or \mathbb{Z}/2\mathbb{Z} \xrightarrow{0} \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0, which is not exact at the first nonzero term since the kernel is \mathbb{Z}/2\mathbb{Z} but the image is 0. This failure is quantified by the first left derived functor, where \operatorname{Tor}_1^R(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}/2\mathbb{Z}) = \mathbb{Z}/2\mathbb{Z} \neq 0. Thus, right exact functors like tensor miss kernel information in exact sequences.^[8]^[10] These examples illustrate higher-order obstructions in exact sequences, where ordinary functors detect only the zeroth-order approximation of homological data, necessitating a hierarchy of derived functors to account for successive deviations from exactness. In early 20th-century algebraic topology, such limitations surfaced in homology computations, as Emmy Noether's 1925 shift from Betti numbers to homology groups exposed the need for more refined algebraic tools to handle inconsistencies across different homology theories.^[11]

Need for higher-order approximations

Ordinary functors between abelian categories often fail to preserve exactness, particularly when applied to short exact sequences, leading to defects in the resulting sequences. Derived functors address this by providing a sequence of higher-order functors, denoted L_i(F) for left derived functors or R^i(F) for right derived functors, where i \geq 0, that systematically capture these exactness failures. The zeroth derived functor L_0(F) coincides with the original functor F up to natural isomorphism when F is right exact, while R^0(F) coincides with F when F is left exact; the higher ones L_i(F) or R^i(F) for i > 0 encode "next-order" corrections, revealing deeper homological information about the objects involved.^[12] The universal property of derived functors positions them as initial objects in the category of certain functorial systems equipped with natural transformations that restore exactness on appropriate resolutions. Specifically, for a right derived functor R^\bullet F, any other cohomological \delta-functor G^\bullet extending F (i.e., with G^0 \cong F) admits a unique natural transformation from R^\bullet F to G^\bullet compatible with the connecting morphisms. This universality ensures that derived functors provide a canonical, minimal extension of F that measures its deviation from exactness in a functorial manner.^[13] Derived functors form a special class of \delta-functors, which are sequences of additive functors connected by natural transformations \delta^n: G^n(C) \to G^{n+1}(A) for short exact sequences $0 \to A \to B \to C \to 0, producing long exact sequences and satisfying compatibility with morphisms of exact sequences. A \delta-functor is half-exact if its zeroth component is left exact, connected if the transformations form long exact sequences, and effaceable if higher functors vanish on injective objects; derived functors are precisely the universal effaceable \delta-functors. This framework previews how derived functors generalize and unify such systems.^[14] Philosophically, derived functors encode homological invariants—such as module dimensions or topological connectivity—in a fully functorial way, transforming ad hoc computations into systematic tools for studying algebraic and geometric structures. By resolving objects via chain complexes, they approximate non-exact functors to yield exact ones on the derived level, thereby revealing intrinsic properties independent of choices.^[12]

Construction

Projective resolutions for left derived functors

In an abelian category \mathcal{A}, an object P is called projective if the covariant Hom-functor \Hom_{\mathcal{A}}(P, -): \mathcal{A} \to \Ab is exact, meaning it preserves exact sequences.^[15] Equivalently, for every epimorphism A \twoheadrightarrow B in \mathcal{A} and every morphism P \to B, there exists a lift P \to A making the diagram commute.^[15] Projective objects play a central role in homological algebra, particularly in resolving objects to compute derived functors.^[8] To construct the left derived functors L_i F of an additive covariant functor F: \mathcal{A} \to \mathcal{B} between abelian categories, where \mathcal{A} has enough projectives (meaning every object admits a surjection from a projective), begin with an object A \in \mathcal{A}.^[16] A projective resolution of A is an exact sequence

\cdots \to P_1 \to P_0 \to A \to 0

in \mathcal{A}, where each P_i is projective and the sequence is exact at each P_i for i \geq 0.^[17] Such resolutions exist under the sufficient projectives assumption and can be constructed inductively: start with a surjection P_0 \twoheadrightarrow A from a projective P_0, then resolve the kernel similarly, and continue.^[16] The deleted (or augmented) resolution is the chain complex P_\bullet: \cdots \to P_1 \to P_0 \to 0, obtained by removing A. Apply F to obtain the complex F(P_\bullet) in \mathcal{B}, and define

