1 Definition and basic idea
The Ext functor is a homological construction that measures how one object can be assembled from another through extensions. It appears most naturally in abelian categories, especially in the category of modules over a ring and in the category of abelian groups. In these settings, Ext records both explicit extension classes and hidden obstructions to splitting exact sequences.
At an intuitive level, Hom describes direct maps between objects, while Ext detects more subtle relationships that only emerge when one studies exact sequences and resolutions. This makes Ext a central tool for understanding structure beyond ordinary morphisms.
1.1 Extensions of modules and abelian groups
Given modules or abelian groups \(A\) and \(B\), an extension is a short exact sequence \[ 0 \to B \to E \to A \to 0. \] Such a sequence shows how \(E\) is built from \(B\) and \(A\). If the sequence splits, then \(E\) is essentially a direct sum of the two pieces. If it does not split, the extension encodes nontrivial gluing data.
In many problems, one seeks to classify all extensions of one object by another up to an appropriate notion of equivalence. Ext provides the natural algebraic framework for doing this.
1.2 From Hom to Ext
The functor Hom assigns to a pair of objects the set or abelian group of morphisms between them. However, Hom alone does not see information that is hidden by lack of exactness. When a functor fails to preserve exact sequences, its higher derived functors measure the defect.
Ext arises from this idea by extending Hom into a family of functors that capture first-order and higher-order obstruction data. In this sense, Ext can be viewed as the derived version of Hom.
1.3 Ext as a derived functor
Ext is defined as the right derived functor of Hom in an abelian category with enough projective or injective objects. The basic construction begins by replacing one argument with a projective or injective resolution and then applying Hom degree by degree.
The resulting cohomology groups are the Ext groups, written \(\mathrm{Ext}^n\). The zeroth group recovers ordinary Hom, while the higher groups measure increasingly subtle extension phenomena.
2 Construction of Ext
There are two standard ways to build Ext: through projective resolutions and through injective resolutions. Both methods lead to canonically isomorphic results when the ambient category has enough projectives and enough injectives. This independence from the chosen resolution is one of the key strengths of the theory.
2.1 Projective resolutions
To compute \(\mathrm{Ext}^n(A,B)\) using projective resolutions, one chooses an exact sequence \[ \cdots \to P_1 \to P_0 \to A \to 0 \] in which each \(P_i\) is projective. Applying \(\mathrm{Hom}(-,B)\) produces a cochain complex whose cohomology in degree \(n\) is \(\mathrm{Ext}^n(A,B)\).
This approach is especially useful in module categories, where projective resolutions are often constructed explicitly.
2.2 Injective resolutions
Alternatively, one may resolve the second variable by an injective resolution \[ 0 \to B \to I^0 \to I^1 \to \cdots. \] Applying \(\mathrm{Hom}(A,-)\) yields a cochain complex, and its cohomology again gives \(\mathrm{Ext}^n(A,B)\).
Injective resolutions are often convenient in categories where injective objects are easier to handle than projective ones, such as categories of sheaves.
2.3 Equivalent definitions
Several equivalent descriptions of Ext exist, each emphasizing a different aspect of the theory. These formulations agree under suitable hypotheses and help connect Ext to other branches of algebra.
2.3.1 Via chain complexes
In the language of chain complexes, Ext can be realized as cohomology of a Hom complex built from a resolution. This viewpoint makes the functorial behavior of Ext transparent and connects it naturally to derived categories and spectral sequences.
2.3.2 Via Yoneda extensions
Ext can also be defined in terms of equivalence classes of extensions. In degree 1, the group \(\mathrm{Ext}^1(A,B)\) classifies short exact sequences of \(A\) by \(B\) up to equivalence. Higher degrees can be described using longer exact sequences, often called Yoneda extensions.
This definition is purely categorical and does not depend on explicit resolutions, making it conceptually important.
3 Ext groups
The Ext functor produces abelian groups or modules, depending on the context. These groups encode extension classes in different degrees and organize them into a graded family.
