1 Construction of connecting morphisms
1.1 From a short exact sequence of complexes
Let \[ 0\to A^\bullet \xrightarrow{i} B^\bullet \xrightarrow{p} C^\bullet \to 0 \] be a short exact sequence of cochain complexes (or chain complexes, with indices adjusted accordingly). Exactness at each degree means \(i^n\) is injective, \(p^n\) is surjective, and \(\operatorname{im} i^n=\ker p^n\). Choose an element \(z\in C^n\) with \(d_C(z)=0\) (so \(z\) is a cocycle). Surjectivity of \(p^n\) allows picking \(y\in B^n\) such that \(p^n(y)=z\). Applying the differential in \(B^\bullet\) yields \(d_B(y)\in B^{n+1}\). Since \(p^{n+1}(d_B(y))=d_C(p^n(y))=d_C(z)=0\), the element \(d_B(y)\) lies in \(\ker p^{n+1}=\operatorname{im} i^{n+1}\). Thus there is \(x\in A^{n+1}\) with \(i^{n+1}(x)=d_B(y)\). One then defines the connecting map on cohomology by sending the class \([z]\in H^n(C^\bullet)\) to the class \([x]\in H^{n+1}(A^\bullet)\). A routine check shows that \(x\) is a cocycle and that the resulting cohomology class is independent of the choice of lift \(y\).
1.2 Snake lemma viewpoint
The same construction can be phrased using the snake lemma. Consider, for each \(n\), the commutative diagram with exact rows \[ 0\to A^n \xrightarrow{i^n} B^n \xrightarrow{p^n} C^n \to 0, \qquad 0\to A^{n+1} \xrightarrow{i^{n+1}} B^{n+1} \xrightarrow{p^{n+1}} C^{n+1}\to 0, \] together with the differentials \(d_A^n, d_B^n, d_C^n\). Restrict the diagram to subobjects of cycles and boundaries (e.g. \(Z^n(-)=\ker d^n\), \(B^n(-)=\operatorname{im} d^{n-1}\)). The snake lemma produces a connecting morphism between the corresponding kernels/quotients; after identifying these with cohomology groups, one obtains the connecting map in the long exact sequence.
1.3 Diagram chase and naturality
The connecting morphism is defined by diagram chase: lift a cocycle from \(C^\bullet\) to \(B^\bullet\), apply the differential, and read off the resulting element in \(A^\bullet\) using exactness. Naturality follows because the procedure is compatible with morphisms of short exact sequences of complexes. Concretely, given a commutative diagram of short exact sequences, the induced morphisms on long exact sequences intertwine the connecting maps. This compatibility can be tracked degreewise by following how chosen lifts correspond under the vertical morphisms.
1.4 Mapping cone construction
A second method uses the mapping cone. Starting from a short exact sequence \[ 0\to A^\bullet \to B^\bullet \to C^\bullet \to 0, \] one can identify \(B^\bullet\) (up to isomorphism in the derived category) with a cone that fits a distinguished triangle \[ A^\bullet \to B^\bullet \to C^\bullet \xrightarrow{\delta} A^\bullet[1]. \] The connecting morphism \(\delta\) is the structural morphism of this triangle. Applying the cohomology functor produces the boundary maps in the associated long exact sequence. This perspective clarifies that connecting morphisms are canonical in a homotopical sense: they arise from the triangle determined by the short exact sequence.
1.5 Choice of signs and degree conventions
Sign issues depend on whether one uses chain complexes or cochain complexes, as well as on grading conventions for cones and totalizations. For cochain complexes (differential of degree \(+1\)), the connecting map shifts degree by \(+1\). For chain complexes (differential of degree \(-1\)), the shift is by \(-1\). In practice, once a consistent convention is fixed for differentials and cone signs, the connecting morphism is determined uniquely and the long exact sequence is exact with the correct grading. Many presentations avoid explicit sign formulas by defining the connecting map through the lift-and-differentiate construction, where the sign is absorbed by the grading choice.
