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Homological algebra

Homological algebra is a branch of that studies algebraic structures, such as modules, groups, and rings, through the lens of homological invariants derived from chain complexes and exact sequences in abelian categories. It provides tools to prove existence theorems, detect obstructions to solutions, and compute invariants that reveal deep relationships between objects, often by measuring "holes" in algebraic data analogous to topological holes. At its core, the subject formalizes the use of chain complexes—sequences of abelian groups or modules connected by homomorphisms called differentials satisfying d^2 = 0—to define homology groups H_n(C) = \ker(d_n)/\im(d_{n+1}), which capture essential structural information. The origins of homological algebra trace back to the late 19th century with early work on homology in topology, but it crystallized during World War II through efforts to unify ad hoc computations in algebraic topology and group cohomology. Pioneering contributions came from mathematicians like Samuel Eilenberg, who in 1944 defined singular homology for topological spaces using chain complexes, and Henri Cartan, whose seminars in the 1940s and 1950s fostered collaborative advancements. The field was formally established in 1956 with the publication of Homological Algebra by Cartan and Eilenberg, which introduced derived functors like Ext and Tor, projective and injective resolutions, and the systematic study of exact sequences in the category of modules. Subsequent developments, including Alexander Grothendieck's 1957 Tohoku paper on abelian categories and derived functors, expanded the framework to sheaf cohomology and spectral sequences, influencing commutative algebra and algebraic geometry. Central concepts include exact sequences, where the image of each map equals the of the next, enabling long exact sequences in that relate the invariants of composed structures, such as in the or . Derived functors generalize and : left-derived functors like measure tensor product exactness, while right-derived functors like Ext quantify extensions between modules. Resolutions, such as projective resolutions \cdots \to P_1 \to P_0 \to M \to 0, allow computing these functors by applying them to free approximations of modules. In broader settings, homological algebra employs for cochain complexes with differentials increasing degree, yielding groups H^n(C) = \ker(d^n)/\im(d^{n-1}) that detect extensions and deformations, as in Hochschild cohomology for associative algebras. Homological algebra's importance lies in its applications across mathematics: in , it computes homotopy groups via ; in , it characterizes global dimension and regularity of rings; and in , it underpins sheaf theory and derived categories for studying stacks and motives. Modern extensions include triangulated categories and , where exact triangles replace short exact sequences to model infinite resolutions. These tools not only unify disparate areas but also drive computational advances, such as in and data analysis via .

Historical Development

Origins in Topology and Geometry

Homological algebra traces its roots to the late , emerging from efforts to classify through invariants that capture their . In 1895, introduced the foundational ideas of in his seminal paper "Analysis Situs," where he defined Betti numbers as numerical invariants measuring the number of "holes" in manifolds of various dimensions. These Betti numbers, denoted b_k(M) = \dim H_k(M; \mathbb{Q}) for a manifold M, provided a way to distinguish topologically distinct spaces, such as orientable and non-orientable surfaces, by associating algebraic counts to geometric features like connectivity and voids. The early saw refinements to Poincaré's intuitive approach, particularly through the development of , which offered a rigorous combinatorial method for computation. In their 1935 book Topologie, and formalized simplicial complexes as triangulations of spaces and defined homology groups using chains of simplices, boundaries, and cycles, enabling explicit calculations for polyhedra and manifolds. This framework built directly on Poincaré's work by providing algebraic tools to compute Betti numbers via integer matrices and their ranks. Parallel to these advances, early cohomological concepts arose from analytic perspectives on topology. In 1931, proved his establishing an isomorphism between the cohomology groups of a smooth manifold, computed using closed differential forms modulo exact ones, and its groups with real coefficients, thus bridging and . This result highlighted how of forms over cycles yields topological invariants, influencing later algebraic developments. Illustrative computations from these early theories reveal the power of homological invariants. For the 1-sphere S^1, representing a circle, the homology groups are H_0(S^1; \mathbb{Z}) \cong \mathbb{Z} (capturing connectedness), H_1(S^1; \mathbb{Z}) \cong \mathbb{Z} (detecting the single loop), and H_n(S^1; \mathbb{Z}) = 0 for n \geq 2. Similar calculations for higher spheres S^n yield H_k(S^n; \mathbb{Z}) \cong \mathbb{Z} for k = 0, n and zero otherwise, while the 2-torus T^2 has H_0(T^2; \mathbb{Z}) \cong \mathbb{Z}, H_1(T^2; \mathbb{Z}) \cong \mathbb{Z}^2, and H_2(T^2; \mathbb{Z}) \cong \mathbb{Z}, reflecting its two independent loops and surface orientation. These examples, computed via simplicial methods, underscore the topological distinctions among basic geometric objects. These topological and geometric origins laid the groundwork for the algebraic formalization of homological methods in subsequent decades.