L_i F(A) = H_i(F(P_\bullet)),

the i-th homology group of F(P_\bullet).^[18] This construction is independent of the choice of projective resolution: if P_\bullet \to A and P'_\bullet \to A are two resolutions, there exists a chain map between them inducing quasi-isomorphisms after applying F, yielding isomorphic homology.^[8] Normalization ensures the resolution is acyclic in positive degrees when F is left exact, with H_i(F(P_\bullet)) = 0 for i > 0, so L_i F(A) = 0 for i > 0 and L_0 F(A) \cong F(A).^[18] In general, L_0 F \cong F, while the higher L_i F (for i > 0) measure the failure of F to be left exact.^[16] A canonical example arises with the tensor product functor F = - \otimes_R M on modules over a ring R, which is right exact.^[19] Taking a projective resolution P_\bullet \to N of an R-module N and applying F yields H_i(P_\bullet \otimes_R M) = \Tor_i^R(N, M), the i-th Tor functor, which vanishes for i > 0 if N (or M) is flat and captures torsion phenomena otherwise.^[19] For instance, if R = \mathbb{Z} and N = \mathbb{Z}/n\mathbb{Z}, a free resolution $0 \to \mathbb{Z} \xrightarrow{\times n} \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} \to 0 gives \Tor_i^\mathbb{Z}(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z}/m\mathbb{Z}) \cong \mathbb{Z}/\gcd(n,m)\mathbb{Z} for i=1 and 0 otherwise.^[8]

Injective resolutions for right derived functors

In an abelian category \mathcal{A} with enough injective objects, an object I is injective if the contravariant Hom-functor \Hom_{\mathcal{A}}(-, I): \mathcal{A}^{\mathrm{op}} \to \mathrm{Ab} is exact.^[8] For a left exact additive functor F: \mathcal{A} \to \mathcal{B} between abelian categories, where \mathcal{A} has enough injectives, the right derived functors R^i F are constructed using injective resolutions. Given an object A \in \mathcal{A}, an injective resolution is an exact sequence $0 \to A \xrightarrow{\epsilon} I^0 \to I^1 \to \cdots with each I^i injective; the deleted cochain complex is then $0 \to I^0 \to I^1 \to \cdots, and R^i F(A) := H^i(F(I^\bullet)), the i-th cohomology group of the cochain complex F(I^\bullet).^[8] This yields the dual normalization to projective resolutions for left derived functors: the cohomology satisfies H^i(F(I^\bullet)) = 0 for i > 0 if F is right exact, and the construction is independent of the choice of resolution since any two injective resolutions of A become chain homotopy equivalent after augmentation.^[8] In particular, R^0 F \cong F, with the natural isomorphism induced by the augmentation A \to I^0, while the higher right derived functors R^i F for i > 0 measure the failure of F to be right exact.^[8] Unlike left derived functors, which use projective resolutions to approximate right exact functors in homology-like settings, right derived functors via injective resolutions are suited to cohomology-like theories.^[8]

Basic Properties

Uniqueness up to isomorphism

The uniqueness of derived functors ensures that their values are well-defined, independent of the choice of resolution used in their construction. For a right exact functor F: \mathcal{A} \to \mathcal{B} between abelian categories, where \mathcal{A} has enough projectives, the left derived functors L_i F are computed by taking the homology of F applied to a projective resolution P_\bullet \to A of an object A \in \mathcal{A}. If P_\bullet \to A and P'_\bullet \to A are two such resolutions, the comparison theorem guarantees the existence of a chain map \phi: P_\bullet \to P'_\bullet, unique up to chain homotopy, that is the identity on A. This map induces a quasi-isomorphism F(P_\bullet) \to F(P'_\bullet), yielding a natural isomorphism H_i(F(P_\bullet)) \cong H_i(F(P'_\bullet)) for all i, so L_i F(A) is independent of the resolution choice.^[20] A key ingredient in establishing this independence is the notion of F-acyclic complexes. A chain complex C_\bullet in \mathcal{A} is said to be F-acyclic if H_i(F(C_\bullet)) = 0 for all i > 0; projective objects are always F-acyclic for any additive F, since F applied to the trivial resolution yields no higher homology. More generally, the derived functor L_i F(A) can be computed using any F-acyclic resolution of A, not just projective ones, as long as the resolution is a quasi-isomorphism to

A{{grok:render&&&type=render_inline_citation&&&citation_id=0&&&citation_type=wikipedia}}

. This flexibility relies on the fact that quasi-isomorphisms between F-acyclic complexes induce isomorphisms on the derived functors, ensuring canonical representatives via the triangulated structure of the derived category. In the specific case of the right derived functors of the Hom functor, such as \operatorname{Ext}^n_{\mathcal{A}}(A, -), uniqueness follows from their representability and the universal property of \delta-functors. These functors arise as Yoneda extensions: an element of \operatorname{Ext}^n_{\mathcal{A}}(A, B) corresponds to an equivalence class of n-fold extensions of B by A, up to congruence under the Yoneda product. Yoneda's lemma implies that the representing object for \operatorname{Ext}^n_{\mathcal{A}}(A, -) is unique up to natural isomorphism, as it embeds the category into the functor category via hom-sets, ensuring that any two such derived functors are naturally isomorphic. This representability confirms that \operatorname{Ext}^n is the universal cohomological \delta-functor satisfying the required exactness properties.^[21]

Long exact sequences

One of the most fundamental properties of derived functors is their behavior under short exact sequences in abelian categories. Specifically, for a right exact functor F: \mathcal{A} \to \mathcal{B} between abelian categories with enough projectives, and a short exact sequence $0 \to A' \to A \to A'' \to 0 in \mathcal{A}, the left derived functors L_i F (for i \geq 0) form a long exact sequence:

\cdots \to L_{i+1} F(A'') \to L_i F(A') \to L_i F(A) \to L_i F(A'') \to L_{i-1} F(A') \to \cdots \to L_0 F(A') \to L_0 F(A) \to L_0 F(A'') \to 0.

