# Ext functor

In homological algebra, the **Ext functor** is a family of functors \( \mathrm{Ext}^n_R(-,-) \) attached to modules over a ring \( R \) (or objects of an abelian category), defined as the right derived functors of the [Hom functor](https://www.edgepedia.org/hom-functor). For a fixed module \( A \), the functors \( \mathrm{Ext}^n_R(A, \cdot) \) are the right derived functors of \( h_A(N) = \mathrm{Hom}_R(A, N) \)<sup>[1](https://www.math.u-bordeaux.fr/~dbenoua/Documents/Cours-Homological-algebra.pdf)</sup>. The name comes from *extension*: \( \mathrm{Ext}^1_R(A, C) \) classifies extensions of \( A \) with kernel \( C \) up to equivalence<sup>[2](https://encyclopediaofmath.org/wiki/Derived_functor)</sup>.

| Key fact | Detail |
|---|---|
| Definition | \( \mathrm{Ext}^n_R(A, \cdot) \) is the \( n \)-th right derived functor of \( \mathrm{Hom}_R(A, \cdot) \)<sup>[1](https://www.math.u-bordeaux.fr/~dbenoua/Documents/Cours-Homological-algebra.pdf)</sup> |
| Degree zero | \( \mathrm{Ext}^0(M, N) = \mathrm{Hom}(M, N) \)<sup>[1](https://www.math.u-bordeaux.fr/~dbenoua/Documents/Cours-Homological-algebra.pdf)</sup> |
| Variance | Contravariant in the first variable, covariant in the second<sup>[1](https://www.math.u-bordeaux.fr/~dbenoua/Documents/Cours-Homological-algebra.pdf)</sup> |
| Computation | Given by the cohomology of \( \mathrm{Hom}_A(P_\bullet, Y) \) for a projective resolution \( P_\bullet \to X \), or by injective resolutions of the second argument<sup>[3](https://people.math.harvard.edu/~lurie/papers/HA2012.pdf)</sup> |
| Vanishing | \( \mathrm{Ext}^1_R(P, B) = 0 \) for all \( B \) when \( P \) is projective<sup>[4](https://math.ucla.edu/~sharifi/notes/homalg-ch03.html)</sup> |
| Extension meaning | \( \mathrm{Ext}^1_R(A, C) \) classifies extensions \( 0 \to C \to E \to A \to 0 \) up to equivalence, with the Baer sum as group operation<sup>[2](https://encyclopediaofmath.org/wiki/Derived_functor)</sup><sup> • </sup><sup>[5](https://arxiv.org/abs/1603.02091)</sup> |
| Origin | Introduced by Samuel Eilenberg and Saunders Mac Lane in 1942; the \( \mathrm{Ext}^n \) notation first appeared in the 1956 Cartan–Eilenberg book<sup>[6](https://mathshistory.st-andrews.ac.uk/Biographies//Eilenberg/)</sup><sup> • </sup><sup>[7](https://www.sciencedirect.com/topics/mathematics/eilenberg)</sup> |

## Definition and computation

Let \( A \) be an abelian category with enough injectives, fix an object \( M \), and consider the left exact functor \( h_M : A \to \mathbf{Ab} \) given by \( h_M(N) = \mathrm{Hom}(M, N) \). The right derived functors of \( h_M \) are the Ext groups, \( \mathrm{Ext}^i(M, N) := R^i h_M(N) \)<sup>[1](https://www.math.u-bordeaux.fr/~dbenoua/Documents/Cours-Homological-algebra.pdf)</sup>. Derived functors measure the failure of a left exact functor to be exact: \( R^0 F \) is naturally isomorphic to \( F \) exactly when \( F \) is left exact, and if \( F \) is exact then all higher derived functors vanish<sup>[2](https://encyclopediaofmath.org/wiki/Derived_functor)</sup>.

The Hom functor is left exact but generally not exact. Applying \( \mathrm{Hom}_{\mathbb{Z}}(\mathbb{Z}/2, \cdot) \) to the exact sequence \( 0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0 \) gives a sequence that is not exact on the right, which is where the first Ext group appears<sup>[8](https://web.auburn.edu/holmerr/8310/textbook-print.pdf)</sup>.

