Draft lesson.

Short exact extensions and the Baer group

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

Level: advanced. Prerequisites: abelian categories, pushouts and pullbacks, mapping cones, quasi-isomorphism roofs and exact Hom sequences of triangles.

A short exact sequence contains more information than its middle object. With its two endpoints fixed, it determines a degree-one derived morphism. We prove that this morphism classifies the sequence, construct every possible class, and identify addition with the pullback-and-pushout construction called the Baer sum. None of these results assumes enough injectives or enough projectives.

Conventions and prerequisites

Let \(\mathcal A\) be an abelian category. We fix objects \(A,B\), regard them as degree-zero complexes, and write

Complexes are cohomological, with \(X[1]^j=X^{j+1}\) and differential \(-d_X\). Composition is written from right to left. We work in an ambient universe where the localization and its Hom groups are defined; no additional smallness assertion about the collection of all middle objects is needed.

Triangles after quasi-isomorphism localization, Sections 2, 4 and 6 proves both exact Hom sequences, completion of a morphism of triangles and the distinguished triangle of a short exact sequence. Cohomological bounds and truncation adjunctions, Sections 2–4 proves the bounded roof comparisons and the natural full faithfulness of the heart:

For the exactness arguments below, we use the kernel, cokernel and biproduct universal properties of an abelian category. In particular every monomorphism is the kernel of its cokernel. We give the required pushout and middle-isomorphism checks explicitly, without an element argument in the general category.

An extension of B by A is a specified exact row \(0\to A\xrightarrow{i}E\xrightarrow{p}B\to0\). Two extensions are equivalent when there is an isomorphism of middle objects commuting with both specified maps and with the identity on each endpoint. An isomorphism of their middle objects alone need not be an equivalence.

1. The normalized class and its actual roof

For an extension \(e=(i,p)\), form the complex \(K_e\) with \(A\) in degree minus one, \(E\) in degree zero and differential \(i\). It is the cone of \(i: A[0]\to E[0]\). Its quotient and projection maps are

All other components vanish. Both are chain maps: \(pi=0\), and the target of \(f_e\) has no degree-zero term. Exactness of the row gives \(H^{-1}K_e=0\), \(H^0K_e=B\), and zero cohomology in every other degree. Thus \(s_e\) is a quasi-isomorphism. Define

This is precisely the boundary of the distinguished row triangle \(A\xrightarrow{i}E\xrightarrow{p}B\xrightarrow{\delta_e}A[1]\). Indeed the cone triangle ends in the projection \(K_e\to A[1]\); replacing \(K_e\) by its quotient quasi-isomorphism to \(B\) gives (1.2). This fixes the sign of the class.

Lemma 1.1 (naturality). Suppose \((a,h,b)\) is a commutative diagram of short exact rows, with \(a: A\to A'\), \(h: E\to E'\) and \(b: B\to B'\). Then

Proof. The cone map \(k: K_e\to K_{e'}\) has components \(a\) and \(h\). The equation \(hi=i'a\) makes it a chain map, and the other square gives \(s_{e'}k=bs_e\). Its numerator satisfies \(f_{e'}k=a[1]f_e\). Invert both quotient quasi-isomorphisms to obtain (1.3). \(\square\)

In particular equivalent extensions have the same class.

Proposition 1.2. The class \(\delta_e\) is zero if and only if the extension splits.

Proof. Apply \(\operatorname{Hom}_D(B,-)\) to its row triangle. The exact segment is

The image of \(1_B\) under the last arrow is \(\delta_e\). If this class vanishes, exactness produces \(s: B\to E\) in the derived category with \(ps=1_B\). By (0.2) it is an actual morphism in the heart with the same equation. Conversely an actual section produces this zero image.

For completeness, a section gives the usual splitting isomorphism \(A\oplus B\to E\), with components \(i\) and \(s\). Since \(p(1_E-sp)=0\), the kernel property gives a unique \(r: E\to A\) with \(ir=1_E-sp\). Its equations are \(ri=1_A\) and \(rs=0\), so \((r,p)\) is the inverse. Thus the splitting is a splitting of the specified row. \(\square\)

2. Constructing every class by a pushout

Theorem 2.1. Every \(\alpha: B\to A[1]\) in the derived category is the class of an extension of \(B\) by \(A\).

Proof. Write \(\alpha\) as a roof \(B\xleftarrow{q}U\xrightarrow{f}A[1]\). Smart upper truncation at zero replaces \(U\) by a complex with zero terms above zero. Its cohomology is that of \(B\), because \(q\) is a quasi-isomorphism. Smart lower truncation at minus one is therefore a further quasi-isomorphism. The component \(f^{-1}\) kills \(\operatorname{im}d_U^{-2}\), by the chain-map equation, so it descends through this lower truncation; the denominator to \(B\) descends as well. We now have a two-term roof

Here \(d\) is monic, \(q: M\to B\) is epic and \(\ker q=\operatorname{im}d\); these follow from the cohomology isomorphism. The sole numerator component is \(f: L\to A\).