This sequence is natural in the objects A', A, A'', meaning that any commutative diagram of short exact sequences induces a commutative diagram of long exact sequences. The theorem holds analogously for additive functors that are left exact, using injective resolutions to define the right derived functors.^[8] Dually, for a left exact functor G: \mathcal{A} \to \mathcal{B} with enough injectives, and the same short exact sequence $0 \to A' \to A \to A'' \to 0, the right derived functors R^i G (for i \geq 0) yield a long exact sequence:

$0 \to R^0 G(A') \to R^0 G(A) \to R^0 G(A'') \to R^1 G(A') \to R^1 G(A) \to R^1 G(A'') \to \cdots.

This cohomology version extends indefinitely to the right, and the maps are again natural. These long exact sequences transform the failure of exactness in the original functor into a precise homological measurement, allowing computations of higher derived terms inductively from lower ones.^[22] The connecting homomorphisms in these sequences, denoted \delta_i: L_i F(A'') \to L_{i-1} F(A') for left derived functors (and similarly \delta^i: R^i G(A'') \to R^{i+1} G(A') for right derived), arise from the snake lemma applied to the commutative diagrams of resolutions. Specifically, given projective resolutions of A', A, and A'', the short exact sequence of resolutions induces a long exact sequence in homology via the functor F, where the connecting maps are composed from the boundaries in the resolution complexes. This construction ensures the exactness at each term, linking the derived functors across the original sequence.^[8] Several vanishing conditions simplify these sequences. If the functor F (or G) is exact, then all higher derived functors vanish: L_i F = 0 for i > 0 (and similarly R^i G = 0 for i > 0), reducing the long exact sequence to the short exact sequence $0 \to F(A') \to F(A) \to F(A'') \to 0. Additionally, if A' is projective (for left derived functors), then L_i F(A') = 0 for all i > 0, making the sequence split into short exact segments starting from L_0 F. Dually, if A'' is injective (for right derived functors), then R^i G(A'') = 0 for all i > 0, terminating the sequence early at each cohomological degree. These conditions highlight how resolutions and exactness properties control the non-triviality of the derived functors.^[22]

Examples

Ext functors

In the category of modules over a ring R, the Ext functors provide a concrete realization of right derived functors applied to the Hom functor. Specifically, for R-modules A and B, the i-th Ext functor is defined as \operatorname{Ext}^i_R(A, B) = R^i \Hom_R(A, -)(B), where R^i denotes the i-th right derived functor, computed using an injective resolution of B.^[8] This construction arises because \Hom_R(A, -) is left exact, and its right derived functors capture the obstructions to exactness in higher degrees.^[10] Dually, \operatorname{Ext}^i_R(A, B) can also be expressed as the i-th left derived functor L_i \Hom_R(-, B)(A), obtained via a projective resolution of A.^[8] This equivalence holds by the general theory of derived functors in abelian categories with enough projectives and injectives, ensuring that both approaches yield isomorphic groups.^[10] The contravariant nature in the first argument and covariant in the second reflects the bifunctorial structure of Hom. For i=0, \operatorname{Ext}^0_R(A, B) \cong \Hom_R(A, B), recovering the original functor as the zeroth derived functor.^[8] In degree i=1, \operatorname{Ext}^1_R(A, B) classifies equivalence classes of short exact sequences of the form $0 \to B \to E \to A \to 0up to [congruence](/page/Congruence), where two extensions are congruent if they fit into a [commutative diagram](/page/Commutative_diagram) with identity maps onAandB$.^[8] The group operation is given by the Baer sum, which splices extensions via a pullback and pushout construction.^[10] Explicit computations illustrate these functors in the category of \mathbb{Z}-modules. For instance, \operatorname{Ext}^1_\mathbb{Z}(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z}) \cong \mathbb{Z}/n\mathbb{Z}, obtained by applying \Hom_\mathbb{Z}(-, \mathbb{Z}) to the projective resolution $0 \to \mathbb{Z} \xrightarrow{\cdot n} \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} \to 0, yielding a [cokernel](/page/Cokernel) isomorphic to \mathbb{Z}/n\mathbb{Z}after accounting for exactness. Moreover, ifAis a free\mathbb{Z}-module, then \operatorname{Ext}^i_\mathbb{Z}(A, B) = 0for alli > 0and anyB$, since free modules are projective and higher derived functors vanish on projectives.^[8] The representability of \operatorname{Ext}^i(A, -) aligns with Yoneda's extension theory, where \operatorname{Ext}^i_R(A, B) consists of equivalence classes of (i+1)-fold Yoneda extensions $0 \to B \to E_0 \to \cdots \to E_{i-1} \to A \to 0, with maps between consecutive terms being proper (i.e., the composition to Aor fromB$ is zero).^[8] This Yoneda product induces a ring structure on the Ext groups, compatible with the derived functor construction.^[10]

