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
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.
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
- 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 Extm(C, B) ⊗ Extn(B, A) → Extm+n(C, A).
- It is associative and agrees with composition of derived-functor classes.
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.
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.
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
Defined in any abelian category, including those without enough projectives or injectives. Agrees with the derived-functor definition whenever that exists.
The ring structure on Ext is transparent as splicing, whereas from resolutions it requires constructing chain maps and comparing them.
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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