2 Long exact sequences and where the connecting morphism lives
2.1 Long exact sequence in homology
For a short exact sequence of chain complexes \[ 0\to A_\bullet \to B_\bullet \to C_\bullet \to 0, \] there is a long exact sequence of homology groups \[ \cdots \to H_n(A_\bullet)\to H_n(B_\bullet)\to H_n(C_\bullet)\xrightarrow{\partial} H_{n-1}(A_\bullet)\to H_{n-1}(B_\bullet)\to \cdots, \] where \(\partial\) is the connecting morphism. It “connects” the homology in degree \(n\) of \(C\) to the adjacent degree \(n-1\) of \(A\), reflecting the shift created when one differentiates a lifted cycle and then passes to the quotient defining homology.
2.2 Long exact sequence in cohomology
For cochain complexes \[ 0\to A^\bullet \to B^\bullet \to C^\bullet \to 0, \] the induced long exact sequence has the form \[ \cdots \to H^n(A^\bullet)\to H^n(B^\bullet)\to H^n(C^\bullet)\xrightarrow{\delta} H^{n+1}(A^\bullet)\to H^{n+1}(B^\bullet)\to \cdots. \] Here the connecting morphism \(\delta\) lands one degree higher on \(A^\bullet\), again encoding the differential’s degree and the boundary/cycle relations that define cohomology.
2.3 Degree shifts and indexing
The connecting map always shifts the degree by one, but the direction depends on whether the complex is graded cohomologically or homologically. Indexing conventions (e.g., whether one writes \(d^n\colon C^n\to C^{n+1}\) or uses negative indices) affect the displayed formulas but not the underlying structural content: the connecting morphism lives between cohomology/homology groups of adjacent degrees.
2.4 Boundary maps vs connecting morphisms terminology
Terminology varies. In many texts, the map from \(H^n(C)\) to \(H^{n+1}(A)\) is called the boundary map or connecting homomorphism. In the derived-category viewpoint, it corresponds to the connecting morphism in a distinguished triangle. While the names emphasize different intuitions, they refer to the same canonical arrow in the long exact sequence constructed from the short exact sequence of complexes.
3 Properties of connecting morphisms
3.1 Functoriality and natural transformations
The construction is functorial with respect to morphisms of short exact sequences of complexes. Suppose one has a commutative diagram of short exact sequences \[ 0\to A^\bullet\to B^\bullet\to C^\bullet\to 0 \quad\text{and}\quad 0\to A'^\bullet\to B'^\bullet\to C'^\bullet\to 0 \] with vertical chain maps. The induced cohomology maps form a morphism of long exact sequences, and the connecting morphisms commute with these induced maps. This yields naturality of the connecting homomorphism as a transformation between functors built from cohomology.
3.2 Compatibility with morphisms of short exact sequences
More generally, if one replaces the short exact sequence by an isomorphic one (or applies an exact functor that preserves exactness degreewise), the connecting morphism transforms accordingly. Exactness-preserving operations on the short exact sequence yield corresponding maps on the resulting long exact sequence. This compatibility is essential for using connecting morphisms systematically in computations and proofs.
3.3 Compositions and exactness constraints
The long exact sequence is exact, so compositions of consecutive maps vanish, and kernels coincide with images. In particular, the connecting morphism is characterized by how it fits into exactness at the adjacent groups. Exactness implies constraints such as:
- the image of \(H^n(B)\to H^n(C)\) equals the kernel of \(H^n(C)\xrightarrow{\delta} H^{n+1}(A)\),
- the image of \(H^n(C)\xrightarrow{\delta} H^{n+1}(A)\) equals the kernel of \(H^{n+1}(A)\to H^{n+1}(B)\).
These relations determine how failure of exactness at a chosen level of cycles/boundaries propagates to neighboring degrees.
