Ext functor
In homological algebra, the Ext functor refers to the family of right derived functors of the Hom functor between modules over a ring, quantifying the extent to which the Hom functor fails to be exact and classifying extensions of modules.[1] For modules A and C over a ring R, the groups \operatorname{Ext}^n_R(A, C) are defined for n \geq 0, with \operatorname{Ext}^0_R(A, C) \cong \operatorname{Hom}_R(A, C), and higher groups obtained as the cohomology of the complex \operatorname{Hom}_R(P_\bullet, C), where P_\bullet is a projective resolution of A.[2][3] The Ext functor is contravariant in the first argument and covariant in the second, preserving direct sums in the second variable and direct products in the first.[1] It satisfies \operatorname{Ext}^n_R(A, C) = 0 for all n > 0 if A is projective or C is injective, characterizing these modules in terms of vanishing higher Ext groups.[2][3] For short exact sequences of modules, the Ext functor yields long exact sequences, including connecting homomorphisms that link extensions across the sequence.[1] A key interpretation arises in the first degree: \operatorname{Ext}^1_R(A, C) parametrizes the equivalence classes of short exact sequences $0 \to C \to X \to A \to 0 up to congruence, forming an abelian group under the Baer sum, where split extensions correspond to the zero element.[3] Higher Ext groups \operatorname{Ext}^n_R(A, C) for n > 1 generalize this to n-fold extensions or obstructions in lifting homomorphisms through exact sequences.[1] These functors unify cohomology theories for algebraic structures like groups, Lie algebras, and associative algebras, playing a central role in computing derived functors and spectral sequences in homological algebra.[1]Basic Concepts
Definition
In homological algebra, the Ext functor arises in the context of abelian categories, which are categories equipped with a zero object, finite biproducts, kernels and cokernels for every morphism, and the property that every monomorphism and epimorphism is normal. The Hom functor in an abelian category \mathcal{A}, denoted \mathrm{Hom}_{\mathcal{A}}(A, B) for objects A, B \in \mathcal{A}, assigns to each pair of objects the abelian group of morphisms from A to B, and extends contravariantly in the first argument and covariantly in the second to yield a bifunctor \mathrm{Hom}_{\mathcal{A}} : \mathcal{A}^{\mathrm{op}} \times \mathcal{A} \to \mathrm{Ab}.[4] This functor is left exact, meaning that if $0 \to B' \to B \to B'' \to 0 is a short exact sequence in \mathcal{A}, then $0 \to \mathrm{Hom}_{\mathcal{A}}(A, B') \to \mathrm{Hom}_{\mathcal{A}}(A, B) \to \mathrm{Hom}_{\mathcal{A}}(A, B'') \to 0 is exact for any A \in \mathcal{A}.[1] (Note: page 82 in the PDF.) To extend such left exact functors while measuring their failure to be exact, one defines derived functors using projective or injective resolutions of objects in \mathcal{A}, assuming \mathcal{A} has enough projectives or injectives. Specifically, the n-th Ext group \mathrm{Ext}^n_{\mathcal{A}}(A, B) for n \geq 0 and objects A, B \in \mathcal{A} is defined as the n-th right derived functor of \mathrm{Hom}_{\mathcal{A}}(A, -) evaluated at B, or dually as the n-th left derived functor of \mathrm{Hom}_{\mathcal{A}}(-, B) evaluated at A.[5][1] (Note: Chapter V, Section 3, pp. 82-83 in the PDF.) This construction embeds \mathrm{Ext}^n_{\mathcal{A}}(-, -) into the broader framework of derived functors on the derived category D(\mathcal{A}) of \mathcal{A}, where \mathrm{Ext}^n_{\mathcal{A}}(A, B) \cong \mathrm{Hom}_{D(\mathcal{A})}(A, B) for the n-th shift B.[6] In this derived functor framework, the zeroth Ext group satisfies the universal property of recovering the original Hom functor: \mathrm{Ext}^0_{\mathcal{A}}(A, B) \cong \mathrm{Hom}_{\mathcal{A}}(A, B), establishing an isomorphism $0 \to \mathrm{Hom}_{\mathcal{A}}(A, B) \to \mathrm{Ext}^0_{\mathcal{A}}(A, B) \to 0.[5][7] For n > 0, \mathrm{Ext}^n_{\mathcal{A}}(A, B) vanishes if A is projective, reflecting the exactness of \mathrm{Hom}_{\mathcal{A}}(A, -) in that case.[1] (Note: Chapter VI, Section 1, p. 106 in the PDF.) A concrete illustration occurs in the category \mathrm{Ab} of abelian groups, where \mathrm{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/m\mathbb{Z}, \mathbb{Z}) \cong \mathbb{Z}/m\mathbb{Z} for m \geq 1.[8]Properties
