Ext and Balanced Resolutions

Exactness gives $\ker d^0=\mathrm{im}\iota$; thus $\iota f$ is a degree-zero cocycle, and $f\mapsto \iota f$ is injective because $\iota$ is monic.

If $d^{0}g=0$, then $g$ factors uniquely through $\iota$. Since there are no negative-degree coboundaries, this is the required $H^0$ identification. Precomposition makes it natural in $M$. For $b:N\to N^{\mathrm{\prime }}$, choose a comparison extension between the supplied resolutions; its degree-zero square with the coaugmentations commutes, so the identification is natural in $N$, and the independence lemma makes this map independent of the chosen extension.

A map $h:P_0\to N$ is a zero-cocycle exactly if $h{d}_1=0$, equivalently if it vanishes on $\ker \epsilon =\mathrm{im}{d}_1$.

Thus $h$ factors uniquely as $f\epsilon$; there are no negative-degree coboundaries. This is the asserted isomorphism and is natural under pre- and postcomposition.

Precomposition by $a$ is a cochain map $\mathrm{Hom}(M,I^{\bullet })\to \mathrm{Hom}(M^{\mathrm{\prime }},I^{\bullet })$. A comparison extension of $b$ is a cochain map $I^{\bullet }\to I^{\mathrm{\prime }\bullet }$, hence postcomposition gives the second cochain map.

Homotopic comparison extensions induce the same cohomology map, so the second map is choice-independent. Identity and composition of the comparison maps give the functor laws, with $a$ reversing arrows.

Postcomposition with $v$ is a cochain map $\mathrm{Hom}(P_{\bullet }(M),N)\to \mathrm{Hom}(P_{\bullet }(M),N^{\mathrm{\prime }})$. A comparison lift $\tilde{u}:P_{\bullet }(M^{\mathrm{\prime }})\to P_{\bullet }(M)$ from A morphism has a comparison lift between the supplied projective resolutions gives precomposition ${\tilde{u}}^{\ast }$ in the opposite direction.

These maps commute because pre- and postcomposition commute. Two lifts of $u$ are chain-homotopic by Projective comparison maps are unique up to chain homotopy. Precomposing with the homotopy gives a cochain homotopy between the two induced maps on $\mathrm{Hom}(-,N)$, so they induce the same map on $H^n$. The comparison identity and composition laws hold up to such homotopy, while postcomposition is strictly functorial. Hence the maps on $H^n$ define the asserted bifunctor.

Apply Positive right derived functors vanish on injective objects to the supplied injective-resolution datum and the additive left exact functor $\mathrm{Hom}(M,-)$. It compares the supplied resolution of $J$ with the length-zero injective resolution and gives vanishing in every positive degree.

By Ext via an injective resolution of the second variable, that right-derived group is precisely ${\mathrm{Ext}}_I^n(M,J)$, so it is zero for $n>0$.

For the additive functor $F={\mathrm{Hom}}_{\mathcal{A}}(-,N)$ on ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}$, the supplied projective resolution of $P$ in $\mathcal{A}$ is an injective resolution in ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}$, and $P$ is injective there. Thus the cohomology in [L1] is the corresponding right-derived construction, and [L2] compares it with the length-zero resolution of $P$.

The Hom cochain complex of that length-zero resolution is concentrated in degree zero, so its cohomology is zero for $q>0$. The comparison in step 1.1 therefore gives ${\mathrm{Ext}}_P^q(P,N)=0$.

For $f\in K^{p,q}$, both mixed composites equal $d_I^q\circ f\circ d_{P,p+1}$, so $hv(f)=vh(f)$. The resolution identities also give $h^2(f)=f\circ d_{P,p+1}{d}_{P,p+2}=0$ and $v^2(f)=d_I^{q+1}{d}_I^q\circ f=0$.

Since the horizontal degree increases from $p$ to $p+1$ after applying $h$, $D^{2}f=h^{2}f+(-1{)}^{p+1}vhf+(-1{)}^{p}hvf+v^{2}f=0.$

Adjoin $C^p$ in vertical degree $-1$. Compatibility makes this an augmented double complex, and the cone of $C^{\bullet }\to \mathrm{Tot}K$ is its signed total complex up to shift. In total degree $n$, only the finitely many columns $0\le p\le n+1$ occur; filtering by the largest horizontal degree has successive quotients equal to shifts of the exact augmented columns.