Algebraic Formalization and Key Contributors

The algebraic formalization of homological methods emerged in the mid-20th century as researchers abstracted topological concepts into a purely algebraic framework, emphasizing categories, functors, and derived constructions applicable beyond . This shift was catalyzed by and Saunders Mac Lane's seminal 1945 paper, which introduced the notions of categories, functors, and natural transformations specifically to unify and rigorize the study of theories in . Their work provided a categorical language that facilitated the comparison of different constructions, laying the groundwork for homological algebra as an independent discipline. During and the immediate postwar years, efforts to unify computations in and accelerated this abstraction. In 1944, Eilenberg defined for topological spaces using chain complexes of singular simplices, providing a purely algebraic tool independent of specific triangulations. Concurrently, was developed, with contributions from mathematicians like Beno Eckmann and Gerhard Hochschild in the early , formalizing invariants for group actions on modules. These advances, discussed in seminars led by in during the , bridged and . Building on this momentum, the 1950s saw the introduction of derived functors, a key innovation that generalized homology computations to arbitrary abelian categories. and played a central role, with their 1956 book Homological Algebra serving as the foundational text that systematically defined chain complexes in abstract terms and established derived functors as right or left derived versions of additive functors between module categories. In this work, they notably introduced the Ext functors, which classify extensions of modules and provide a homological measure of non-split extensions. Alexander Grothendieck further extended these ideas in the 1950s through his work on sheaf cohomology, adapting homological techniques to sheaves of abelian groups on topological spaces and algebraic varieties. His 1957 Tohoku paper formalized abelian categories with enough injectives, enabling the definition of sheaf cohomology via derived functors and bridging algebraic and geometric applications. These contributions by Eilenberg, Mac Lane, Cartan, and Grothendieck transformed homological algebra into a versatile tool, influencing fields from commutative algebra to differential geometry.

Core Concepts

Chain Complexes

A chain complex in homological algebra is a sequence of abelian groups or modules (C_n)_{n \in \mathbb{Z}}, together with homomorphisms known as boundary maps d_n: C_n \to C_{n-1} for each n, satisfying the condition d_{n-1} \circ d_n = 0 for all n. This composition rule ensures that the image of each boundary map lies in the of the subsequent map, forming the foundational structure for computing algebraic invariants. The indexing convention in chain complexes is homological, where the degree n corresponds to chains in C_n and boundaries decrease the index by one, distinguishing it from cohomological indexing. A cochain complex provides the contravariant analogue, consisting of a sequence of abelian groups or modules (C^n)_{n \in \mathbb{Z}} equipped with coboundary maps d^n: C^n \to C^{n+1} such that d^{n+1} \circ d^n = 0 for all n. Here, the maps increase the index, aligning with applications in cohomology theory. A prominent example is the singular chain complex of a X, denoted (C_n(X), \partial_n), where each C_n(X) is the generated by the set of singular n-simplices, which are continuous maps \sigma: \Delta^n \to X from the standard n-simplex \Delta^n to X. Elements of C_n(X) are finite formal integer linear combinations of these simplices. In simplicial complexes, the boundary maps are explicitly defined on the generators: for an oriented n-simplex \sigma = [v_0, v_1, \dots, v_n], the is given by \partial_n(\sigma) = \sum_{i=0}^n (-1)^i \sigma|_{[v_0, \dots, \hat{v}_i, \dots, v_n]}, where \sigma|_{[v_0, \dots, \hat{v}_i, \dots, v_n]} denotes the restriction of \sigma to the i-th face obtained by omitting the i-th . This alternating sum ensures the nilpotency condition \partial_{n-1} \circ \partial_n = 0, as faces of faces cancel in pairs.

Homology and Cohomology Groups

Homological algebra extracts invariants from chain complexes through homology groups and from cochain complexes through cohomology groups. For a chain complex C_\bullet = (\dots \to C_{n+1} \xrightarrow{d_{n+1}} C_n \xrightarrow{d_n} C_{n-1} \to \dots), where each C_n is an abelian group and d_n d_{n+1} = 0, the nth homology group is defined as H_n(C) = \ker d_n / \operatorname{im} d_{n+1}, consisting of homology classes of n-cycles modulo n-boundaries. Similarly, for a cochain complex C^\bullet = (\dots \to C^{n-1} \xrightarrow{d^{n-1}} C^n \xrightarrow{d^n} C^{n+1} \to \dots), with d^n d^{n-1} = 0, the nth cohomology group is \hat{H}^n(C) = \ker d^n / \operatorname{im} d^{n-1}. These constructions are functorial: a chain map f: C_\bullet \to C'_\bullet induces a f_*: H_n(C) \to H_n(C') for each n, making a covariant from the of chain complexes to graded abelian groups. In contrast, a cochain map f: C^\bullet \to C'^\bullet induces f^*: \hat{H}^n(C') \to \hat{H}^n(C), rendering contravariant. Chain homotopy equivalence between maps preserves these induced maps on and , ensuring that homotopy equivalent complexes yield isomorphic invariants. A key property arises from short exact sequences of chain complexes. Given $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0, the induced sequence in \dots \to H_n(A) \to H_n(B) \to H_n(C) \to H_{n-1}(A) \to \dots is long exact, connecting the homology groups across the sequence and providing a tool to relate invariants of related complexes. This exactness motivates the study of as a means to detect exactness and extensions in algebraic structures. The universal coefficient theorem relates homology with arbitrary coefficients to integer homology. For a chain complex C_\bullet and abelian group G, there is a natural short exact sequence $0 \to H_n(C; \mathbb{Z}) \otimes G \to H_n(C; G) \to \operatorname{Tor}_1^\mathbb{Z}(H_{n-1}(C; \mathbb{Z}), G) \to 0, which splits (but not naturally), yielding the isomorphism H_n(C; G) \cong H_n(C; \mathbb{Z}) \otimes G \oplus \operatorname{Tor}_1^\mathbb{Z}(H_{n-1}(C; \mathbb{Z}), G). A proof sketch proceeds by considering the chain complex C_\bullet \otimes G, where the homology H_n(C \otimes G) fits into the short exact sequence above via the five-lemma applied to the long exact sequences from tensoring the defining exact sequences for H_n(C; \mathbb{Z}) and using the right exactness of tensor and left exactness of \operatorname{Hom}; the Tor term arises from the resolution of G. Examples illustrate these invariants in topological settings. The cellular homology of real projective space \mathbb{RP}^n yields H_k(\mathbb{RP}^n; \mathbb{Z}) = \mathbb{Z} for k=0, \mathbb{Z}/2\mathbb{Z} for $0 < k < n with k odd, \mathbb{Z} for k=n if n odd, and 0 otherwise, computed from its CW structure with one cell per dimension up to n and boundary maps of degree 2. For classifying spaces, the cohomology ring of complex projective space \mathbb{CP}^\infty is H^*(\mathbb{CP}^\infty; \mathbb{Z}) \cong \mathbb{Z} with |x|=2, where x is the generator in degree 2 corresponding to the hyperplane class; this polynomial structure reflects the H-space multiplication and is derived from the cell structure with cells in even dimensions.