Tor functors

The Tor functors are the left derived functors of the tensor product functor in the category of modules over a ring R. Specifically, for R-modules A and B, the i-th Tor functor is defined as \Tor_i^R(A, B) = L_i(-\otimes_R B)(A), computed by taking a projective resolution P_\bullet \to A of A and forming the homology of the chain complex P_\bullet \otimes_R B.^[22] This construction measures the failure of the tensor product to be exact, as the tensor functor -\otimes_R B is right exact but not necessarily left exact.^[22] The zeroth Tor functor recovers the tensor product: \Tor_0^R(A, B) \cong A \otimes_R B.^[23] The first Tor functor, \Tor_1^R(A, B), detects the extent to which A fails to be flat over R; in particular, A is flat if and only if \Tor_1^R(A, -) = 0.^[24] A representative example is \Tor_1^\mathbb{Z}(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z}/m\mathbb{Z}) \cong \mathbb{Z}/\gcd(n,m)\mathbb{Z}, which captures the shared torsion between the cyclic groups.^[23] Over a field k, all higher Tor functors vanish, \Tor_i^k(M, N) = 0 for i > 0, since every k-module is free and the tensor product is exact.^[24] For change of rings, if \varphi: R \to S is a ring homomorphism, there is a natural isomorphism under suitable conditions, such as when S is flat over R; for instance, \Tor_n^R(A, C) \cong \Tor_n^S(A \otimes_R S, C) if \Tor_q^R(A, S) = 0 for q > 0.^[22] More generally, for modules M over R \otimes_S and N over S, the Tor functors over the tensor product ring relate via spectral sequences to iterated Tor computations over R and S.^[25] In group homology, the Tor functors compute the homology of a group G with coefficients in a \mathbb{Z}G-module M via H_n(G, M) \cong \Tor_n^{\mathbb{Z}G}(\mathbb{Z}, M), where \mathbb{Z} is the trivial module.^[26] In particular, H_2(G, \mathbb{Z}) \cong \Tor_2^{\mathbb{Z}G}(\mathbb{Z}, \mathbb{Z}) is the Schur multiplier of G, which classifies central extensions of G up to equivalence.^[26] Long exact sequences arise from short exact sequences of modules, preserving the exactness properties of the tensor functor.^[23]

Sheaf cohomology

Sheaf cohomology provides a framework for measuring the extent to which local sections of a sheaf on a topological space X fail to glue into global sections, extending the algebraic machinery of derived functors to geometric settings. For an abelian sheaf \mathcal{F} on X, the sheaf cohomology groups H^i(X, \mathcal{F}) are defined as the i-th right derived functors R^i \Gamma(X, -)(\mathcal{F}) of the global sections functor \Gamma(X, -): \operatorname{Ab}(X) \to \operatorname{Ab}, where \operatorname{Ab}(X) denotes the category of abelian sheaves on X.^[27] The functor \Gamma(X, -) is left exact, ensuring the existence of these right derived functors, which are computed by resolving \mathcal{F} with an injective resolution $0 \to \mathcal{F} \to \mathcal{I}^\bullet in \operatorname{Ab}(X) and taking cohomology of the complex \Gamma(X, \mathcal{I}^\bullet).^[28] This approach leverages the fact that injective sheaves on X are acyclic for \Gamma(X, -) under suitable topological assumptions on X, such as paracompactness.^[12] A practical method for computing these derived functors is via Čech cohomology, which approximates sheaf cohomology by considering cohomology with respect to an open cover of X. For a cover \mathcal{U} = \{U_\alpha\} of X, the Čech cohomology groups \check{H}^i(\mathcal{U}, \mathcal{F}) are defined using the cochain complex of sections over intersections of the cover elements, and under conditions like fine covers or when \mathcal{F} is acyclic on the cover refinements, these coincide with the derived functor groups H^i(X, \mathcal{F}).^[28] This approximation is particularly useful in topology and algebraic geometry for explicit calculations, as it reduces the problem to combinatorial data from the cover, though it may require passing to the direct limit over all covers for the full sheaf cohomology.^[29] In algebraic geometry, sheaf cohomology reveals key structures on varieties. For instance, on a scheme X, the first cohomology group H^1(X, \mathcal{O}_X^*), where \mathcal{O}_X^* is the sheaf of units in the structure sheaf \mathcal{O}_X, classifies isomorphism classes of line bundles on X up to tensor product, forming the Picard group \operatorname{Pic}(X).^[30] Another fundamental result is the vanishing of higher cohomology for quasi-coherent sheaves on affine schemes: if X = \operatorname{Spec}(A) is affine and \mathcal{F} is quasi-coherent, then H^i(X, \mathcal{F}) = 0 for all i > 0, a theorem originally due to Cartan in the analytic setting and extended to schemes (Cartan's theorem).^[31] This vanishing underscores the affine nature of schemes and facilitates computations by reducing global questions to local ring theory.^[32] Sheaf cohomology also connects sheaf theory to classical topological invariants. On smooth manifolds, the de Rham cohomology, computed from the complex of differential forms, is isomorphic to the sheaf cohomology H^i(X, \mathbb{R}) of the constant sheaf \mathbb{R}, via the de Rham theorem, which uses the de Rham complex as a resolution of the constant sheaf.^[33] Similarly, sheaf cohomology with constant integer coefficients agrees with singular cohomology on paracompact spaces, providing a unified perspective where geometric and analytic cohomologies emerge as special cases of derived functors applied to constant sheaves.^[34]