Starting with one column and adjoining the others, the short exact sequences of successive filtered complexes show inductively that every finite truncation is acyclic. These truncations stabilize degreewise, so the full cone is acyclic. Hence $C^{\bullet }\to \mathrm{Tot}K$ is a quasi-isomorphism.

Interchange the two indices of $K$. Multiplying the component in bidegree $(p,q)$ by $(-1{)}^{pq}$ identifies its signed total complex with the total complex after the interchange; the augmented rows become exact augmented columns with compatible edge maps.

Apply the exact-column assembly lemma to the interchanged double complex. Transporting its quasi-isomorphism back through the sign identification gives $E^{\bullet }\to \mathrm{Tot}K$.

Each $P_p$ is projective, so $\mathrm{Hom}(P_p,-)$ is exact. Applying it to the augmented injective resolution gives the displayed exact column. Naturality of Hom shows that these augmentations commute with precomposition by $d_P$.

The objects in vertical degree $-1$ are $\mathrm{Hom}(P_p,N)$, with horizontal differential given by precomposition by $d_P$. They therefore form $\mathrm{Hom}(P_{\bullet },N)$ and give the asserted natural edge map to the total complex.

Each $I^q$ is injective, so $\mathrm{Hom}(-,I^q)$ is exact. Applying it to the augmented projective resolution gives the displayed exact row. Naturality of Hom shows that these augmentations commute with postcomposition by $d_I$.

The objects in horizontal degree $-1$ are $\mathrm{Hom}(M,I^q)$, with vertical differential given by postcomposition by $d_I$. They therefore form $\mathrm{Hom}(M,I^{\bullet })$ and give the asserted natural edge map to the total complex.

The augmented columns are exact after applying $\mathrm{Hom}(P_p,-)$, and the augmented rows are exact after applying $\mathrm{Hom}(-,I^q)$. Both augmentations commute with the other differential.

Finite-diagonal acyclic assembly applied first to columns and then to rows makes both edge-to-total maps quasi-isomorphisms. Taking cohomology yields the displayed isomorphism; the maps themselves are retained for the later naturality proof.

For fixed resolutions, the two edge maps defining the balance zigzag are induced by the augmentations $P_{\bullet }\to M$ and $N\to I^{\bullet }$, so they involve no comparison lift. After replacing a projective or injective resolution, choose a comparison map over or under the resolved object. These maps give a morphism between the two Hom double complexes and commute with both edge augmentations.

Any two projective comparison maps are chain-homotopic, and any two injective comparison maps are cochain-homotopic, by the two comparison uniqueness theorems. Applying $\mathrm{Hom}$ turns either homotopy into a homotopy of the corresponding total-complex maps. Hence the induced maps on all three cohomologies in the edge-to-total zigzag are independent of the chosen lifts, and the balance isomorphism is independent of those choices.

Choose the projective and injective comparison maps for $u$ and $v$. Projective-resolution Ext has the stated bifunctor variance and Injective-resolution Ext has the stated bifunctor variance give the two routes around the naturality square.

The chosen comparison maps induce a morphism of Hom double complexes. Because its squares with the projective augmentation $P_{\bullet }\to M$ and the injective coaugmentation $N\to I^{\bullet }$ commute, both edge-to-total quasi-isomorphisms commute with it. Taking cohomology makes the balance square commute. The independence lemma removes the chosen comparison maps, proving naturality in both variables.

By Projective comparison maps exist choose $f:P^{\mathrm{\prime }}\to P$ lifting $1_M$, and let $c_{P,P^{\mathrm{\prime }}}=H^n(f^{\ast })$ on the Hom cochain complexes. Two such $f,g$ differ by $dh+hd$ by Projective comparison maps are unique up to chain homotopy. Precomposition by $h$, in degree $n$ sending $\varphi$ to $\varphi h_{n-1}$, is a cochain homotopy between $f^{\ast }$ and $g^{\ast }$, so their induced maps on cohomology coincide. Identity comparison gives $c_{P,P}=1$. A composite $P^{\mathrm{\prime }\mathrm{\prime }}\to P^{\mathrm{\prime }}\to P$ lifts $1_M$, so comparison uniqueness proves $c_{P^{\mathrm{\prime }},P^{\mathrm{\prime }\mathrm{\prime }}}{c}_{P,P^{\mathrm{\prime }}}=c_{P,P^{\mathrm{\prime }\mathrm{\prime }}}$. Opposite-direction comparisons consequently induce inverse maps.