Categorical Framework

Abelian Categories

Abelian categories provide the foundational categorical framework for homological algebra, abstracting the properties of the category of abelian groups to allow the development of homology and cohomology in a general setting. An abelian category \mathcal{A} is an additive category equipped with a zero object and finite biproducts, in which every morphism admits both a kernel and a cokernel, and moreover, every monomorphism is the kernel of its cokernel while every epimorphism is the cokernel of its kernel. This structure ensures that concepts like exact sequences can be defined and manipulated uniformly, without relying on the specific properties of underlying sets or groups. Prominent examples include the category \mathbf{Ab} of abelian groups, where objects are abelian groups and morphisms are group homomorphisms; the category R-Mod of left modules over a ring R with unit, with module homomorphisms as morphisms; and the category of sheaves of abelian groups on a topological space, where morphisms are sheaf morphisms preserving the abelian structure. These categories satisfy the abelian axioms, enabling the translation of algebraic techniques to geometric and topological contexts. Key properties of abelian categories include the existence of images for every morphism, realized as the kernel of the cokernel, and the fact that every monomorphism coincides with the kernel of its cokernel. Additionally, the additive structure implies that the Hom-sets \Hom_{\mathcal{A}}(A, B) form abelian groups for all objects A, B \in \mathcal{A}, with composition of morphisms being bilinear over the integers. These features guarantee that subobjects and quotient objects behave analogously to subgroups and quotient groups. To verify that R-Mod is abelian, first note that it is additive: the zero module serves as the zero object, direct sums of modules exist and coincide with biproducts, and \Hom_R(M, N) is an abelian group under pointwise addition with bilinear composition. For any module homomorphism f: M \to N, the kernel \ker f = \{ m \in M \mid f(m) = 0 \} is a submodule (hence the kernel in the categorical sense), and the cokernel is N / \im f, the quotient by the image submodule. Monomorphisms in R-Mod are injective homomorphisms, which are kernels of their cokernels since the cokernel map factors through the inclusion of the image; dually, epimorphisms are surjective and cokernels of their kernels. Exactness in R-Mod, defined via the vanishing of certain kernels or images, is preserved under translations in sequences, as submodules and quotients align with categorical kernels and cokernels. This structure underpins the use of abelian categories in defining exact sequences throughout homological algebra.

Additive Functors and Natural Transformations

In homological algebra, an additive functor between additive categories preserves the additive structure on the hom-sets. Specifically, for a covariant functor F: \mathcal{A} \to \mathcal{B} between additive categories, additivity means that F(f + g) = F(f) + F(g) for all morphisms f, g in \mathcal{A}, and similarly for contravariant functors. Additive functors also preserve the zero morphism and biproducts when they exist, ensuring compatibility with direct sums, which are central to abelian categories. An exact functor further preserves exactness of sequences. A functor F is exact if it maps short exact sequences to short exact sequences, meaning that if $0 \to A \to B \to C \to 0 is exact in \mathcal{A}, then $0 \to F(A) \to F(B) \to F(C) \to 0 is exact in \mathcal{B}. This property is stronger than mere additivity and is essential for maintaining the homological structure across categories. Left exact functors preserve left exact sequences (exact at the first two terms), while right exact functors preserve right exact sequences (exact at the last two terms). Representative examples illustrate these properties in the category of modules over a ring R. The tensor product functor -\otimes_R N: R\text{-Mod} \to R\text{-Mod} is left additive, as it preserves direct sums and hom-addition, but it is right exact rather than fully exact, preserving cokernels but not necessarily kernels in short exact sequences. In contrast, the contravariant Hom functor \text{Hom}_R(-, M): R\text{-Mod}^{\text{op}} \to R\text{-Mod} is also additive and left exact, preserving kernels but not necessarily cokernels. Natural transformations provide a way to compare additive functors while respecting the categorical structure. A natural transformation \eta: F \Rightarrow G between functors F, G: \mathcal{A} \to \mathcal{B} consists of s \eta_X: F(X) \to G(X) for each object X in \mathcal{A}, such that for every f: X \to Y, the diagram \begin{CD} F(X) @>\eta_X>> G(X) \\ @VF(f)VV @VG(f)VV \\ F(Y) @>\eta_Y>> G(Y) \end{CD} commutes, i.e., \eta_Y \circ F(f) = G(f) \circ \eta_X. This commutativity ensures that \eta is compatible with the morphisms in \mathcal{A}, making it a morphism in the . Natural transformations between additive functors are themselves additive, preserving the group structure on components. In abelian categories, the Yoneda lemma establishes a fundamental connection between natural transformations and representable functors, providing a criterion for natural isomorphisms. For functors F: \mathcal{A}^{\text{op}} \to \text{Ab} and an object A in \mathcal{A}, the lemma states that there is a natural isomorphism \text{Nat}(\text{Hom}_{\mathcal{A}}(A, -), F) \cong F(A), where the left side is the set of natural transformations from the representable functor \text{Hom}_{\mathcal{A}}(A, -) to F, and the isomorphism sends \eta \mapsto \eta_A(\text{id}_A). This bijection is natural in both A and F, meaning it respects morphisms in \mathcal{A} and natural transformations between such F. The contravariant version similarly holds: \text{Nat}(F, \text{Hom}_{\mathcal{A}}(-, A)) \cong F(A). The lemma implies that objects are determined up to isomorphism by their representable functors, and it underpins the uniqueness of representing objects in abelian categories.