Naturality and Variations

Natural transformations

Derived functors preserve the naturality inherent in the original functors. Specifically, if \eta: F \to G is a natural transformation between additive functors F, G: \mathcal{A} \to \mathcal{B} from an abelian category \mathcal{A} to an abelian category \mathcal{B}, then it induces natural transformations L_i(\eta): L_i F \to L_i G on the left derived functors for each i \geq 0, and similarly R^i(\eta): R^i F \to R^i G on the right derived functors. This follows from the universal property of derived functors as \delta-functors: since \eta provides a natural transformation on degree 0 that commutes with the connecting morphisms in long exact sequences, it extends uniquely to the entire system of derived functors.^[16] The functoriality of derived functors extends to morphisms in the domain category. For a morphism f: A \to B in \mathcal{A}, the induced map F(f): F(A) \to F(B) lifts to a chain map between resolutions, yielding natural maps L_i F(f): L_i F(A) \to L_i F(B) and R^i F(f): R^i F(A) \to R^i F(B), with L_0 F(f) \cong F(f) and R^0 F(f) \cong F(f). Moreover, naturality holds with respect to resolutions: a chain map \phi: P_\bullet \to Q_\bullet between projective (or injective) resolutions of objects in \mathcal{A} induces a commutative diagram on the homology (or cohomology) groups after applying the functor, ensuring that the derived functors are well-defined independently of the choice of resolution up to natural isomorphism.^[35] Composition of derived functors is governed by natural transformations under suitable conditions. If F': \mathcal{B} \to \mathcal{C} is another additive functor and exact (hence L_i F' = 0 for i > 0), then there is a natural isomorphism L_k (F' \circ F) \cong F' \circ L_k F for each k \geq 0, reflecting the compatibility of derived constructions with functor composition. In general, for left exact functors, a canonical natural transformation L F' \circ L F \to L(F' \circ F) exists, arising from the composition of the augmentation maps to the derived functors; analogous results hold for right derived functors.^[36] The edge maps and connecting homomorphisms in long exact sequences associated to short exact sequences are themselves natural transformations. For a short exact sequence $0 \to A \to B \to C \to 0 in \mathcal{A}, the long exact sequence \cdots \to R^i F(A) \to R^i F(B) \to R^i F(C) \to R^{i+1} F(A) \to \cdots features connecting maps \delta^i: R^i F(C) \to R^{i+1} F(A) that are natural in the sense that any commutative diagram of short exact sequences induces a commutative diagram of long exact sequences, preserving the edge inclusions and projections. This naturality ensures the robustness of derived functors in homological computations.^[35]