By Injective comparison maps exist choose $g:I\to I^{\mathrm{\prime }}$ extending $1_N$ and let $c_{I,I^{\mathrm{\prime }}}=H^n(g_{\ast })$. The uniqueness theorem Injective comparison maps are unique up to cochain homotopy supplies a homotopy between any two such extensions; postcomposition with it is a Hom cochain homotopy. Thus this map is independent of the extension. Identity extensions and composites give the two injective comparison laws, and reverse extensions give inverses, exactly as in step 1.1.

The chosen comparison maps induce the augmentation-compatible Hom double-complex squares in The Ext balance isomorphism is independent of resolution comparison data. Both edge-to-total maps commute with them. Taking cohomology and inverting the appropriate edge isomorphism proves $c_{I,I^{\mathrm{\prime }}}{\beta }_{P,I}={\beta }_{P^{\mathrm{\prime }},I^{\mathrm{\prime }}}{c}_{P,P^{\mathrm{\prime }}}$. The comparison laws of steps 1.1–2.1 and this square reduce successive changes, including mixed projective/injective changes, to direct comparison. Naturality in module morphisms follows from The Ext balance isomorphism is natural in both variables and comparison uniqueness: the two composite lifts in each change-of-resolution naturality square lift the same object morphism and hence induce equal cohomology maps.

Apply Right derived functors form a cohomological delta functor to the left exact functor $\mathrm{Hom}(M,-)$; it supplies the displayed long exact sequence for the injective construction.

Replace its terms by balanced Ext using The balanced Ext bifunctor. Delta-functor naturality gives naturality in the short exact sequence, while precomposition in $M$ gives the stated contravariant naturality.

Regard ${\mathrm{Hom}}_{\mathcal{A}}(-,N)$ as a left exact functor ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}\to \mathbf{A}\mathbf{b}$. A projective resolution in $\mathcal{A}$ is an injective resolution in ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}$, so Right derived functors form a cohomological delta functor on the opposite category gives the displayed order and connecting maps.

Translating the short exact sequence to the opposite category gives the three Hom terms in the displayed order and the maps ${\mathrm{\partial }}^q$. The balanced Ext bifunctor identifies the right-derived groups with Ext, and delta-functor naturality gives contravariant naturality in the short exact sequence.

To check covariance in $N$, fix a projective horseshoe $0\to P^{\mathrm{\prime }}\to P\to P^{\mathrm{\prime }\mathrm{\prime }}\to 0$ for the given short exact sequence, as supplied by The horseshoe lemma for projective resolutions. Its degreewise splitting makes $0\to \mathrm{Hom}(P^{\mathrm{\prime }\mathrm{\prime }},N)\to \mathrm{Hom}(P,N)\to \mathrm{Hom}(P^{\mathrm{\prime }},N)\to 0$ a short exact sequence of cochain complexes. Postcomposition with $v:N\to N^{\mathrm{\prime }}$ gives a morphism from this sequence to the analogous one with coefficients $N^{\mathrm{\prime }}$. By Naturality of the cohomology connecting morphism, all connecting squares commute. These are the horseshoe connecting maps used by the right-derived theorem in step 1.1. The comparison isomorphisms transporting the middle horseshoe resolution to the supplied one are induced by precomposition, which commutes with postcomposition by $v$. Thus the transported connecting maps, and hence the balanced Ext sequence of step 2.1, are covariantly natural in $N$.

For $0\to N^{\mathrm{\prime }}\to N\to N^{\mathrm{\prime }\mathrm{\prime }}\to 0$, fix $P\to M$ and choose an injective horseshoe $0\to I^{\mathrm{\prime }}\to I\to I^{\mathrm{\prime }\mathrm{\prime }}\to 0$. There are three short exact sequences of cochain complexes: $\mathrm{Hom}(P,N^{\ast })$, $\mathrm{Hom}(M,I^{\ast })$, and $T(P,I^{\ast })$, where $\ast$ denotes the three terms of the short exact sequence, not cochain degree. The first is exact by projectivity of each $P_p$; the second and third are exact because the horseshoe is split in each degree, and total diagonals are finite. The coaugmentations and augmentation give two morphisms of short exact sequences from the edge sequences to the total sequence.