The Ext functors possess a bifunctorial nature, being contravariant in the first argument and covariant in the second. Specifically, for abelian groups (or modules over a ring) A, A', B, B' and morphisms f: A \to A', g: B' \to B, there is an induced natural transformation \operatorname{Ext}^n(f, g): \operatorname{Ext}^n(A', B') \to \operatorname{Ext}^n(A, B) for each n \geq 0.[9] This functoriality ensures that \operatorname{Ext}^\bullet(-, -) behaves compatibly with the category structure, forming a \delta-functor in the sense of homological algebra.[1] A key algebraic property is dimension shifting, which arises from short exact sequences. For instance, consider a short exact sequence $0 \to K \to P \to A \to 0wherePis projective; then there is an isomorphism\operatorname{Ext}^{n+1}(A, B) \cong \operatorname{Ext}^n(K, B)for alln \geq 0and anyB.[](https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/weibel-homv2.pdf) Similarly, in the second variable, if $0 \to B \to E \to I \to 0 with I injective, the long exact sequence implies isomorphisms under vanishing conditions on intermediate terms, such as \operatorname{Ext}^{n+1}(A, B) \cong \operatorname{Ext}^n(A, I) when \operatorname{Ext}^k(A, E) = 0 for relevant k.[1] These shifts facilitate computations by relating higher Ext groups to lower ones via resolutions. Applying the functor \operatorname{Hom}(A, -) to a short exact sequence $0 \to B' \to B \to B'' \to 0 yields a five-term [exact sequence](/page/Exact_sequence) $0 \to \operatorname{Hom}(A, B') \to \operatorname{Hom}(A, B) \to \operatorname{Hom}(A, B'') \to \operatorname{Ext}^1(A, B') \to \operatorname{Ext}^1(A, B), reflecting the left exactness of \operatorname{Hom}(A, -).[9] This sequence captures the initial deviation from exactness and is a direct consequence of the derived functor construction. More generally, the Ext functors satisfy a long exact sequence property: for a short exact sequence $0 \to A' \to A \to A'' \to 0$ of abelian groups (or modules), the sequence \cdots \to \operatorname{Ext}^n(A', B) \to \operatorname{Ext}^n(A, B) \to \operatorname{Ext}^n(A'', B) \to \operatorname{Ext}^{n+1}(A', B) \to \cdots is exact for all n \geq 0 and any B.[9][1] This long exact sequence is fundamental for analyzing how extensions behave under module homomorphisms and underpins many inductive arguments in homological algebra. The naturality of these constructions extends to compatibility with direct sums. In the category of modules over a ring, \operatorname{Ext}^n(\bigoplus_i A_i, B) \cong \prod_i \operatorname{Ext}^n(A_i, B) for arbitrary (possibly infinite) direct sums in the first variable, reflecting the contravariant additivity of \operatorname{Hom}(-, B).[9] In the second variable, \operatorname{Ext}^n(A, \bigoplus_j B_j) \cong \bigoplus_j \operatorname{Ext}^n(A, B_j), holding under the AB3 axiom for the category (small direct sums exact).[1] These isomorphisms hold for finite sums without qualification and are essential for decomposing computations in categories with good direct sum properties. As a concrete illustration, consider the category of vector spaces over a field k. Here, every vector space is both projective and injective, so \operatorname{Ext}^n(V, W) = 0 for all n \geq 1 and any vector spaces V, W.[9] In particular, \operatorname{Ext}^1(V, W) = 0 implies that every short exact sequence $0 \to W \to E \to V \to 0$ splits, confirming that extensions are trivial in this semisimple category.[1]Extensions and Their Classification
Equivalence of Extensions