Half-exact and connected functors

In homological algebra, half-exact functors provide a framework for extending the construction of derived functors beyond fully exact cases, particularly when the original functor preserves only partial exactness. A half-exact functor (also called middle exact) is one that maps short exact sequences $0 \to A \to B \to C \to 0 to sequences F(A) \to F(B) \to F(C) exact at F(B), i.e., \operatorname{im}(F(A) \to F(B)) = \ker(F(B) \to F(C)), though not necessarily at the ends.^[37] This property allows the definition of right derived functors R^i F using injective resolutions, even if the functor fails to preserve all kernels, as long as acyclic objects exist to ensure the resolution's homology aligns appropriately. Such functors are common in settings like sheaf theory or module categories where full exactness is not assumed, enabling broader applications in cohomology computations.^[37] Connected functors, often arising in cohomological contexts, are those for which the derived functors R^i F vanish in negative degrees, i.e., R^i F = 0 for i < 0, with R^0 F \cong F. This condition ensures that the family \{R^i F\} forms a well-behaved system, particularly in producing long exact sequences from short exact sequences via connecting homomorphisms.^[38] For an exact sequence $0 \to M' \to M \to M'' \to 0, the connectivity yields a long exact sequence \cdots \to R^{i} F(M') \to R^{i} F(M) \to R^{i} F(M'') \to R^{i+1} F(M') \to \cdots, preserving the homological structure without artifacts in lower degrees.^[38] This property is crucial for uniqueness theorems and natural transformations between derived systems, as it aligns the functor with standard cohomological delta-functors. Variations in derived functor definitions distinguish between the total derived functor, which applies F to an entire resolution complex and yields a complex in the target category, and the individual derived functors, which extract the homology groups R^i F(A) = H^i(RF(A)) of that total complex. The total derived functor RF operates on the derived category, capturing the full homological information, while the individual R^i F focus on specific degrees for computational purposes. Effaceability plays a key role in these constructions: a functor S^n is effaceable if, for every object M and n > 0, there exists an acyclic resolution M \to I^\bullet such that S^n(I^\bullet) = 0, ensuring that higher derived functors can be "effaced" or zeroed out on sufficiently nice objects like injectives.^[39] This effaceability condition guarantees the uniqueness of the derived functor up to natural isomorphism and facilitates the extension to half-exact or connected settings without introducing ambiguities.^[39]

Generalizations

Delta functors

A δ-functor (or delta-functor) between two abelian categories \mathcal{A} and \mathcal{B} is a sequence of additive functors T^i: \mathcal{A} \to \mathcal{B} for i \in \mathbb{Z}, equipped with natural transformations \delta^i: T^i(M) \to T^{i+1}(N) (connecting homomorphisms), such that for every short exact sequence $0 \to M \to N \to P \to 0 in \mathcal{A}, the sequence

\cdots \to T^i(M) \to T^i(N) \to T^i(P) \xrightarrow{\delta^i} T^{i+1}(M) \to T^{i+1}(N) \to T^{i+1}(P) \to \cdots

is exact in \mathcal{B}.^[1] A δ-functor T^\bullet is half-exact if its zeroth functor T^0 is left exact, and it is effaceable if for every i > 0 and object A \in \mathcal{A}, there exists a monomorphism A \hookrightarrow I into an injective object I such that T^i(I) = 0. By Yoneda's lemma applied to the category of δ-functors, every half-exact δ-functor is effaceable and thus unique up to unique natural isomorphism; any two such δ-functors with isomorphic zeroth components coincide.^[40] Derived functors provide a concrete realization of δ-functors: for a left-exact functor F: \mathcal{A} \to \mathcal{B}, the right derived functors R^i F form a δ-functor with R^0 F \cong F, satisfying the long exact sequence axiom via the properties of projective or injective resolutions.^[1] The concept of δ-functors was introduced by Henri Cartan and Samuel Eilenberg in their 1956 monograph Homological Algebra, which axiomatized the long exact sequence property to unify various homology and cohomology theories across algebra and topology.

Derived categories

The derived category D(\mathcal{A}) of an abelian category \mathcal{A} is constructed as the localization of the homotopy category of chain complexes in \mathcal{A} with respect to the quasi-isomorphisms, resulting in a triangulated category where the objects are chain complexes and the morphisms are represented by roofs of chain maps, modulo homotopy and quasi-isomorphisms, corresponding to hyperhomology classes.^[41] This framework, introduced by Verdier, refines the classical notion of derived functors by embedding them into a categorical setting that captures homological information globally rather than degree by degree.^[42] In the derived category, right derived functors extend additively to total functors Rf: D(\mathcal{A}) \to D(\mathcal{B}) between derived categories, where Rf is the total right derived functor of an additive functor f: \mathcal{A} \to \mathcal{B}, satisfying the compatibility H^i(Rf(C)) \cong R^i f(H^i(C)) for any complex C in D(\mathcal{A}).^[43] This representation treats derived functors as morphisms between complexes, modernizing the classical resolution-based approach by working directly with the localized category.^[44] A key advantage of this perspective is that it circumvents the ambiguity in choosing projective or injective resolutions for individual objects, as the derived category identifies complexes up to quasi-isomorphism, thereby ensuring functoriality without resolution-dependent choices. Moreover, it facilitates the composition of derived functors and the construction of inverse limits in a coherent manner, enabling applications in areas like sheaf theory and algebraic geometry where classical sequences become cumbersome.^[45] For instance, in the derived category D(\mathrm{Mod}_R) of modules over a commutative ring R, the classical Ext functors recover as

\Ext^i_R(A, B) \cong \Hom_{D(\mathrm{Mod}_R)}(A{{grok:render&&&type=render_inline_citation&&&citation_id=0&&&citation_type=wikipedia}}, B)