3.4 Independence of auxiliary choices
Although the definition uses a choice of lift of a cocycle (or cycle), the output cohomology/homology class does not depend on that choice. If a different lift is chosen, the difference lies in the subcomplex \(A^\bullet\) and its differential becomes a coboundary/cycle boundary in the appropriate degree. Consequently, the connecting morphism is canonical, not merely well-defined for one fixed selection of lifts.
3.5 Behavior under quasi-isomorphisms
If a morphism of short exact sequences induces quasi-isomorphisms on two of the complexes, then the induced map on the third is also a quasi-isomorphism under standard hypotheses, and the connecting maps on cohomology are compatible. While the connecting morphism itself depends on the specific chain-level models, its effect on cohomology is stable under quasi-isomorphisms when the long exact sequences are identified through the induced cohomology isomorphisms.
4 Computing connecting morphisms in examples
4.1 Chain complexes from modules and resolutions
A common computational setup arises from resolutions. Let \(0\to A \to B \to C\to 0\) be an exact sequence of modules and take projective (or injective) resolutions to form complexes. Connecting morphisms in Ext or cohomology can then be interpreted as boundary maps resulting from short exact sequences of complexes built from these resolutions. This approach turns the abstract definition into explicit formulas in terms of chosen lifts within a resolution.
4.2 Easy computations in low degrees
In low degrees, one can often compute \(\delta\) directly. For instance, in a cochain complex, elements of \(H^0(C^\bullet)\) are cocycles in degree \(0\) modulo coboundaries. A cocycle in degree \(0\) is a differential-closed element, and the connecting morphism is obtained by lifting it to degree \(0\) in \(B^\bullet\), differentiating, and then projecting to degree \(1\) in \(A^\bullet\). When degrees are sparse or differentials are simple, the computation reduces to checking kernels and images in a small range.
4.3 Connecting morphisms for standard short exact sequences
Standard constructions include short exact sequences like
- \(0\to \ker p \to B \xrightarrow{p} \operatorname{im} p\to 0\),
- quotient sequences \(0\to M'\to M\to M/M'\to 0\),
- sequences formed from direct sums or subcomplexes.
When these are assembled into complexes and fitted into the general framework, the connecting morphism can be tracked by the induced differentials and the identifications of subobjects as cycles/boundaries.
4.4 Interpretation via boundaries and cycles
Conceptually, the connecting morphism measures how a cocycle in \(C^\bullet\) fails to lift to a cocycle in \(B^\bullet\). If a cocycle \(z\in Z^n(C^\bullet)\) lifts to an element \(y\in B^n\) with \(d_B(y)=0\), then the connecting image is trivial. If every lift has nonzero differential, the differential of a lift becomes an element of \(A^{n+1}\) whose cohomology class records the obstruction. This “failure-of-lifting” view provides both intuition and a practical computational method.
5 Conceptual interpretations
5.1 “Change of degree” across exact sequences
The degree shift in a connecting morphism reflects a structural boundary phenomenon: differentiation moves one step in the complex, while exactness moves information between subcomplexes and quotients. Thus the connecting map can be understood as converting a closed object in one degree into a closed object in the adjacent degree by passing through the differential and then restricting to the appropriate kernel.
5.2 Measuring obstruction to lifting
A cocycle in \(C^\bullet\) always has lifts to \(B^\bullet\) because the quotient map is surjective degreewise. However, being closed is not automatic for the lift. The connecting morphism records the cohomology class of the differential of a lifted cocycle, which vanishes exactly when the cocycle lifts to a cocycle. In this sense, connecting morphisms quantify the obstruction to strengthening a lift from “preimage” to “cycle.”
5.3 Relation to derived functors
Connecting morphisms appear naturally in long exact sequences associated with derived functors. For example, Ext and Tor arise as cohomology groups of complexes constructed from resolutions; short exact sequences of modules induce long exact sequences in Ext/Tor, whose connecting maps match the boundaries induced by the resolution-level constructions. This compatibility places connecting morphisms within the broader formalism of homological algebra.