For $0\to M^{\mathrm{\prime }}\to M\to M^{\mathrm{\prime }\mathrm{\prime }}\to 0$, fix $N\to I$ and choose a projective horseshoe $0\to P^{\mathrm{\prime }}\to P\to P^{\mathrm{\prime }\mathrm{\prime }}\to 0$. The three short exact sequences are $\mathrm{Hom}(P^{\ast },N)$, $\mathrm{Hom}(M^{\ast },I)$, and $T(P^{\ast },I)$, all ordered with double-prime first and prime last. The first and third are exact by the degreewise splitting; the second is exact by injectivity of each $I^q$. Again the augmentation and coaugmentation give morphisms from both edge sequences to the total sequence.

In each case let $a$ be the projective-edge map to the total and $b$ the injective-edge map. By [F3], $H(a)$ and $H(b)$ commute with the connecting maps of their respective sequences and the total sequence. They are isomorphisms by [F1]. Hence $\beta =H(b{)}^{-1}H(a)$ also commutes with connecting maps. These are the actual balance maps, not merely some degreewise natural isomorphism. The total differential is $h+(-1{)}^{p}v$; both edge maps are cochain maps with the page's unsigned Hom differentials, so these are commuting squares with no additional sign.

The horseshoe middle resolutions may differ from the fixed ones. Transport their cohomology to the supplied data by comparison isomorphisms. By [F3] this is exactly how the derived connecting maps are defined, and [F1] makes balance commute with these comparisons. Therefore the squares proved in step 2.1 hold for the supplied resolutions as well. In degree zero both augmentations identify the common cocycles with Hom, giving its identity identification.

Apply [L1] to $0\to \mathrm{\Omega }M\to P_0\to M\to 0$. Its relevant terms are ${\mathrm{Ext}}^q(P_0,N)\to {\mathrm{Ext}}^q(\mathrm{\Omega }M,N)\to {\mathrm{Ext}}^{q+1}(M,N)\to {\mathrm{Ext}}^{q+1}(P_0,N)$.

Since $P_0$ is projective, the outer groups vanish for $q\ge 1$ by Positive projective-resolution Ext vanishes on a projective first variable, giving the displayed natural isomorphism. The degree-zero end of the same long exact sequence is exactly the displayed five-term sequence.

The long exact sequence in the second variable contains ${\mathrm{Ext}}^q(M,I)\to {\mathrm{Ext}}^q(M,\mathrm{\Sigma }N)\to {\mathrm{Ext}}^{q+1}(M,N)\to {\mathrm{Ext}}^{q+1}(M,I)$.

The outer groups vanish for $q\ge 1$ because $I$ is injective, so the middle arrow is an isomorphism. At $q=0$ the same sequence gives exactly the printed low-degree segment.

Choose $a_M:Q_{\bullet }(M)\to P_{\bullet }(M)$ lifting $1_M$. A reverse comparison is its homotopy inverse since both composites lift the identity and [F2] compares them with identity chain maps.

Precomposition gives $a_M^{\ast }:\mathrm{Hom}(P_{\bullet }(M),N)\to \mathrm{Hom}(Q_{\bullet }(M),N)$. A homotopy $a-b=dh+hd$ induces the cochain homotopy $s^n(f)=f\circ h_{n-1}$, with $s^0=0$. Thus $H^n(a_M^{\ast })$ is a choice-independent isomorphism. Its source is ${\mathrm{Ext}}_P^n(M,N)$ by Ext via a projective resolution of the first variable, hence balanced Ext by The balanced Ext bifunctor.

For $u:M^{\mathrm{\prime }}\to M$, choose lifts $P(u)$ and $Q(u)$ by [F1]. Precomposition defines their cohomology actions independently of the lifts by [F2]. Identities and composition follow because composites lift the object composites. The maps $P(u)a_{M^{\mathrm{\prime }}}$ and $a_{M}Q(u)$ both lift $u$, so [F2] makes them homotopic. Precomposition and cohomology therefore give the required contravariant naturality square in $M$.

Postcomposition by $v:N\to N^{\mathrm{\prime }}$ commutes exactly with precomposition by $a_M$. This gives naturality in $N$ and hence both variables. For a single supplied resolution $Q_{\bullet }(M)$, steps 1.1–2.1 already give the canonical objectwise isomorphism. No simultaneous class-wide choice of comparison maps is required.

Choose $a_N:I^{\bullet }(N)\to J^{\bullet }(N)$ extending $1_N$. Its reverse comparison is a homotopy inverse: both composites extend the identity, so [F2] compares them to the identity cochain maps.