, where

A{{grok:render&&&type=render_inline_citation&&&citation_id=0&&&citation_type=wikipedia}}

and B denote the complexes concentrated in degrees 0 and i, respectively, illustrating how hyperhomology morphisms encode extension groups.^[21]

References

[1]
[PDF] Derived Functors and Tor - Purdue Math
Derived functors, introduced by Cartan and Eilenberg, are defined as RiFM = Hi(F(I•)) and extend to additive functors from A ! B with R0F = F.
[2]
[PDF] An Introduction to Derived Functors - Rohil Prasad
Dec 11, 2015 · This paper is a short exposition to the basic theory of derived functors. The reader should be familiar with the basic language of category ...
[3]
[PDF] derived functors and homological dimension - UT Math
This paper overviews the basic notions of abelian categories, exact functors, and chain complexes. It will use these concepts to define derived functors, prove ...
[4]
None
Summary of each segment:
[5]
Section 12.5 (00ZX): Abelian categories—The Stacks project
An abelian category is a category satisfying just enough axioms so the snake lemma holds. An axiom (that is sometimes forgotten) is that the canonical map
[6]
[PDF] An Introduction to Categories and Homological Algebra - IMJ-PRG
A particular class of categories plays a central role: the additive categories and among them the abelian categories, in which one can perform homological ...
[7]
[PDF] intro to sheaves and abelian categories
Jun 11, 2017 · A morphism between sheaves F and G is a morphism of presheaves between. F and G . The resulting category of sheaves of abelian groups on X is ...
[8]
[PDF] AN INTRODUCTION TO HOMOLOGICAL ALGEBRA
This book is an introduction to homological algebra, covering topics such as complexes, derived functors, and Tor and Ext.<|control11|><|separator|>
[9]
[PDF] rotman.pdf
A complex (or chain complex) is such a functor with the property that ... J.J. Rotman, An Introduction to Homological Algebra, Universitext,. 37. DOI ...
[10]
[PDF] Homological Algebra
Homological Algebra. By HENRI CARTAN and SAMUEL EILENBERG. 20. The ... not covered by the specializations. An important example is. Hilbert's theorem ...
[11]
[PDF] HISTORY OF HOMOLOGICAL ALGEBRA Charles A. Weibel ...
First, he introduced the singular chain complex S(X) of a topological space, and then he defined H∗(X; A) and H∗(X; A) to be the homology and cohomology of the.
[12]
[PDF] The Derived Functor Approach to Sheaf Cohomology
ABSTRACT. Sheaves and their cohomology have transformed the study of com- plex and algebraic geometry over the last eighty years. The classical formulation.Missing: early 20th
[13]
Definition 12.12.3 (010S)—The Stacks project
We say F is a universal \delta -functor if and only if for every \delta -functor G = (G^ n, \delta _ G) and any morphism of functors t : F^0 \to G^0 there ...
[14]
12.12 Cohomological delta-functors - Stacks Project
A cohomological delta-functor from \mathcal{A} to \mathcal{B} is given by a collection of additive functors F^n and morphisms \delta _{A \to B \to C} for short ...
[15]
projective object in nLab
### Definition of Projective Object in Abelian Categories
[16]
[PDF] Derived Functors
We will see that derived functors, when they exist, are indeed universal 5- functors. For this we need the concept of projective and injective resolutions.
[17]
projective resolution in nLab
### Projective Resolutions in Abelian Categories
[18]
derived functor in homological algebra in nLab
### Definition and Construction of Left Derived Functors Using Projective Resolutions
[19]
Tor in nLab
Oct 9, 2024 · In the context of homological algebra, the Tor Tor -functor is the derived tensor product: the left derived functor of the tensor product of R R ...<|control11|><|separator|>
[20]
https://www-users.cse.umn.edu/~garrett/m/algebra/leftder.pdf
[21]
Section 13.27 (06XP): Ext groups—The Stacks project
Given two Yoneda extensions E, E' of the same degree then E is equivalent to E' if and only if \delta (E) = \delta (E'). Proof. Let \xi : B[0] \to A[i] be an ...
[22]
[PDF] HOMOLOGICAL ALGEBRA
... Homological Algebra, by Henri Cartan and Samuel Eilenberg. Page 3. HOMOLOGICAL ... In par ticular, the process could be iterated and thus a sequence of functors.
[23]
[PDF] Some Common Tor and Ext Groups
Feb 12, 2009 · Common Tor and Ext groups include G⊗H, Tor(G,H), Hom(G,H), and Ext(G,H) where G and H are Z, Z/n, or Q. Z/n ⊗ H is H/nH, and Tor(Z/n,H) is nH.Missing: ≅ | Show results with:≅
[24]
[PDF] Cn−1
The algebra by means of which the Tor functor is derived from tensor products has a very natural generalization in which abelian groups are replaced by ...<|control11|><|separator|>
[25]
None