Exactness and Diagrams

Exact Sequences

In homological algebra, an exact sequence is a sequence of morphisms between objects in an abelian category, denoted as \dots \to A_{n-1} \xrightarrow{f_{n-1}} A_n \xrightarrow{f_n} A_{n+1} \to \dots, that is exact at A_n if the image of f_{n-1} equals the kernel of f_n, i.e., \operatorname{im} f_{n-1} = \ker f_n. This condition ensures that each morphism captures precisely the "relations" or "boundaries" imposed by the previous one, providing a framework for relating subobjects, quotients, and extensions within the category. Exact sequences generalize the notion of exactness in chain complexes, where the differential satisfies d^2 = 0, and play a central role in measuring deviations from injectivity or surjectivity. A short exact sequence is a finite exact sequence of the form $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0, where the map f is injective (so \ker f = 0) and g is surjective (so \operatorname{im} g = C), implying that A is isomorphic to the of g and C is isomorphic to the of f. In the category of modules over a , this means A embeds as a submodule of B with isomorphic to C. Such sequences encode extension problems, where B is an extension of C by A, and their equivalence under congruence (isomorphisms making commutative ) is analyzed via diagram chasing techniques that verify inclusions and quotients by navigating arrows in commutative diagrams. Long exact sequences extend this idea infinitely in one or both directions, such as \dots \to A_n \to A_{n+1} \to \dots, maintaining exactness at each term, or arising as \dots \to H_n(X) \to H_n(Y) \to H_{n-1}(Z) \to \dots from a short exact sequence of chain complexes $0 \to Z_\bullet \to X_\bullet \to Y_\bullet \to 0, where H_\bullet denotes groups. These sequences capture global relations across degrees, often via connecting homomorphisms that link at consecutive levels. A short exact sequence $0 \to A \to B \to C \to 0 splits if there exists a retraction B \to A or a C \to B making the diagram commute, equivalently if B \cong A \oplus C as objects in the . Not all short exact sequences split; for instance, the sequence $0 \to \mathbb{Z} \xrightarrow{\times n} \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} \to 0, where the first map is multiplication by n and the second is the canonical projection, is but does not split for n > 1, as \mathbb{Z} is not a direct summand of itself in this way. Diagram chasing in such examples confirms the exactness by showing that elements in the image match those annihilated by the next map, and establishes of extensions up to by ensuring any two extensions differing by a are congruent.

Snake Lemma and Five Lemma

The is a fundamental result in homological algebra that constructs a long from a of abelian groups (or morphisms in an ) with exact rows. Consider a of the form \begin{CD} 0 @>>> X @>{\alpha}>> Y @>{\beta}>> Z @>>> 0 \\ @. @V{f}VV @V{g}VV @V{h}VV @. \\ 0 @>>> X' @>>{\alpha'}> Y' @>>{\beta'}> Z' @>>> 0 \end{CD} where the rows are exact sequences. The snake lemma asserts that there exists a connecting \delta: \ker h \to \coker f such that the sequence $0 \to \ker f \to \ker g \to \ker h \xrightarrow{\delta} \coker f \to \coker g \to \coker h \to 0 is . This lemma was first established in the context of modules over rings, with the often visualized as a "snake" due to the zigzagging connecting map. The connecting \delta is constructed explicitly via chasing: for z \in \ker h, there exists y \in Y such that \beta(y) = z; since h(z)=0, the image g(y) lies in the image of \alpha', so y is in the image of \alpha up to adjustment, yielding the class in \coker f. at each term follows from the exactness of the rows and commutativity, ensuring that kernels and cokernels align appropriately. If the first vertical map f is injective, the initial segment $0 \to \ker f \to \ker g \to \ker his [exact](/page/Ex'Act) at the first two terms; similarly, if\beta'is surjective, the final segment to\coker his [exact](/page/Ex'Act). The full proof relies on chasing elements through the [diagram](/page/Diagram) to verify [exactness](/page/Ex'Act) at\ker hand\coker f$, with the snake shape illustrating the path of the argument. The five lemma complements the snake lemma by providing a criterion for isomorphisms in commutative diagrams of exact sequences. Specifically, consider a commutative diagram in an \begin{CD} 0 @>>> A_1 @>>> A_2 @>>> A_3 @>>> A_4 @>>> A_5 @>>> 0 \\ @. @V{f_1}VV @V{f_2}VV @V{f_3}VV @V{f_4}VV @V{f_5}VV @. \\ 0 @>>> B_1 @>>> B_2 @>>> B_3 @>>> B_4 @>>> B_5 @>>> 0 \end{CD} with exact rows. If f_1, f_2, f_4, and f_5 are isomorphisms, then f_3 is also an . More generally, if f_2 and f_4 are isomorphisms, f_1 is surjective, and f_5 is injective, then f_3 is an . This result holds by applying the to appropriate short exact sequences derived from the diagram, chasing to show that f_3 is both injective and surjective. These lemmas find key applications in establishing uniqueness properties and computations in homological algebra. The five lemma is instrumental in proving the uniqueness (up to chain homotopy equivalence) of projective resolutions: given two projective resolutions of a M, a chain map between them induces isomorphisms on via the lemma applied to the augmented complexes, ensuring the resolutions are equivalent. Similarly, the facilitates the computation of Ext groups; for a short $0 \to A \to B \to C \to 0, applying it to the Hom complexes with a projective resolution of Cyields a long [exact sequence](/page/Exact_sequence) in\Ext^( -, N), allowing \Ext^(C, N)to be derived from known values onAandB$.