Define \(E\) by the cokernel

The first map is monic, since its second component is the negative of a monomorphism. Moreover \(jd=if\). There is a unique \(p: E\to B\) with \(pc=(0,q)\).

We check monicity of \(i\) by its kernel. If \(x: X\to A\) has \(ix=0\), the kernel of \(c\) is \((f,-d)\), so \(\iota_Ax=(f,-d)y\) for a unique \(y: X\to L\). Its second component gives \(dy=0\), hence \(y=0\), and its first gives \(x=0\). Thus \(\ker i=0\) and \(i\) is monic.

Next \(p\) is the cokernel of \(i\). If \(h: E\to Z\) kills \(i\), its composite with \(c\) has zero A-component, so it is uniquely \(g\operatorname{pr}_M\) for a map \(g: M\to Z\). The equation \(hc(f,-d)=0\) gives \(gd=0\). Since \(q\) is the cokernel of \(d\), there is a unique \(\bar g: B\to Z\) with \(g=\bar gq\). Consequently \(hc=\bar gpc\), and cancellation of the epimorphism \(c\) gives \(h=\bar gp\). Uniqueness also follows by cancelling \(q\), or the epic composite \(pc=(0,q)\). This proves the full cokernel property, and an abelian monomorphism is the kernel of its cokernel. Therefore

is exact. The cokernel presentation (2.2) is the pushout of \(d\) along \(f\), since a pair of maps out of \(A,M\) descends precisely when their composites with \(f,d\) agree.

The map \(k: (L\to M)\to K_e=(A\to E)\) has components \(f,j\). It is a chain map because \(jd=if\). It satisfies \(s_e k=q\) and \(f_e k=f\), and it is a quasi-isomorphism: both source and target have only degree-zero cohomology, identified with \(B\), and it induces the identity there. Therefore the roof (2.1) and (1.1) represent the same derived arrow. This proves \(\delta_e=\alpha\). \(\square\)

3. Equal classes and the fixed endpoints

Theorem 3.1. Two extensions of \(B\) by \(A\) have the same class if and only if they are equivalent with the identities on \(A\) and \(B\). Consequently (1.2) induces a bijection from their equivalence classes to \(\operatorname{Ext}^1(B,A)\).

Proof. One direction is Lemma 1.1. Suppose \(\delta_e=\delta_{e'}\). Twice rotate their distinguished row triangles, giving

and the same expression for \(e'\). The identities of \(B\) and \(A[1]\) make a commutative square at the first arrows. The morphism-of-triangles axiom, proved in the prerequisite, completes it to an arrow \(h[1]: E[1]\to E'[1]\). The two further squares give \(hi=i'\) and \(p'h=p\); the minus signs on each side cancel. Shift back and apply (0.2) to get an actual \(h: E\to E'\) in \(\mathcal A\).

Here is the middle-isomorphism check. If \(x: X\to E\) is killed by \(h\), then \(px=p'hx=0\), so \(x=iy\) for some \(y: X\to A\). Now \(0=hx=i'y\), and monicity of \(i'\) gives \(y=0\), so \(x=0\). Thus \(h\) is monic. If \(z: E'\to Z\) kills \(h\), it kills \(i'=hi\); hence \(z=wp'\) by the cokernel property. Then \(0=zh=wp\), and epicity of \(p\) gives \(w=0\), so \(z=0\). Subtracting any two arrows shows that this is precisely epicity of \(h\). A morphism both monic and epic in an abelian category is an isomorphism by its coimage-to-image comparison. Thus this is the required extension equivalence. Surjectivity was Theorem 2.1, completing the bijection. \(\square\)

The theorem asserts existence of a middle isomorphism, not uniqueness. The automorphisms preserving both endpoint identifications are computed in Exercise 3.

4. Pullbacks, pushouts and addition

For \(b: B'\to B\), the pullback of \(e\) is the row with middle object \(E\times_B B'\), maps \(A\to E\times_B B'\) and projection to \(B'\). For \(a: A\to A'\), its pushout has middle object \(A'\amalg_A E\). Both rows are short exact. The pushout assertion was proved by (2.2)–(2.3), taking \(d=i\) and \(f=a\); the pullback assertion follows by the same kernel/cokernel argument in the opposite abelian category. In particular it applies to arbitrary endpoint maps, without monicity or epicity restrictions on \(a,b\).

Their structural diagrams and Lemma 1.1 give

These formulas also prove functoriality on extension equivalence classes. Identities and composites agree because they agree after the bijection of Theorem 3.1. The two endpoint operations commute on classes, since \(a[1](\delta_e b)=(a[1]\delta_e)b\).

