The balanced Ext bifunctor

Assume the Axiom of Dependent Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain. Let $\mathcal{A}$ be an abelian category with enough projectives and enough injectives, and fix supplied projective and injective resolution data on all objects of $\mathcal{A}$. For each $n\ge 0$, define ${\mathrm{Ext}}_{\mathcal{A}}^n(M,N)$ to mean either ${\mathrm{Ext}}_P^n(M,N)$ or ${\mathrm{Ext}}_I^n(M,N)$, identified by Projective and injective constructions of Ext agree for supplied resolutions. The independence and naturality statements in The Ext balance isomorphism is independent of resolution comparison data and The Ext balance isomorphism is natural in both variables apply under the stated DC assumption. This is a bifunctor contravariant in $M$ and covariant in $N$, up to the canonical identifications below; it is not a definition by equality of the two complexes.

For projective resolutions $P,P^{\mathrm{\prime }}$ of the same $M$, use Projective comparison maps exist under DC to choose $f:P^{\mathrm{\prime }}\to P$ lifting the identity. Precomposition induces a map $c_{P,P^{\mathrm{\prime }}}:H^n\mathrm{Hom}(P,N)\to H^n\mathrm{Hom}(P^{\mathrm{\prime }},N)$. Any two such $f$ are chain-homotopic by Projective comparison maps are unique up to chain homotopy. If $f-g=dh+hd$, precomposition by $h$ is the cochain homotopy between the two Hom maps: in degree $n$ it sends a cochain $\varphi$ to $\varphi h_{n-1}$. Thus the induced map on cohomology is independent of $f$. An identity comparison induces the identity. If $P^{\mathrm{\prime }\mathrm{\prime }}$ is a third resolution, the composite of comparisons $P^{\mathrm{\prime }\mathrm{\prime }}\to P^{\mathrm{\prime }}\to P$ lifts the same identity as a direct comparison; uniqueness up to homotopy gives $c_{P^{\mathrm{\prime }},P^{\mathrm{\prime }\mathrm{\prime }}}{c}_{P,P^{\mathrm{\prime }}}=c_{P,P^{\mathrm{\prime }\mathrm{\prime }}}$. In particular opposite-direction comparisons induce inverse isomorphisms.

For injective resolutions $I,I^{\mathrm{\prime }}$ of $N$, use Injective comparison maps exist to obtain $g:I\to I^{\mathrm{\prime }}$ under $N$. Postcomposition gives $c_{I,I^{\mathrm{\prime }}}$ on $H^n\mathrm{Hom}(M,I)$. The cochain-homotopy uniqueness of Injective comparison maps are unique up to cochain homotopy makes this map independent of $g$: postcomposing with a homotopy gives a Hom cochain homotopy. Identities and composites of the comparison maps prove $c_{I,I}=1$ and $c_{I^{\mathrm{\prime }},I^{\mathrm{\prime }\mathrm{\prime }}}{c}_{I,I^{\mathrm{\prime }}}=c_{I,I^{\mathrm{\prime }\mathrm{\prime }}}$, including invertibility.

Write ${\beta }_{P,I}$ for the balance isomorphism from the projective presentation to the injective presentation. The augmentation-compatible Hom double-complex squares in the cited independence and naturality results give $c_{I,I^{\mathrm{\prime }}}{\beta }_{P,I}={\beta }_{P^{\mathrm{\prime }},I^{\mathrm{\prime }}}{c}_{P,P^{\mathrm{\prime }}}\mathrm{.}$ This square, together with the two comparison composition laws, proves coherence of changing either presentation, including successive mixed changes. It is the change map $c_{P,P}$ or $c_{I,I}$ that is an identity when its resolution is unchanged. The balance map ${\beta }_{P,I}$ between different presentations need not be a literal identity. No uniqueness theorem for arbitrary delta-functor morphisms is used in this justification.

The specified resolution systems are hypotheses, not a global selection deduced here from DC. For fixed resolutions and explicitly supplied comparison maps and homotopies, the Hom and cohomology calculations above are choice-free; DC is used for existence of the arbitrary comparison data.