5.4 Role in spectral sequences (motivation)
Spectral sequences are organized by filtrations that produce exact couples and successive approximations to homology or cohomology. Connecting morphisms emerge as the differential data that governs how classes persist or die when moving between pages. While the specifics depend on the construction, the overarching theme is that boundaries between filtrations generate maps resembling connecting morphisms, controlling propagation of information across degrees.
6 Special cases and variants
6.1 Connecting morphism from 0→A→B→C→0 of modules
Given a short exact sequence of modules \[ 0\to A \to B \to C \to 0, \] one can place these modules into short exact sequences of complexes concentrated in degree \(0\). For instance, interpret them as cochain complexes with only degree \(0\) nonzero. The resulting connecting morphism then lands in degree \(1\) cohomology (or homology, depending on convention). In derived contexts, this boundary map becomes the class associated to the extension in Ext, linking the purely algebraic extension to the connecting morphism mechanism.
6.2 Connecting morphism in Ext and Tor contexts
Ext and Tor fit into long exact sequences when a short exact sequence of modules is present and one argument is resolved projectively or injectively. The connecting morphism in these long exact sequences is built from the same “lift and differentiate” logic, but implemented within the chosen resolution. This yields explicit maps between Ext groups or Tor groups in adjacent degrees.
6.3 Cohomological connecting morphisms in Ext-classes
For extensions classified by Ext, the Yoneda interpretation provides a conceptual bridge: an element of Ext can be represented by a short exact sequence of modules (or by a derived morphism between complexes). The connecting morphism appearing in long exact sequences corresponds to composition with these extension classes in the derived category. In that setting, the connecting morphism is not an arbitrary map but rather the boundary induced by the extension structure itself.
6.4 Variants for graded or filtered complexes
For graded complexes, one performs the connecting construction degreewise, respecting internal gradings alongside cochain degree. For filtered complexes, one obtains connecting morphisms on associated graded objects or on the pages of spectral sequences, where exactness may hold only up to filtration. These variants preserve the core principle: exact sequences of complexes yield canonical maps between adjacent cohomological degrees, now refined by internal grading or filtration level.
7 Common identities and lemmas
7.1 Compatibility with long exact sequence maps
Connecting morphisms satisfy compatibility relations with the other maps in the long exact sequence induced by a short exact sequence of complexes. In particular, the long exact sequence can be assembled so that every square involving two consecutive maps commutes, reflecting the functorial construction from kernels and cokernels. These commutativity properties are often used to verify naturality in diagrams and to track how maps behave under compositions.
7.2 Five lemma–style consequences
While the five lemma concerns when certain vertical maps in a commutative diagram are isomorphisms, its homological applications often involve connecting maps. In diagrams of short exact sequences of complexes, the existence and naturality of connecting morphisms enable proofs that isomorphisms in two degrees force isomorphisms in the neighboring degree, using exactness of the long exact sequence as the main input.
7.3 Explicit boundary formulas when available
In settings where differentials and lifts are explicit—such as complexes coming from concrete module presentations—one can write formulas for the connecting morphism in terms of chosen representatives. Typically, one expresses a class in \(H^n(C)\) by a cocycle \(z\), chooses a lift \(y\) in \(B^n\), computes \(d_B(y)\), and identifies the corresponding element \(x\) in \(A^{n+1}\). The boundary map is then the cohomology class of \(x\). These formulas may depend on grading conventions but are straightforward in practice when explicit chain models are given.
7.4 Homotopy invariance considerations
Connecting morphisms depend on the chain-level short exact sequence, but their induced behavior on cohomology is homotopy-invariant in the sense that quasi-isomorphic replacements lead to the same maps once cohomology is identified. If one modifies the complexes by chain homotopies compatible with the short exact sequence structure, the connecting morphism changes only by the equivalence dictated by the induced maps on cohomology. This stability supports the use of connecting morphisms in derived and homotopical arguments.