Given two extensions \(e_1,e_2\) with endpoints \(A,B\), first take their direct-sum extension with endpoints \(A\oplus A,B\oplus B\). Pull it back along the diagonal \(\Delta_B: B\to B\oplus B\), then push it out along the codiagonal \(\nabla_A: A\oplus A\to A\). This extension is their Baer sum \(e_1+_{\mathrm B}e_2\). Its middle object is described concretely by the pullback \(E_1\times_B E_2\) and the subsequent pushout of its kernel map from \(A\oplus A\).

Theorem 4.1. The Baer sum satisfies

It therefore makes the extension equivalence classes an abelian group, naturally isomorphic to (0.1). Its zero is the split row and its negative is pushout along \(-1_A\).

Proof. The roof of the direct-sum extension is the direct sum of its two cone roofs. Its denominator is the sum of the two quotient quasi-isomorphisms and its numerator the sum of the two projections. Thus its boundary is the block-diagonal arrow \(\delta_{e_1}\oplus\delta_{e_2}\). Apply (4.1) twice:

The last equality is the biproduct definition of addition in an additive category. The derived category is additive, as proved by the localization prerequisite. The bijection of Theorem 3.1 now transfers its associativity and commutativity to exactly the stated Baer operation; it is not an unrelated operation defined only abstractly. Proposition 1.2 identifies the zero class with the split extension. Formula (4.1) for \(a=-1_A\) shows that its pushout gives the additive inverse. Thus all group axioms follow for the constructed extensions themselves, up to the specified endpoint-preserving equivalence. \(\square\)

This group is contravariant in \(B\) and covariant in \(A\). The endpoint actions preserve sums because composition of derived morphisms is bilinear.

5. Four graded exercises with complete solutions

Exercise 1 — A cyclic presentation with fixed endpoints (introductory)

For integers \(m\ge2,n\ge1\), classify extensions of \(\mathbb Z/m\) by \(\mathbb Z/n\). Give explicit middle presentations, determine equivalence with the endpoint identities, and identify each presentation's derived class.

Solution. Put \(A=\mathbb Z/n\), with generator \(t\). Choose a lift \(e\) of the generator of \(B=\mathbb Z/m\). Then \(me=a t\) for some \(a\in\mathbb Z/n\). The resulting presentation is

To verify exactness directly, realize \(E_a\) as \(\mathbb Z^2/\langle(n,0),(-a,m)\rangle\). A relation with zero second coordinate has coefficient zero on \((-a,m)\); therefore the first generator has exactly the relations \(nt=0\), proving injectivity of \(i\). If \(p([b,c])=0\), write \(c=m\ell\), and the second relation gives \([b,c]=[b+a\ell,0]\), so its kernel is the image of \(A\). The map \(p\) is surjective. Conversely, for any extension, the chosen lifts generate it: subtract a multiple of \(e\) from an arbitrary element to get an element of its kernel. They obey (5.1), and the resulting map from \(E_a\) is an isomorphism by the endpoint-identity kernel and cokernel argument of Theorem 3.1.

A map preserving endpoints from \(E_a\) to \(E_{a'}\) must take \(t\) to \(t'\) and \(e\) to \(e'+b t'\), for some \(b\in A\). Its relation holds exactly when \(a=a'+mb\) in \(A\). This condition is also sufficient; the map then preserves both endpoints and is an isomorphism. Thus the classes are \(A/mA\simeq\mathbb Z/d\), where \(d=\gcd(m,n)\).

The free resolution \(P=(\mathbb Z\xrightarrow{m}\mathbb Z)\) of \(B\), in degrees minus one and zero, computes derived Hom by Hereditary categories, Lemma 3.1. Map it to \(K_{E_a}\) by the degree-zero lift \(1\mapsto e\) and degree-minus-one map \(1\mapsto a\). The equation \(me=i(a)\) makes this a chain map. It gives the same quotient to \(B\), so (1.2) is represented by the numerator \(a: \mathbb Z\to A\). A homotopy changes it by \(mb\); hence its derived class is precisely \(a+mA\). In particular \(a\in mA\) is exactly the split condition. When \(n=1\), \(A=0\), and this states that the only extension is the row with middle object \(B\), as required.

Exercise 2 — The middle group does not determine the extension (intermediate)

Fix a prime \(p\ge3\). For \(A=B=\mathbb Z/p\), show that all nonzero extension classes have isomorphic middle groups but are distinct with the endpoint identities. Describe the orbits if both endpoint identifications may be changed by automorphisms.