Applying $\mathrm{Hom}(M,-)$ carries a homotopy $a-b=dh+hd$ to the homotopy $f\mapsto h\circ f$. Hence $H^n\mathrm{Hom}(M,a_N)$ is an isomorphism independent of $a_N$. The definition Ext via an injective resolution of the second variable identifies its source with ${\mathrm{Ext}}_I^n(M,N)$, which The balanced Ext bifunctor identifies with balanced Ext.

For $u:N\to N^{\mathrm{\prime }}$, choose comparison maps $I(u)$ and $J(u)$ extending $u$. Define their actions on cohomology by postcomposition. Independence follows from [F2]; identity and composition laws follow because comparison composites extend the corresponding object composites. Moreover $J(u)a_N$ and $a_{N^{\mathrm{\prime }}}I(u)$ both extend $u$, so [F2] makes them homotopic. Applying Hom and cohomology gives precisely the naturality square in $N$.

For $v:M^{\mathrm{\prime }}\to M$, precomposition by $v$ commutes exactly with postcomposition by $a_N$. This proves contravariant naturality in $M$ and therefore naturality in both variables. The same construction at a single $N$ uses only the individual resolution $J^{\bullet }(N)$, proving the final assertion without a global choice of comparison maps.

Apply $F$ to a projective resolution of $M$. Exactness preserves its augmentation exactness and the preservation hypothesis makes it a projective resolution of $FM$. The adjunction Under local smallness, transposition gives the natural hom-set bijection, and conversely identifies its Hom cochain complex into $X$ with the original Hom cochain complex into $GX$.

Taking cohomology and using The balanced Ext bifunctor gives the claimed isomorphism. The dual argument applies the stated injective preservation to the adjoint construction; no assertion is made without these hypotheses.

Applying ${\mathrm{Hom}}_{\mathbb{Z}}(-,\mathbb{Z})$ gives $0\to \mathbb{Z}\stackrel{n}{\to }\mathbb{Z}\to 0$, in cohomological degrees $0,1$.

Its first cohomology is $\mathrm{coker}(n:\mathbb{Z}\to \mathbb{Z})=\mathbb{Z}\mathrm{/}n$. Ext can be computed from any projective resolution of the first variable identifies this cohomology with ${\mathrm{Ext}}_{\mathbb{Z}}^1(\mathbb{Z}\mathrm{/}n,\mathbb{Z})$. Since $n>1$, the quotient is nonzero.

The injective construction is the cohomology of $\mathrm{Hom}(M,I(N))$ and requires a supplied injective resolution $N\hookrightarrow I(N)$; the projective construction similarly requires a supplied resolution $P(M)\twoheadrightarrow M$. Neither complex is specified before that datum is supplied.

Thus the raw resolution constructions are not definitions of a choice-free object merely from $M$ and $N$. Choice-independence is a later comparison result, so it cannot make the claim “defined before supplying resolutions” true.

By definition the two groups are $H^q\mathrm{Hom}(P_{\bullet },N)$ and $H^q\mathrm{Hom}(M,I^{\bullet })$. They are cohomologies of different complexes, with no equality map included in either definition.

The finite-diagonal double-complex argument supplies a natural isomorphism between these groups. Because that isomorphism is the conclusion of the balance theorem rather than a definitional identity, the asserted statement is false.

Precomposition sends $f:M\to I^q(N)$ to $f\circ a:M^{\mathrm{\prime }}\to I^q(N)$, hence gives ${\mathrm{Ext}}^q(M,N)\to {\mathrm{Ext}}^q(M^{\mathrm{\prime }},N)$; its direction is opposite to $a$.

Postcomposition by the comparison induced by $b$ gives ${\mathrm{Ext}}^q(M,N)\to {\mathrm{Ext}}^q(M,N^{\mathrm{\prime }})$. Thus only the second variable is covariant, while the first is contravariant, disproving covariance in both.

The group $\mathbb{Q}\mathrm{/}\mathbb{Z}$ is divisible and therefore injective, but it occurs as the quotient, hence as the first argument of the extension. If the sequence split, a retraction $\mathbb{Q}\to \mathbb{Z}$ would restrict to the identity on $\mathbb{Z}$.

Every homomorphism $\mathbb{Q}\to \mathbb{Z}$ is zero, so the extension is nonsplit and yields $0\mathrm{\ne }{\mathrm{Ext}}^1(\mathbb{Q}\mathrm{/}\mathbb{Z},\mathbb{Z})$. Injectivity instead forces positive Ext to vanish in the second variable, and projectivity does so in the first.