Derived Constructions

Derived Functors

Derived functors provide a systematic way to quantify the failure of an additive functor to preserve exactness in abelian categories, serving as universal approximations to these deviations. In the context of homological algebra, they extend the notion of homology to functors, allowing the construction of long exact sequences from short exact ones. For an additive covariant functor F: \mathcal{A} \to \mathcal{B} between abelian categories where \mathcal{A} has enough injective objects and F is left exact, the right derived functors R^i F are defined using an injective resolution of an object A \in \mathcal{A}. An injective resolution of A is an exact sequence $0 \to A \to I^0 \to I^1 \to \cdots where each I^i is injective in \mathcal{A}. Applying F yields a complex F(I^\bullet), and R^i F(A) = H^i(F(I^\bullet)), the i-th cohomology group of this complex. Similarly, for a right exact functor F where \mathcal{A} has enough projective objects, the left derived functors L_i F(A) = H_i(F(P_\bullet)), with P_\bullet \to A \to 0 a projective resolution where each P_i is projective. The zeroth derived functors recover the original under appropriate exactness conditions: R^0 F \cong F for left exact F, and L_0 F \cong F for right exact F. If F is , all higher derived functors vanish, i.e., R^i F = [0](/page/0) and L_i F = [0](/page/0) for i > 0, reflecting that exact functors preserve all exact sequences without deviation. Vanishing also occurs for special objects: R^i F(A) = [0](/page/0) for i > 0 if A is injective, and L_i F(A) = [0](/page/0) for i > 0 if A is projective. These functors are independent of the choice of resolution, as different resolutions are equivalent, ensuring that the total complexes obtained by applying F yield isomorphic groups. Derived functors form δ-functors, satisfying exactness axioms that produce long exact sequences from short exact sequences in \mathcal{A}, such as \cdots \to R^i F(A') \to R^i F(A) \to R^i F(A'') \to R^{i+1} F(A') \to \cdots for $0 \to A' \to A \to A'' \to [0](/page/0). The construction extends to contravariant functors by dualizing resolutions, using projectives for right derived functors and injectives for left derived. A key result is the theorem, which describes how derived functors behave under ring homomorphisms: if \phi: R \to S is a ring map and F is induced accordingly, then under flatness conditions on S over R, R^i F_S \cong R^i F_R \otimes_R S, preserving the derived structure across rings. For example, the left derived functors of the functor F(A) = A \otimes_R M yield the Tor groups, L_i F(A) = \Tor_i^R(A, M), measuring the exactness failure of tensoring with M. Specific instances like Ext and Tor functors arise as derived functors of Hom and , respectively.

Ext and Tor Functors

In homological algebra, the Ext and functors arise as of the and functors, respectively, capturing obstructions to exactness in these operations. Specifically, for a R and R-modules A and B, the \operatorname{Ext}^i_R(A, B) is the i-th right of \operatorname{Hom}_R(A, -) applied to B, computed as the i-th of the complex \operatorname{Hom}_R(P_\bullet, B), where P_\bullet \to A is a projective resolution of A. Similarly, the \operatorname{Tor}_i^R(A, B) is the i-th left of -\otimes_R B applied to A, given by the i-th of the complex P_\bullet \otimes_R B. A key property of the is its interpretation in terms of extensions: \operatorname{Ext}^1_R(A, B) classifies short exact sequences of the form $0 \to B \to E \to A \to 0 up to , where two extensions are congruent if there exists an between them commuting with the maps to and from A and B. The group structure on \operatorname{Ext}^1_R(A, B) is induced by the Baer sum on extensions: given two extensions $0 \to B \xrightarrow{i_1} E_1 \xrightarrow{p_1} A \to 0 and $0 \to B \xrightarrow{i_2} E_2 \xrightarrow{p_2} A \to 0, their Baer sum is the extension $0 \to B \to E_1 \oplus E_2 / \Delta \to A \to 0, where \Delta = \{(b, -b) \mid b \in B\} is the of the on B, with b \mapsto (i_1(b), i_2(b)) and induced by (e_1, e_2) \mapsto (p_1(e_1), p_2(e_2)). This makes \operatorname{Ext}^1_R(A, B) into an , reflecting the additive structure of extension classes. Both functors satisfy long exact sequences arising from short exact sequences of modules. For a short exact sequence $0 \to A' \to A \to A'' \to 0, applying the derived functors yields the long exact sequence \cdots \to \operatorname{Ext}^i_R(A'', B) \to \operatorname{Ext}^i_R(A, B) \to \operatorname{Ext}^i_R(A', B) \to \operatorname{Ext}^{i+1}_R(A'', B) \to \cdots for the , and analogously for with homological indexing. These sequences encode how extensions and tensor products behave under exactness, providing tools to compute higher derived terms from lower ones. Concrete computations illustrate these functors' behavior over the integers R = \mathbb{Z}. For cyclic groups, \operatorname{Ext}^1_\mathbb{Z}(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z}) \cong \mathbb{Z}/n\mathbb{Z}, arising from the projective $0 \to \mathbb{Z} \xrightarrow{\cdot n} \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} \to 0 and applying \operatorname{Hom}_\mathbb{Z}(-, \mathbb{Z}), whose cohomology in degree 1 is the of multiplication by n on \mathbb{Z}. Likewise, \operatorname{Tor}_1^\mathbb{Z}(\mathbb{Z}/m\mathbb{Z}, \mathbb{Z}/n\mathbb{Z}) \cong \mathbb{Z}/\gcd(m,n)\mathbb{Z}, obtained from the same tensored with \mathbb{Z}/n\mathbb{Z}, where the homology in degree 1 is the of multiplication by n on \mathbb{Z}/m\mathbb{Z}, isomorphic to the gcd subgroup. Dimension shifting isomorphisms relate Ext and Tor across degrees and arguments. For instance, if $0 \to K \to P \to A \to 0 is a short exact sequence with P projective, then there is an isomorphism \operatorname{Ext}^{i+1}_R(A, B) \cong \operatorname{Ext}^i_R(K, B) for i \geq 0, derived from the long exact sequence by vanishing of terms involving the projective P. A balancing isomorphism further connects the functors: \operatorname{Tor}_i^R(A, B) \cong \operatorname{Tor}_i^R(B, A), reflecting the symmetry of the tensor product. These isomorphisms facilitate computations by shifting resolutions or swapping variables.