Solution. Formula (5.1) now has \(pt=0\) and \(pe=a t\), where \(a\in\mathbb F_p\). For \(a\ne0\), the equation \(t=a^{-1}pe\) shows that \(e\) generates the middle group. It has order \(p^2\): its p-th multiple is the nonzero element \(a t\), and its \(p^2\)-th multiple is zero. Thus each such middle group is cyclic of order \(p^2\). Nevertheless \(mA=pA=0\), so Exercise 1 shows that two such extensions are endpoint-preserving equivalent only if their parameters are equal. Since \(p\ge3\), there are at least two distinct nonzero examples with the same abstract middle group.

An automorphism of either endpoint is multiplication by a nonzero scalar. Pullback and pushout act on the class by precomposition and postcomposition, respectively, by (4.1); under the free-resolution calculation each multiplies \(a\) by that scalar. Thus there are exactly two orbits: zero and the nonzero elements of \(\mathbb F_p\). The zero orbit has middle group \((\mathbb Z/p)^2\); the nonzero orbit has middle group \(\mathbb Z/p^2\). These orbit statements allow changed identifications and do not assert an equivalence with the identities.

Exercise 3 — Every automorphism of a fixed extension (advanced)

Prove that automorphisms of an extension fixing both endpoints form a group naturally isomorphic to \(\operatorname{Hom}_{\mathcal A}(B,A)\) under addition. Compute these automorphisms for the nonsplit row \(0\to\mathbb Z/p\xrightarrow{\bar1\mapsto p}\mathbb Z/p^2\to\mathbb Z/p\to0\).

Solution. For \(\phi: B\to A\), put \(T_\phi=1_E+i\phi p\). It fixes \(i\) and \(p\), since \(pi=0\). Also \((i\phi p)^2=0\), so \(T_{-\phi}\) is its inverse. Multiplication gives \(T_\phi T_\psi=T_{\phi+\psi}\).

Conversely, any endpoint-preserving automorphism \(h\) has \(p(h-1_E)=0\), so \(h-1_E=i t\) for a unique \(t: E\to A\), by the kernel property of \(i\). Its equation \((h-1_E)i=0\) implies \(ti=0\), since \(i\) is monic. The cokernel property of \(p\) now gives a unique \(\phi: B\to A\) with \(t=\phi p\). Thus \(h=T_\phi\), with unique \(\phi\). The same proof shows that any endomorphism fixing the endpoints is automatically invertible. This establishes the asserted natural group parametrization in an arbitrary abelian category.

In the displayed cyclic row, \(\operatorname{Hom}(B,A)=\mathbb F_p\). The parameter \(c\in\mathbb F_p\) gives multiplication by \(1+cp\) on \(\mathbb Z/p^2\), because \(i\phi p(\bar1)=cp\). These are p distinct automorphisms, and their multiplication adds the parameters modulo \(p\), since the extra product term is divisible by \(p^2\).

Exercise 4 — Nonsplit extensions whose sum splits (advanced)

For a prime \(p\), let \(e\) be the nonsplit row in Exercise 3. Compute the Baer sum of r copies, for each integer \(r\ge1\), directly from its pullback-and-pushout presentation. Determine exactly when that sum splits.

Solution. The repeated pullback is the subgroup of \((\mathbb Z/p^2)^r\) whose coordinates have the same reduction modulo \(p\). Write \(g=(1,\ldots,1)\) and let \(h_j\) have \(p\) in coordinate \(j\) and zero in the others. These elements generate the subgroup: subtract a common lift of the reduction, and each residual coordinate is divisible by \(p\). Its presentation has relations

To see that these are all the relations, the group they present maps surjectively to the subgroup just described. Its quotient by the subgroup generated by the \(h_j\) is cyclic of order at most \(p\); that subgroup has order at most \(p^r\). Hence the presented group has at most \(p^{r+1}\) elements. The actual pullback has exactly \(p^{r+1}\) elements, because there are p choices of common reduction and independently p lifts in each coordinate. A surjection between these finite groups is consequently an isomorphism.

The pushout along the codiagonal of the kernel identifies all the \(h_j\) with a single kernel generator \(t\). Thus the Baer sum has presentation \(\langle t,g\mid pt=0,\ pg=r t\rangle\), exactly \(E_{\bar r}\) of (5.1). By Exercise 1 and Proposition 1.2 it splits precisely when \(r=0\) in \(\mathbb F_p\). In particular p copies of the nonsplit row have split Baer sum. For \(r=1\), the original row is recovered; for \(p=2,r=2\), the middle group of the sum is \((\mathbb Z/2)^2\). This computes the constructed sum, not merely its abstract group-law value.

Freely accessible references

The following sources provide further reading. All proof steps used here are supplied above or in the linked prerequisite lessons.

• The Stacks Project, Extensions, Section 12.6, tag 010I: specified extension rows, pullback, pushout and Baer addition.
• The Stacks Project, Ext groups, Section 13.27, tag 06XP: derived extension classes and the relation to exact sequences.

Original text: CC0 1.0 Universal. Written by OpenAI GPT-6.1 Sol.