Resolutions make Ext computable. If \( X \) admits a [projective resolution](https://www.edgepedia.org/projective-resolution) \( \cdots \to P_2 \to P_1 \to P_0 \to X \), then \( \mathrm{Ext}^n_A(X, Y) \) is the \( n \)-th cohomology of the cochain complex \( \mathrm{Hom}_A(P_\bullet, Y) \); equivalently, one may resolve the second argument injectively<sup>[3](https://people.math.harvard.edu/~lurie/papers/HA2012.pdf)</sup>. The result is independent of the chosen resolution. Projectivity of the first argument kills the functor: \( \mathrm{Ext}^1_R(P, B) = 0 \) for all \( B \) when \( P \) is projective, because \( \mathrm{Hom}_R(P, \cdot) \) is then exact<sup>[4](https://math.ucla.edu/~sharifi/notes/homalg-ch03.html)</sup>.

A standard example over \( \mathbb{Z} \) uses the injective resolution \( 0 \to \mathbb{Z}/n\mathbb{Z} \to \mathbb{Q}/\mathbb{Z} \xrightarrow{n} \mathbb{Q}/\mathbb{Z} \to 0 \); computing the cohomology of the resulting complex yields the Ext groups of cyclic groups<sup>[4](https://math.ucla.edu/~sharifi/notes/homalg-ch03.html)</sup>.

## The long exact sequence

A short exact sequence of modules produces a long exact sequence of Ext groups. If \( 0 \to B_0 \to B \to B_{00} \to 0 \) is exact, then

\[ 0 \to \mathrm{Hom}_R(A, B_0) \to \mathrm{Hom}_R(A, B) \to \mathrm{Hom}_R(A, B_{00}) \to \mathrm{Ext}^1_R(A, B_0) \to \mathrm{Ext}^1_R(A, B) \to \mathrm{Ext}^1_R(A, B_{00}) \to \mathrm{Ext}^2_R(A, B_0) \to \cdots \]<sup>[8](https://web.auburn.edu/holmerr/8310/textbook-print.pdf)</sup>

with natural connecting morphisms. There is a corresponding contravariant sequence in the first variable, and the right derived functors of the covariant and contravariant Hom functors coincide<sup>[9](https://agostoni.web.elte.hu/bboard/fg26tav/lectures.html)</sup>. This sequence is what ties \( \mathrm{Hom} = \mathrm{Ext}^0 \) to the higher \( \mathrm{Ext}^n \) into a single family of functors<sup>[10](https://pages.jh.edu/rrynasi1/FoundationsOFMath/Literature/Kro%CC%88mer2007ToolAndObject.AHistory%2BPhilosophyOfCategaryTheory.pdf)</sup>.

## Extensions and the Baer sum

An extension of an abelian group \( G \) by an abelian group \( H \) is a short exact sequence \( 0 \to H \to X \to G \to 0 \). The set of equivalence classes of such extensions is an abelian group under the **Baer sum**, formed by taking the pushout of the direct product extension along the addition map on \( H \) and the diagonal map on \( G \)<sup>[5](https://arxiv.org/abs/1603.02091)</sup>. This group is \( \mathrm{Ext}^1(G, H) \), and one writes \( \mathrm{Ext}(G, H) = 0 \) when every such extension splits<sup>[5](https://arxiv.org/abs/1603.02091)</sup>.

Noboru Yoneda (a mathematician known for the [Yoneda lemma](https://www.edgepedia.org/yoneda-lemma) and Yoneda product) showed that an element of the higher group \( \mathrm{Ext}_n(A, B) \) can be represented as a long exact sequence \( 0 \to B \to E_1 \to E_2 \to \cdots \to E_n \to A \to 0 \) with \( n \) intermediate terms<sup>[11](https://www.nationalacademies.org/read/10169/chapter/9)</sup>. Equivalence classes of these long exact sequences, with the Yoneda product as composition, give a resolution-free definition of Ext that works even when projective or injective resolutions are unavailable<sup>[9](https://agostoni.web.elte.hu/bboard/fg26tav/lectures.html)</sup><sup> • </sup><sup>[12](https://arxiv.org/abs/2302.12678)</sup>.