### Summary of Tor Functors from the Document
[26]
On the change of rings in the homological algebra. - Project Euclid
$$Tor_{n}^{\varphi}(A, C)=H_{n}(N(\Phi^{(1)}))$ . From the exact sequence. $0\rightarrow X^{\prime}\otimes_{A}Y^{\prime}\rightarrow ...
[27]
[PDF] Group Homology and Cohomology
tor, and that —H is right adjoint to an exact functor. ... The group H2(G; Z) is called the Schur multiplier of G in honor of Schur, who first investigated the.
[28]
Section 21.2 (01FT): Cohomology of sheaves—The Stacks project
21.2 Cohomology of sheaves in other words, it is the ith right derived functor of the global sections functor. The family of functors H^ i(\mathcal{C}, -) ...
[29]
[PDF] Sheaf Cohomology 1. Computing by acyclic resolutions
Feb 19, 2005 · That is, sheaf cohomology is an example of a right derived functor. Here we note that the global-sections functor Γ(X, ∗) is left-exact.
[30]
[PDF] introduction to algebraic geometry, class 21
(Experts in Cech cohomology can later check that line bundles are parametrized be H1(X,O. ∗. X), where O∗. X is the sheaf of invertible functions and the ...
[31]
[PDF] Sheaf Cohomology - UCSB Math
We will first show the group of line bundles is isomorphic to H1(X,O×) and ... This defines a sheaf O{L} for any line bundle L over X. The sheaf O{L} ...
[32]
[PDF] 1 The theorem, and a bogus proof - Kiran S. Kedlaya
Apr 25, 2009 · Let X be an affine scheme and let F be a quasicoherent sheaf on X. Then. Hi(X, F)=0 for i > 0, that is, F is acyclic for sheaf cohomology. Here ...
[33]
Section 30.3 (01XE): Vanishing of cohomology—The Stacks project
30.3 Vanishing of cohomology. We have seen that on an affine scheme the higher cohomology groups of any quasi-coherent sheaf vanish (Lemma 30.2.2).
[34]
From Sheaf Cohomology to the Algebraic de Rham Theorem - arXiv
Feb 23, 2013 · Grothendieck's algebraic de Rham theorem asserts that the singular cohomology of Y with complex coefficients can be computed from the complex of sheaves.
[35]
Sheaves, Cohomology, and the de Rham Theorem - SpringerLink
The approach will be to exhibit both the de Rham cohomology and the differentiable singular cohomology as special cases of sheaf cohomology and to use a basic ...
[36]
[PDF] HOMOLOGICAL ALGEBRA Romyar Sharifi - UCLA Mathematics
These notes provide an introduction to homological algebra and the category theory that underpins its modern structure. Central to homological algebra is ...
[37]
13.22 Composition of right derived functors - Stacks Project
We can compute the right derived functor of a composition. Suppose that \mathcal{A}, \mathcal{B}, \mathcal{C} be abelian categories.
[38]
[PDF] Homological Theory of Exact Categories - Uni Bielefeld
Apr 1, 2025 · ... half exact for all objects X and Y in A (here: A functor is half exact if applied to a short exact sequence it gives a sequence which is ...
[39]
[PDF] a cohomology theory for - commutative algebras. i1
Hn and Hn. Proposition. 4. Hn is a connected functor, i.e. if 0—>M'—»M—»M"—»0 is an exact sequence of R-modules, then there are connecting homomor- phisms ...
[40]
[PDF] Basics of Homological Algebra
The group cohomology Hi(G, −) is the i-th right-derived functor of F. ... An injective resolution for I is simply 0 → I → I → 0 and so 0 → IG → IG ...
[41]
Why do universal δ-functors annihilate injectives?
Mar 30, 2012 · This is because the R∗T0 is also a universal delta functor, and hence (by universality) must be the same as T∗. In fact, universal delta ...Does this square of δ-functors anti-commute? - Math Stack ExchangeApplying a functor to a homotopy of chain maps yields an ...More results from math.stackexchange.comMissing: property | Show results with:property
[42]
13.15 Derived functors on derived categories - Stacks project
In practice derived functors come about most often when given an additive functor between abelian categories.
[43]
[PDF] Verdier-ams.pdf
q. The derived category of A will be a category DpAq with the same objects as KpAq but with a much larger class of morphisms ...
[44]
[PDF] Notes on Derived Categories and Derived Functors
Keller, Derived categories and their uses, in Handbook of Algebra, Vol. ... Historically, it was not always clear how to construct some important derived functors.
[45]
[1501.06731] Introduction to Derived Categories - arXiv
Jan 27, 2015 · Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the "old" homological algebra (of derived functors ...<|control11|><|separator|>
[46]
[PDF] Derived categories and their uses - School of Mathematics
In section 11, we formulate Verdier's definition of the derived category [56] in the context of exact categories. In section 12, we give a sufficient ...