A short exact extension of the module A by the module B is a short exact sequence of the form $0 \to B \xrightarrow{i} E \xrightarrow{p} A \to 0, where i is injective, p is surjective, and \ker p = \operatorname{im} i.[10] Two such extensions $0 \to B \xrightarrow{i} E \xrightarrow{p} A \to 0 and $0 \to B \xrightarrow{i'} E' \xrightarrow{p'} A \to 0 are equivalent if there exists an isomorphism \gamma: E \to E' such that the diagram \begin{CD} 0 @>>> B @>i>> E @>p>> A @>>> 0 \\ @. @| @V\gamma VV @| \\ 0 @>>> B @>>i'> E' @>>p'> A @>>> 0 \end{CD} commutes, meaning \gamma \circ i = i' and p' \circ \gamma = p. This equivalence relation is defined via commutative diagrams with identity maps on B and A, ensuring the middle terms are isomorphic while preserving the exactness.[1][10] There is a natural bijection between the set of equivalence classes of these extensions and the group \operatorname{Ext}^1(A, B). This isomorphism, established by Baer, maps each equivalence class to an element of \operatorname{Ext}^1(A, B) via a connecting homomorphism derived from projective resolutions of A.[1] Specifically, given an extension, one constructs a lift of the identity on A through a projective resolution P_\bullet \to A, yielding a cohomology class in \operatorname{Ext}^1(A, B); conversely, every element in \operatorname{Ext}^1(A, B) arises from such a lift, corresponding to a unique equivalence class of extensions by the Yoneda lemma.[1][10] A morphism between two extensions induces a map in \operatorname{Ext}^1(A, B) by composing with the connecting homomorphism, preserving the group structure. Baer's theorem confirms this correspondence is bijective, ensuring every class in \operatorname{Ext}^1(A, B) lifts to an extension and morphisms act functorially.[1] For example, in the category of abelian groups, the group \operatorname{Ext}^1(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z}) is isomorphic to \mathbb{Z}/n\mathbb{Z}, classifying extensions $0 \to \mathbb{Z} \to E \to \mathbb{Z}/n\mathbb{Z} \to 0. The zero class corresponds to the split extension E \cong \mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}, while the generator class [11] yields the nonsplit extension where the inclusion \mathbb{Z} \to \mathbb{Z} is multiplication by n, so E = \mathbb{Z} with quotient by n\mathbb{Z}; other classes are scalar multiples, all with middle term isomorphic to \mathbb{Z}.[10][1] If \operatorname{Ext}^1(A, B) = 0, every extension is equivalent to the split extension E \cong B \oplus A, as the vanishing implies the existence of a section s: A \to E such that p \circ s = \operatorname{id}_A. This provides a criterion for uniqueness up to isomorphism: the extension splits precisely when its class in \operatorname{Ext}^1(A, B) is zero.[10][1]Baer Sum
The Baer sum provides an addition operation on the set of equivalence classes of extensions in an abelian category, endowing \operatorname{Ext}^1(A, B) with the structure of an abelian group.[12] Given two extensions \xi: 0 \to B \xrightarrow{i} E \xrightarrow{p} A \to 0 and \eta: 0 \to B \xrightarrow{j} F \xrightarrow{q} A \to 0, the direct sum yields the extension $0 \to B \oplus B \to E \oplus F \to A \oplus A \to 0 with maps (i, j) and (p, q). To obtain the Baer sum \xi + \eta: 0 \to B \to E \oplus_A F \to A \to 0, first form the pushout of E \oplus F along the codiagonal map B \oplus B \to B (given by (b_1, b_2) \mapsto b_1 + b_2), resulting in an extension of A \oplus A by B; then take the pullback along the diagonal A \to A \oplus A (given by a \mapsto (a, a)), yielding the middle term E \oplus_A F as the fiber product over A. The inclusion into the Baer sum is the diagonal (i, j): B \to E \oplus F, and the projection is the codiagonal (p, q): E \oplus F \to A.[13][12] This construction is independent of the choices made in forming the direct sum and the pullback-pushout, as equivalent extensions yield equivalent Baer sums via natural isomorphisms of direct sums and the functoriality of pullbacks and pushouts in abelian categories.