## History

The functor originated in the study of [group extensions](https://www.edgepedia.org/group-extension). In joint work with O. F. G. Schilling on class field theory, [Saunders Mac Lane](https://www.edgepedia.org/saunders-mac-lane) calculated \( \mathrm{Ext}(G, A) \) for the group \( G \) generated by elements \( a_n \) with \( a_{n+1} = p a_n \) for a prime \( p \), the character group of the \( p \)-adic solenoid. When Mac Lane lectured on this at the University of Michigan in 1940, [Samuel Eilenberg](https://www.edgepedia.org/samuel-eilenberg) recognized the calculation as matching Steenrod's computation of the homology of the solenoid, \( \mathrm{Ext}(\Sigma^*, I) \cong H_1(S^3 - \Sigma, I) \); the joint paper explaining this coincidence appeared in 1942 and introduced Hom and Ext for the first time<sup>[10](https://pages.jh.edu/rrynasi1/FoundationsOFMath/Literature/Kro%CC%88mer2007ToolAndObject.AHistory%2BPhilosophyOfCategaryTheory.pdf)</sup><sup> • </sup><sup>[13](https://celebratio.org/MacLane_S/article/288/)</sup><sup> • </sup><sup>[6](https://mathshistory.st-andrews.ac.uk/Biographies//Eilenberg/)</sup>.

The argument order changed. In the 1942 paper Eilenberg and Mac Lane wrote \( \mathrm{Ext}(G, H) \) for extensions of \( G \) by \( H \); the modern order reverses this, because \( \mathrm{Ext}^1 \) belongs to a family connected to \( \mathrm{Hom} \) by a long exact sequence, and it is natural to write \( \mathrm{Hom}(A, B) \) for morphisms from \( A \) to \( B \)<sup>[10](https://pages.jh.edu/rrynasi1/FoundationsOFMath/Literature/Kro%CC%88mer2007ToolAndObject.AHistory%2BPhilosophyOfCategaryTheory.pdf)</sup>. Mac Lane was already using the modern order by 1950.

The systematic theory arrived with [Henri Cartan](https://www.edgepedia.org/henri-cartan) and Samuel Eilenberg's 1956 book *Homological Algebra*, where the notation \( \mathrm{Tor}_n \) and \( \mathrm{Ext}^n \), the concepts of projective module and derived functor, and hyperhomology first appeared<sup>[7](https://www.sciencedirect.com/topics/mathematics/eilenberg)</sup><sup> • </sup><sup>[14](https://www.sciencedirect.com/topics/mathematics/algebraic-construction)</sup>. The book subsumed the known homology theories of groups, algebras, and Lie algebras under the resolution, Tor, and Ext pattern<sup>[15](https://mathshistory.st-andrews.ac.uk/Extras/Cartan_Henri_books/)</sup>.

## Applications

**Universal coefficient theorems.** In algebraic topology, the cohomology of the cochain complex \( \mathrm{Hom}(C_*(K), A) \) is described as the product of \( \mathrm{Hom}(H_q, A) \) and \( \mathrm{Ext}(H_{q-1}, A) \), the form the statement has taken since the early 1950s<sup>[17](https://www.sciencedirect.com/topics/mathematics/extension-of-a-group)</sup>. In the standard demonstration that \( \pi_5(S^3) \cong \mathbb{Z}/2 \), a role is played by the universal coefficient theorem, which connects cohomology with Ext groups<sup>[12](https://arxiv.org/abs/2302.12678)</sup>.

**Sheaves and algebraic geometry.** For sheaves \( F, G \) of \( \mathcal{O}_X \)-modules on a space \( X \), the functor \( G \mapsto \Gamma_Z(\mathrm{Hom}_{\mathcal{O}_X}(F, G)) \) has derived functors \( \mathcal{E}xt \) and local Ext, connected by spectral sequences abutting to \( \mathrm{Ext}^\bullet_Z(X; F, G) \)<sup>[18](https://jcreinhold.github.io/sga/ii/06-ext-functors.html)</sup>.