Computational Tools

Spectral Sequences

Spectral sequences provide a powerful for computing the or groups of a filtered chain or cochain , by successively refining approximations through a sequence of pages, each equipped with s of increasing length. Introduced by in his foundational work on the of representations, a arises from a filtered cochain (C^*, d) where the \{F^p C^n\} is a decreasing sequence of subcomplexes satisfying F^p C^n \supset F^{p+1} C^n and \bigcap_p F^p C^n = 0, \bigcup_p F^p C^n = C^n. The associated graded complex is \mathrm{gr}^p C^n = F^p C^n / F^{p+1} C^n, and the E_0-page is defined as E_0^{p,q} = H^{p+q}(\mathrm{gr}^p C^*), the of the graded pieces with respect to the induced d_0 of degree 1 preserving the grading. Subsequent pages are obtained by taking cohomology with respect to successively longer differentials: the r-th differential d_r: E_r^{p,q} \to E_r^{p+r, q-r+1} has bidegree (r, 1-r), and the (r+1)-page is E_{r+1}^{p,q} = H^{p,q}(E_r^*, d_r), the cohomology of the r-th page. Under suitable boundedness conditions, such as the filtration being bounded below (i.e., F^p C^n = 0 for p \ll 0), the spectral sequence converges to the cohomology of the total complex, meaning there exists a finite filtration on H^n(C^*, d) whose associated graded is \mathrm{gr}_p H^n(C^*, d) \cong E_\infty^{p, n-p}. First-quadrant spectral sequences, where E_r^{p,q} = 0 for p < 0 or q < 0, often arise in applications and enjoy strong convergence properties by the complete convergence theorem. Edge homomorphisms play a crucial role in extracting information from the , providing maps from the of the total to the terms on the edges of the pages. Specifically, the F^p C^* \hookrightarrow C^* induces a map H^n(C^*, d) \to H^n(F^p C^*, d), and the quotient C^*/F^{p+1} C^* \twoheadrightarrow \mathrm{gr}^p C^* induces H^n(\mathrm{gr}^p C^*, d_0) \to E_0^{p,n-p}; these compose to edge maps H^n(C^*, d) \to E_0^{p,n-p} \to E_r^{p,n-p} for r \geq 1, and in the limit, to E_\infty^{p,n-p}. These maps are natural and compatible across pages, allowing one to relate global invariants to local computations along the . A prominent example is the (also known as the Leray-Serre spectral sequence) for the of a Serre fibration F \to E \to B with fiber F, base B, and total space E, assuming the fibration is orientable or working with coefficients where needed. The E_2-page is E_2^{p,q} = H^p(B; \mathcal{H}^q(F; \mathbb{Z})), the of the base with local coefficients given by the of the fiber, and it converges to H^{p+q}(E; \mathbb{Z}). This sequence was developed by to compute the of Eilenberg-MacLane spaces and fibrations. In the case of the Hopf fibration S^1 \to S^3 \to S^2, the Serre spectral sequence computes H^*(S^3; \mathbb{Z}) from H^*(S^2; \mathbb{Z}) and H^*(S^1; \mathbb{Z}). Since the base S^2 is simply connected, local coefficients are trivial, so E_2^{p,q} = H^p(S^2; \mathbb{Z}) \otimes H^q(S^1; \mathbb{Z}), which is \mathbb{Z} at (0,0), (0,1), (2,0), (2,1), and $0elsewhere in low degrees. The nontrivial differential isd_2: E_2^{0,1} \to E_2^{2,0}, an isomorphism given by multiplication by the Euler class (of degree &#36;2), yielding E_3^{0,1} = 0 and E_3^{2,0} = 0; the term at (2,1) has d_2: E_2^{2,1} \to E_2^{4,0} = 0 and survives to E_\infty^{2,1} \cong \mathbb{Z}. Thus, E_\infty^{p,q} has \mathbb{Z} at (0,0) and (2,1), confirming H^n(S^3; \mathbb{Z}) = \mathbb{Z} for n=0,3 and $0otherwise, with the(2,1)term providing the graded piece\mathrm{gr}_2 H^3(S^3; \mathbb{Z}) \cong \mathbb{Z}$. This computation highlights how differentials detect the twisting in the fibration. The Grothendieck spectral sequence generalizes this to the composition of derived functors in abelian categories, providing a tool to compute the derived functors of a composite F \circ G from those of F and G separately. For left exact functors F, G: \mathcal{A} \to \mathcal{B} between abelian categories with enough projectives, and an object A \in \mathcal{A}, the E_2-page is E_2^{p,q} = R^p F (R^q G(A)), and it abuts to R^n (F \circ G)(A), converging under conditions like R^q G(A) = 0 for q \gg 0. This sequence, introduced in Grothendieck's seminal Tôhoku paper, underpins many computations in algebraic geometry and homological algebra, such as local-to-global Ext spectral sequences.