[12] The Baer sum is associative and commutative because direct sums are both, and the diagonal and codiagonal maps satisfy the required naturality conditions; the identity element is the equivalence class of the split extension $0 \to B \to A \oplus B \to A \to 0 (with inclusion to the second factor and projection from the first), and the inverse of [\xi] is [-\xi], obtained similarly by replacing the codiagonal on B with the difference map (b_1, b_2) \mapsto b_1 - b_2. Thus, the set of equivalence classes of extensions acquires a unique abelian group structure with this operation.[12][13] The zero element in this group structure corresponds precisely to the equivalence class of the split short exact sequence.[12] For a representative example in the category of abelian groups, consider \operatorname{Ext}^1_\mathbb{Z}(\mathbb{Z}/m\mathbb{Z}, \mathbb{Z}/n\mathbb{Z}) \cong \mathbb{Z}/\gcd(m,n)\mathbb{Z}, where the Baer sum of equivalence classes corresponds to addition in this group.[12]Computing Ext Groups
Projective Resolutions
One standard method for computing the Ext groups \operatorname{Ext}^n_{\mathcal{A}}(A, B) in an abelian category \mathcal{A} with enough projective objects, such as the category of modules over a unital ring R, involves constructing a projective resolution of the first argument A. A projective resolution of A is a long exact sequence \cdots \to P_2 \to P_1 \to P_0 \to A \to 0, where each P_i is projective (i.e., \operatorname{Hom}_{\mathcal{A}}(P_i, -) is an exact functor).[14] To compute \operatorname{Ext}^n_{\mathcal{A}}(A, B), delete the term A from the resolution to obtain the projective resolution complex \mathbf{P}_\bullet: \cdots \to P_2 \to P_1 \to P_0 \to 0. Apply the covariant Hom functor \operatorname{Hom}_{\mathcal{A}}(\mathbf{P}_\bullet, B) to yield the cochain complex $0 \to \operatorname{Hom}_{\mathcal{A}}(P_0, B) \xrightarrow{d^0} \operatorname{Hom}_{\mathcal{A}}(P_1, B) \xrightarrow{d^1} \operatorname{Hom}_{\mathcal{A}}(P_2, B) \to \cdots, where the differential d^n: \operatorname{Hom}_{\mathcal{A}}(P_n, B) \to \operatorname{Hom}_{\mathcal{A}}(P_{n+1}, B) is induced by composition with the resolution map P_{n+1} \to P_n, up to sign: d^n(f) = f \circ (-1)^{n+1} \delta_{n+1} for f \in \operatorname{Hom}_{\mathcal{A}}(P_n, B), with \delta_{n+1}: P_{n+1} \to P_n the resolution differential (the sign convention may vary but does not affect cohomology). Then, \operatorname{Ext}^n_{\mathcal{A}}(A, B) \cong H^n(\operatorname{Hom}_{\mathcal{A}}(\mathbf{P}_\bullet, B)), the nth cohomology group of this complex.[14] The step-by-step process begins with constructing the resolution: start with a surjection P_0 \twoheadrightarrow A from a projective P_0 (often free), set K_0 = \ker(P_0 \to A), then choose a surjection P_1 \twoheadrightarrow K_0 from a projective P_1, and iterate to obtain P_2 \twoheadrightarrow \ker(P_1 \to P_0), ensuring exactness at each P_i by the projectivity of the P_j. After applying \operatorname{Hom}_{\mathcal{A}}(-, B), compute the cohomology at each degree n \geq 0 as H^n = \frac{\ker(d^n: \operatorname{Hom}_{\mathcal{A}}(P_n, B) \to \operatorname{Hom}_{\mathcal{A}}(P_{n+1}, B))}{\operatorname{im}(d^{n-1}: \operatorname{Hom}_{\mathcal{A}}(P_{n-1}, B) \to \operatorname{Hom}_{\mathcal{A}}(P_n, B))}. The augmentation map P_0 \to A and the exactness of the full resolution \mathbf{P}_\bullet \to A \to 0 guarantee that H^0(\operatorname{Hom}_{\mathcal{A}}(\mathbf{P}_\bullet, B)) \cong \operatorname{Hom}_{\mathcal{A}}(A, B) and that the higher cohomology groups are independent of the choice of resolution, aligning with the axiomatic definition of Ext as the right derived functor of \operatorname{Hom}_{\mathcal{A}}(-, B). This follows from the long exact sequence induced by the short exact sequence $0 \to \ker(\epsilon) \to P_0 \xrightarrow{\epsilon} A \to 0, where \epsilon: P_0 \to A is the augmentation, and inductively applying the projectivity to show vanishing of certain connecting homomorphisms.