3.1 The groups Ext^n
For objects \(A\) and \(B\), the groups \(\mathrm{Ext}^n(A,B)\) are the \(n\)th derived functors of Hom. They form a graded collection indexed by nonnegative integers. Each degree has a distinct interpretation, with degree 1 corresponding to extensions and higher degrees to iterated extension data.
In module categories over a ring, these groups are typically modules over the same ring or over the base ring, depending on the setting.
3.2 The degree 0 case
The zeroth Ext group is just Hom: \[ \mathrm{Ext}^0(A,B) \cong \mathrm{Hom}(A,B). \] This reflects the idea that degree 0 contains no obstruction or hidden extension information. It is simply the space of ordinary morphisms.
3.3 Higher Ext groups
Higher Ext groups capture increasingly refined obstruction classes. The first group describes short exact sequences, while \(\mathrm{Ext}^2\) and beyond often appear as obstructions to lifting or extending constructions.
These higher groups are especially important in deformation theory, geometry, and representation theory, where they often control whether local data can be globalized or whether a deformation can be extended further.
4 Fundamental properties
Ext satisfies a collection of structural properties that make it a robust computational and conceptual tool. These include functoriality, exactness properties, additivity, and vanishing results under suitable hypotheses.
4.1 Functoriality
Ext is functorial in both variables, contravariantly in the first and covariantly in the second when defined in the usual convention. A map between input objects induces maps between the corresponding Ext groups.
This naturality ensures that Ext behaves consistently under morphisms and allows it to be used in diagram chases and categorical arguments.
4.2 Long exact sequences
A short exact sequence in either argument gives rise to a long exact sequence in Ext. This is one of the most useful features of the theory, because it allows the computation of unknown groups from known ones.
Long exact sequences also explain how Ext interacts with kernels, cokernels, and connecting morphisms. They are central in many inductive arguments.
4.3 Additivity
Ext is additive with respect to direct sums in each variable, up to the appropriate variance. For example, Ext of a direct sum in one argument often decomposes as a product or direct sum of the corresponding groups.
This property makes computations manageable and reflects the fact that Ext respects the additive structure of the ambient category.
4.4 Vanishing results
Ext groups often vanish in special cases. If the first object is projective, then higher Ext groups with it in the first variable vanish. If the second object is injective, the same occurs in the second variable.
Such vanishing results are foundational, because they characterize projective and injective objects homologically and often simplify complex calculations.
5 Computation methods
Although Ext is abstractly defined, it can often be computed by explicit techniques. The most common methods use resolutions, but change-of-rings formulas and spectral sequences also play major roles.
5.1 Using projective resolutions
A projective resolution converts the problem of computing Ext into the computation of cohomology of a Hom complex. When projective modules have simple descriptions, this method can be carried out directly.
This approach is standard for modules over rings with well-understood projective objects, such as principal ideal domains or polynomial rings in favorable cases.
5.2 Using injective resolutions
Injective resolutions provide an alternative computational route. They are particularly effective in contexts where injective objects have concrete descriptions, such as divisible abelian groups or injective sheaves.
Because the resulting Ext groups are independent of the chosen resolution, one may select whichever method is simpler in a given setting.
5.3 Change of rings
When passing between rings via a homomorphism, Ext groups can be compared using change-of-rings results. These formulas relate Ext over one ring to Ext over another and are often used when a module has structure over a larger or smaller ring.
Such comparisons are valuable in algebraic geometry and representation theory, where base change occurs naturally.
5.4 Spectral sequences
Spectral sequences provide a powerful indirect method for computing Ext. They arise from filtered complexes, double complexes, or compositions of functors, and they can express complicated Ext groups in terms of more accessible data.
Although abstract, spectral sequences often reduce difficult calculations to successive approximations and are indispensable in advanced homological algebra.
6 Relationship with exact sequences
Ext is closely tied to exact sequences, especially short exact sequences. In fact, the first Ext group can be interpreted as a classification space for extensions, and operations on extensions correspond to algebraic operations in Ext.
6.1 Classification of short exact sequences
The group \(\mathrm{Ext}^1(A,B)\) classifies short exact sequences \[ 0 \to B \to E \to A \to 0 \] up to an equivalence relation preserving the ends. The zero element corresponds to split sequences.