Horseshoe Lemma

The Horseshoe Lemma, also known as the simultaneous resolution theorem, constructs a projective for the middle object in a short exact sequence using given projective resolutions of the flanking objects. Consider a short exact sequence of modules $0 \to A' \xrightarrow{i} A \xrightarrow{p} A'' \to 0 over a R, along with projective resolutions P'_\bullet \twoheadrightarrow A' and P''_\bullet \twoheadrightarrow A''. The lemma asserts that there exists a projective resolution P_\bullet \twoheadrightarrow A and chain maps \alpha_\bullet: P'_\bullet \to P_\bullet, \beta_\bullet: P_\bullet \to P''_\bullet such that the sequence of chain complexes $0 \to P'_\bullet \xrightarrow{\alpha_\bullet} P_\bullet \xrightarrow{\beta_\bullet} P''_\bullet \to 0 is short exact in each degree (with P_n = P'_n \oplus P''_n), the original diagram of modules commutes with the augmentations, and this resolution P_\bullet is unique up to chain homotopy equivalence. The proof proceeds inductively by degree, starting from the augmented complexes and leveraging projectivity to lift the inclusion i: A' \to A and projection p: A \to A''. In degree zero, select a projective cover P_0 \twoheadrightarrow A; projectivity allows a lift \tilde{p}_0: P_0 \to P''_0 of p along the surjection P''_0 \twoheadrightarrow A'', and exactness at A ensures a lift \tilde{i}_0: P'_0 \to P_0 of i such that \tilde{p}_0 \circ \tilde{i}_0 = 0. For higher degrees n \geq 1, the previous maps induce a short exact sequence $0 \to \ker(d'_n) \to P'_n \oplus P_n \to \ker(d''_n) \to 0, where projectivity of P'_n \oplus P_nyields the required lifts\tilde{i}_nand\tilde{p}_nmaking the differentials compatible, with\tilde{i}_nand\tilde{p}_nforming the components of\alpha_nand\beta_n$. This construction preserves exactness and homotopy uniqueness due to the acyclic nature of the resolutions. A key application arises in computing left derived functors such as \Tor^R_i(-,B). Applying the tensor functor - \otimes_R B to the short exact sequence of projective complexes yields another short exact sequence of complexes $0 \to P'\bullet \otimes_R B \to P\bullet \otimes_R B \to P''\bullet \otimes_R B \to 0, since projective modules are flat. The associated long exact sequence in [homology](/page/Homology) is \cdots \to \Tor^R_i(A',B) \to \Tor^R_i(A,B) \to \Tor^R_i(A'',B) \to \Tor^R{i-1}(A',B) \to \cdots. Dually, for right derived functors like \Ext^i_R(-,B), an injective version of the [lemma](/page/Lemma) (obtained by passing to the opposite category) yields the long exact sequence \cdots \to \Ext^i_R(A'',B) \to \Ext^i_R(A,B) \to \Ext^i_R(A',B) \to \Ext^{i+1}_R(A'',B) \to \cdots$. These sequences hold because the horseshoe construction induces a short of complexes that remains exact under the relevant functors, preserving the ; direct sum isomorphisms occur when the connecting maps vanish, such as if the original short exact sequence splits. As an illustrative example, consider the short defining a , $0 \to M \xrightarrow{\iota} M \oplus N \xrightarrow{\pi} N \to 0, where \iota(m) = (m,0)and\pi(m,n) = n. Given projective resolutions P'\bullet \twoheadrightarrow MandP''\bullet \twoheadrightarrow N, the Horseshoe Lemma directly yields the resolution (P'\bullet \oplus P''\bullet) \twoheadrightarrow M \oplus Nvia the induced inclusion and projection maps, which are chain maps. Since the sequence splits, the long exact sequences in Tor and Ext degenerate to short exact sequences that split, giving\Tor^R_i(M \oplus N, B) \cong \Tor^R_i(M,B) \oplus \Tor^R_i(N,B)and similarly for Ext. This resolution is homotopy equivalent to any other projective resolution ofM \oplus N$ and demonstrates how the lemma facilitates computations for sums.