[14] A concrete example illustrates this process over the ring R = \mathbb{Z}/p^2\mathbb{Z} for a prime p, with A = B = \mathbb{Z}/p\mathbb{Z} \cong R/(p). The minimal projective resolution of A is infinite and periodic: \cdots \to R \xrightarrow{\times p} R \xrightarrow{\times p} R \xrightarrow{\epsilon} A \to 0, where the differentials \delta_n: P_n \to P_{n-1} are multiplication by p for n \geq 1, and \epsilon: P_0 \to A is the canonical projection R \to R/(pR) (compositions vanish since p^2 = 0 in R). Deleting A yields \mathbf{P}_\bullet: \cdots \to R \xrightarrow{\times p} R \xrightarrow{\times p} R \to 0. Applying \operatorname{Hom}_R(-, B), note that \operatorname{Hom}_R(R, B) \cong B \cong \mathbb{Z}/p\mathbb{Z} for each term, as R-linear maps are determined by the image of $1, which must annihilate p. The induced differentials d^n are zero, because composition with \times p yields f \circ (\times p)(r) = f(r \cdot p) = p \cdot f(r) = 0 \cdot f(r) = 0 in B. Thus, all cohomology groups are H^n \cong \mathbb{Z}/p\mathbb{Z} for n \geq 0, so \operatorname{Ext}^n_R(A, B) \cong \mathbb{Z}/p\mathbb{Z} for all n \geq 0.[14] This approach using projective resolutions is advantageous in module categories where projective (often free) modules are straightforward to construct and manipulate, such as over polynomial rings or principal ideal domains, allowing explicit calculations of extension groups that reveal properties like projective dimension.[14]Injective Resolutions
To compute the Ext groups using injective resolutions, embed the module B into an exact sequence known as an injective resolution:$0 \to B \to I^0 \to I^1 \to I^2 \to \cdots,
where each I^n is an injective module and the sequence is exact. Apply the functor \Hom(A, -) to this resolution, yielding a cochain complex
$0 \to \Hom(A, I^0) \to \Hom(A, I^1) \to \Hom(A, I^2) \to \cdots.
The n-th cohomology group of this complex is isomorphic to \Ext^n(A, B).[15] This approach computes the right derived functors R^n \Hom(A, -)(B) of the left-exact covariant functor \Hom(A, -). By the balance theorem for Ext, these are naturally isomorphic to the left derived functors L_n \Hom(-, B)(A) of the left-exact contravariant functor \Hom(-, B), allowing the same groups to be obtained via projective resolutions of A. The isomorphism arises from dimension-shifting arguments in the homological algebra of abelian categories, ensuring consistency between the two computational methods.[16] The process involves truncating the injective resolution immediately after B, so the relevant cochain complex for cohomology computation begins at \Hom(A, I^0) and proceeds to higher terms without including \Hom(A, B). The cohomology is then calculated as the kernel of the map to the next term modulo the image from the previous term at each degree n \geq 0, with \Ext^0(A, B) \cong \Hom(A, B). This yields a well-defined invariant independent of the choice of resolution, up to natural isomorphism.[15] In the category of sheaves on a Riemann surface, injective resolutions facilitate computations of sheaf Ext groups; for instance, the Dolbeault resolution provides an injective resolution of the structure sheaf \mathcal{O}_X:
$0 \to \mathcal{O}_X \to \mathcal{A}^{0,0}_X \to \mathcal{A}^{0,1}_X \to 0,
where \mathcal{A}^{0,q}_X denotes the sheaf of smooth (0,q)-forms (fine sheaves, hence injective). Applying \Hom(\mathbb{Z}_X, -) and taking cohomology computes \Ext^1(\mathbb{Z}_X, \mathcal{O}_X) \cong H^1(X, \mathcal{O}_X) \cong H^{0,1}(X), the space of conjugate Dolbeault classes, whose dimension equals the genus of the surface. Similar resolutions of the constant sheaf \mathbb{C}_X can be used to compute groups like \Ext^1(\mathcal{O}_X, \mathbb{C}_X).[9] This method is particularly preferable in categories lacking enough projective objects, such as the category of sheaves of abelian groups (or \mathcal{O}_X-modules) on a topological space, where projective sheaves are scarce or nonexistent beyond trivial cases, but enough injective sheaves (e.g., flabby or fine sheaves) always exist to form resolutions.[9]