This interpretation makes Ext a natural language for describing how objects can be glued together without splitting into simpler pieces.
6.2 Baer sum
Baer sum is the operation that turns equivalence classes of extensions into an abelian group. It combines two extensions of the same object by the same object into a new one.
Under this operation, the set of extension classes becomes \(\mathrm{Ext}^1(A,B)\). The group structure is therefore not imposed artificially but arises from the geometry of exact sequences.
6.3 Splicing extensions
Longer extensions can be combined by splicing, which joins exact sequences end to end. This process underlies the Yoneda interpretation of higher Ext groups.
Splicing explains how iterated extension data can be encoded in higher cohomological degrees and why these groups naturally compose.
7 Examples
Concrete examples show how Ext behaves in familiar algebraic settings. These cases often illustrate both the power of the theory and the effect of structural properties of the base category.
7.1 Ext over principal ideal domains
Over a principal ideal domain, many modules admit simple classifications, which makes Ext more accessible. Torsion phenomena are especially important, and Ext often detects them in a precise way.
For finitely generated modules, computations typically reduce to decompositions into cyclic components. This makes Ext a useful companion to the structure theorem for modules over a PID.
7.2 Ext over fields
Over a field, every vector space is projective and injective in the usual module category. As a result, higher Ext groups vanish for vector spaces.
This reflects the semisimplicity of vector spaces: there are no nontrivial extension problems to measure.
7.3 Ext for cyclic groups
For cyclic groups viewed as modules over the integers, Ext groups are closely related to divisibility and torsion. For example, extensions of one cyclic group by another can be computed using standard resolutions.
These examples provide some of the simplest nontrivial instances of Ext and are frequently used in introductory homological algebra.
8 Applications
Ext appears throughout modern mathematics because it packages extension problems in a flexible and computable form. Its applications range from pure algebra to geometry and topology.
8.1 Homological algebra
In homological algebra, Ext is one of the basic derived functors and a prototype for more general constructions. It is used to define and study resolutions, derived functors, and homological dimensions.
Many foundational theorems in the subject can be formulated in terms of Ext, making it a core organizing concept.
8.2 Algebraic topology
In algebraic topology, Ext groups occur in cohomological calculations, especially through universal coefficient theorems and spectral sequences. They help relate homology and cohomology groups and can appear as correction terms in exact formulas.
Thus Ext often bridges algebraic invariants that at first seem separate.
8.3 Algebraic geometry
In algebraic geometry, Ext is used to study sheaves, coherent cohomology, and deformation problems. It describes extension classes of sheaves and helps formulate local-to-global principles.
Sheaf-theoretic Ext groups are especially important in understanding how geometric objects deform or how local data assembles into global structures.
8.4 Representation theory
In representation theory, Ext measures extensions between representations. These groups are essential for understanding whether a representation decomposes and how more complicated representations are built from simpler ones.
They also play a role in block theory, highest weight categories, and the study of derived equivalences.
9 Variants and generalizations
The Ext construction has many extensions beyond the classical setting of modules over rings. These generalizations preserve the central idea of measuring derived Hom information in broader contexts.
9.1 Derived categories
Derived categories provide a natural home for Ext. In this framework, Ext groups can be realized as morphisms between shifted objects in the derived category.
This perspective unifies resolutions, cohomology, and derived functors into a single formalism and is especially useful in modern homological algebra.
9.2 Sheaf Ext
Sheaf Ext is the version of Ext for sheaves of modules or abelian groups. It is used in geometry and topology to study local and global properties of sheaf morphisms and extension classes.
The sheaf-theoretic setting often requires injective resolutions of sheaves, and the resulting Ext groups can vary from local to global versions depending on context.
9.3 Ext in abelian categories
Ext can be defined in any abelian category with enough projectives or injectives, and in many more general settings via derived categories or exact categories. This abstraction shows that the concept depends less on modules specifically and more on the presence of exactness and homological structure.
As a result, Ext serves as a unifying language across large parts of mathematics.