Applications and Extensions

Functoriality in Homology

In homological algebra, functoriality manifests through the construction of induced maps on groups arising from continuous maps between topological spaces or, more generally, chain maps between chain complexes. Given a continuous map f: X \to Y and the associated chain map f_\#: C_*(X) \to C_*(Y) on singular chains, the induced homomorphism f_*: H_n(X) \to H_n(Y) is defined by f_*() = [f_\#(z)] for a cycle z \in Z_n(X), preserving classes and respecting boundaries. Similarly, in , a map f: X \to Y induces f^*: H^n(Y) \to H^n(X) via the contravariant action on cochains, satisfying f^*([\phi]) = [\phi \circ f^\#] for a cocycle \phi \in Z^n(Y). These induced maps ensure that and are functors from the category of topological spaces (or pairs) to abelian groups, preserving composition: (g \circ f)_* = g_* \circ f_*. Naturality in homology extends this to transformations between functors on chain complexes. If \eta: F \to G is a between functors from complexes to complexes, then \eta induces a on : H_n(\eta): H_n(F(\mathcal{C})) \to H_n(G(\mathcal{C})) for any \mathcal{C}, commuting with the actions of chain maps. This property guarantees that commutes with , making it a functor in the derived sense and enabling the comparison of different theories. The Eilenberg-Steenrod axioms formalize the functorial behavior of a theory H_*( \cdot, \cdot; G) on pairs of topological spaces with coefficients in an G, providing a precise for theories like . These axioms include: (1) the dimension axiom, where H_n(pt) = G if n=0 and $0 otherwise for the point space; (2) the axiom, ensuring induced maps depend only on homotopy classes of maps; (3) the exactness axiom, stating that the sequence H_n(A) \to H_n(X) \to H_n(X,A) is exact for a pair (X,A); (4) the additivity axiom, where H_*( \coprod_i X_i ) \cong \bigoplus_i H_*(X_i) for disjoint unions; and (5) the excision axiom, where for a (X; A, B) with \overline{U} \subset \operatorname{int} A for U \subset A open, the inclusion induces an isomorphism H_*(U, U \cap B) \cong H_*(A, A \cap B). satisfies these axioms uniquely up to natural isomorphism for spaces of the type of CW-complexes. The wedge axiom for based spaces extends additivity and is used in uniqueness results for based theories. Excision is a key functorial property captured by a strengthened exactness axiom: for a triad (X; A, B) where A and B are subspaces with \overline{U} \subset \operatorname{int} A for U \subset A open, the inclusion induces an isomorphism H_*(U, U \cap B) \cong H_*(A, A \cap B). This leads to the Mayer-Vietoris sequence, a long exact sequence for the homology of a space X = U \cup V with open covers U, V \subset X: \cdots \to H_n(U \cap V) \xrightarrow{(i_* - j_*)} H_n(U) \oplus H_n(V) \xrightarrow{k_* + l_*} H_n(X) \to H_{n-1}(U \cap V) \to \cdots, where i, j are inclusions into U, V and k, l into X, derived from the short exact sequence of chains $0 \to C_*(U \cap V) \to C_*(U) \oplus C_*(V) \to C_*(U \cup V) \to 0 via the functorial long exact sequence in homology. Functoriality is exemplified in the Künneth theorem, which relates the of a product to tensor products and terms of the factors' homologies. For spaces X, Y with finitely generated free homology groups over \mathbb{Z}, there is a natural H_n(X \times Y) \cong \bigoplus_{p+q=n} H_p(X) \otimes H_q(Y) \oplus \bigoplus_{p+q=n+1} \operatorname{Tor}_1^\mathbb{Z}(H_p(X), H_q(Y)), arising from the Eilenberg-Zilber chain map and the universal coefficient theorem, preserving the functorial structure under product maps. This formula highlights how cross terms via account for torsional interactions in the product .

Connections to Other Fields

Homological algebra finds significant applications in through the development of , a tool introduced in the early 2000s for . computes topological features of data sets, such as point clouds, by tracking the birth and death of homology classes across scales, thereby providing robust summaries of shape and structure in high-dimensional data. This adaptation of classical from to computational settings has enabled applications in fields like and , where it identifies persistent patterns invariant under noise. In , homological algebra underpins the theory of derived categories, first formalized by Grothendieck in his 1957 Tôhoku paper, which axiomatizes resolutions and derived functors for sheaves of modules. These categories, denoted D(\mathcal{A}) for an \mathcal{A}, incorporate complexes up to quasi-isomorphisms and support operations like shifts {{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}} and cones, facilitating computations of . Verdier extended this framework in the by introducing triangulated categories, which formalize the triangulated structure of derived categories through distinguished triangles, enabling exact triangles to model fiber sequences and extensions in geometric contexts. Representation theory leverages homological algebra via Ext groups, which classify extensions of modules and quantify the structure of module categories over algebras. In this setting, Ext quotients describe the of representations, revealing homological dimensions and stability conditions for indecomposable modules. For quiver representations, where a is a and representations assign vector spaces to vertices and maps to arrows, homological methods, including projective resolutions, classify finite-dimensional indecomposables over algebraically closed fields, as established by Gabriel's theorem on quivers of finite type. In , homological algebra determines key invariants like depth and through the vanishing conditions of Ext and functors. Specifically, the depth of a M over a R relates to the highest i such that \operatorname{Ext}^i_R(k, M) \neq 0, where k is the , while measures flatness and tensor products. The Auslander-Buchsbaum formula asserts that for a finitely generated M of finite projective dimension over a commutative Noetherian R, \operatorname{pd}_R M = \operatorname{depth} R - \operatorname{depth} M, linking homological directly to ring and depths. André-Quillen homology provides a homological framework for commutative rings, generalizing to measure infinitesimal deformations and extensions of ring structures. Developed independently by André and Quillen in the late , it uses cotangent complexes to compute homology groups H_*(A/B, M) for commutative A-algebras B with coefficients in a bimodule M. Complementing this, , introduced by Grothendieck in 1966, employs a crystalline site over schemes in positive characteristic to define a Weil cohomology theory compatible with , using divided power envelopes to handle p-adic coefficients. A modern extension appears in motivic cohomology, pioneered by Voevodsky in the 1990s, which constructs a bigraded cohomology theory for algebraic varieties using triangulated categories of motives and sheaves with transfers. This theory bridges algebraic K-theory, étale cohomology, and Chow groups, with motivic cohomology groups H^{p,q}(X, \mathbb{Z}) relating to higher Chow groups and providing a universal cohomology for algebraic cycles.