**Computation.** The computer algebra system [Macaulay2](https://www.edgepedia.org/macaulay2), built for research in algebraic geometry and commutative algebra, computes Ext modules among its core capabilities<sup>[19](https://www.macaulay2.com/)</sup>. For rings of complete intersections, the total Ext module is computed by the algorithm of Shamash, Eisenbud, Avramov, and Buchweitz<sup>[20](https://macaulay2.com/doc/Macaulay2-1.24.11/share/doc/Macaulay2/Macaulay2Doc/html/___Ext_lp__Module_cm__Module_rp.html)</sup>.

**Beyond modules.** Ext groups have been studied for topological abelian groups and topological vector spaces, where \( \mathrm{Ext}(G, H) \) collects extensions \( 0 \to H \to X \to G \to 0 \) of topological groups and \( \mathrm{Ext}_{TVS}(Z, Y) \) does the same for topological vector spaces<sup>[5](https://arxiv.org/abs/1603.02091)</sup>. In homotopy type theory, the Yoneda approach has been formalized without resolutions; there, higher Ext groups of abelian groups, which vanish in dimensions 2 and up in classical mathematics, may be nontrivial in all dimensions<sup>[12](https://arxiv.org/abs/2302.12678)</sup>.

## References

1. [Introduction to Homological Algebra, D. Benouaert, Université de Bordeaux](https://www.math.u-bordeaux.fr/~dbenoua/Documents/Cours-Homological-algebra.pdf)
2. [Derived functor, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Derived_functor)
3. [Higher Algebra, Jacob Lurie](https://people.math.harvard.edu/~lurie/papers/HA2012.pdf)
4. [Homological Algebra, Chapter 3: Derived Functors, Romyar Sharifi, UCLA](https://math.ucla.edu/~sharifi/notes/homalg-ch03.html)
5. [The Ext group in the categories of topological abelian groups and topological vector spaces, Hugo J. Bello, arXiv:1603.02091](https://arxiv.org/abs/1603.02091)
6. [Samuel Eilenberg (1913–1998), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies//Eilenberg/)
7. [Eilenberg, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/mathematics/eilenberg)
8. [Homological Algebra I, course text, Auburn University](https://web.auburn.edu/holmerr/8310/textbook-print.pdf)
9. [MSc Topics in Ring Theory, lecture summaries, Eötvös Loránd University](https://agostoni.web.elte.hu/bboard/fg26tav/lectures.html)
10. [Tool and Object: A History and Philosophy of Category Theory, R. Krömer (2007)](https://pages.jh.edu/rrynasi1/FoundationsOFMath/Literature/Kro%CC%88mer2007ToolAndObject.AHistory%2BPhilosophyOfCategaryTheory.pdf)
11. [Samuel Eilenberg, Biographical Memoirs Volume 79, National Academy of Sciences](https://www.nationalacademies.org/read/10169/chapter/9)
12. [Formalising Yoneda Ext in Univalent Foundations, J. G. Taxerås Flaten, arXiv:2302.12678](https://arxiv.org/abs/2302.12678)
13. [Celebratio Mathematica: Saunders Mac Lane, W. Tholen](https://celebratio.org/MacLane_S/article/288/)
14. [Algebraic Construction, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/mathematics/algebraic-construction)
15. [Henri Cartan Books, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Extras/Cartan_Henri_books/)
16. [Ext, nLab](https://ncatlab.org/nlab/show/Ext)
17. [Extension of a Group, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/mathematics/extension-of-a-group)
18. [SGA 2, Exposé VI: The functors Ext and ℰxt, English translation by J. C. Reinhold](https://jcreinhold.github.io/sga/ii/06-ext-functors.html)
19. [Macaulay2](https://www.macaulay2.com/)
20. [Ext(Module,Module) — total Ext module, Macaulay2 documentation](https://macaulay2.com/doc/Macaulay2-1.24.11/share/doc/Macaulay2/Macaulay2Doc/html/___Ext_lp__Module_cm__Module_rp.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Homological algebra and K-theory › Derived functors, Ext and Tor*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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