Yoneda Ext and n-Fold Extensions

Executive summary

Extn without resolutions

An element of Extⁿ(C, A) can be presented as an exact sequence with n intermediate terms running from A to C, modulo the equivalence generated by maps between such sequences. Splicing two of these gives the Yoneda product, making Ext^*(C, C) a graded ring. The description needs no projectives or injectives, so it defines Ext in categories where resolutions are unavailable.

Learning objectives

Describe an n-fold extension and the equivalence relation.
Splice extensions to form the Yoneda product.
Explain why the equivalence must be generated rather than direct.
Identify the Ext ring and its role.

Section 01n-fold extensions

0 → A → B_n → … → B₁ → C → 0

Two such sequences are congruent when there is a morphism between them fixing A and C. The relevant equivalence is the one generated by congruence, since congruence itself is not symmetric — a morphism in one direction need not have an inverse.

The relation must be generated

For n = 1 the short five lemma makes any congruence an isomorphism, so the relation is already an equivalence. For n ≥ 2 it is not, and taking the generated equivalence relation — chains of morphisms in either direction — is essential. This is a genuine subtlety, not a formality.

Section 02Splicing and the Yoneda product

AlgorithmSplicing two extensionsin: two extensions sharing a middle module → out: their splice

Take an m-fold extension of C by B and an n-fold extension of B by A.
Compose the surjection onto B with the injection out of B to join the sequences at B.
The result is an (m + n)-fold extension of C by A. Exactness at the junction holds because the image of one map is B, which is the kernel of the next.
On equivalence classes this induces the Yoneda product Ext^m(C, B) ⊗ Extⁿ(B, A) → Ext^m+n(C, A).
It is associative and agrees with composition of derived-functor classes.

Use

The Ext algebra

Ext^*_Λ(k, k) for an augmented algebra is a graded ring whose structure encodes deep information — the Steenrod algebra arises this way.

Use

Group cohomology ring

H^*(G, k) = Ext^*_k[G](k, k) is a graded-commutative ring, and its spectrum is the support variety of modular representation theory.

Use

Obstruction theory

A class in Ext² is precisely the obstruction to extending a partial construction, and the Yoneda product composes successive obstructions.

Section 03Why the description matters

RobustnessNo resolutions needed

Defined in any abelian category, including those without enough projectives or injectives. Agrees with the derived-functor definition whenever that exists.

StructureThe product is visible

The ring structure on Ext is transparent as splicing, whereas from resolutions it requires constructing chain maps and comparing them.

Two descriptions, complementary strengths

Resolutions make Ext computable; Yoneda extensions make it interpretable. Most working arguments compute with a resolution and then interpret the answer as an extension class — particularly in degree 2, where the class is an obstruction.

ReferenceFrequently asked questions

Is the Yoneda definition equivalent to the derived functor one?

Yes, whenever the latter is defined — that is, when the category has enough projectives or enough injectives. The isomorphism is natural and respects the products.

Why is the equivalence relation generated rather than direct?

Because a morphism between n-fold extensions need not be invertible for n ≥ 2, so congruence is only a preorder. The equivalence it generates allows chains alternating in direction, which is what makes the classes a group.

What does an element of Ext<sup>2</sup> obstruct?

The extension of a module structure or a partial map over one further stage. In group cohomology H²(G, A) obstructs the existence of an extension realising a given action; in deformation theory it obstructs extending a first-order deformation to second order.

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