Chain complex in an abelian category

Intuition Beginner

A chain complex is a sequence of mathematical objects connected by arrows. The special rule is that two consecutive arrows always cancel to zero.

This rule means every output from one arrow automatically becomes invisible to the next arrow. Homology measures the gap between "things that become invisible" and "things that came from the previous arrow."

In topology, a chain complex can record vertices, edges, faces, and higher-dimensional pieces of a space. In algebra, the same idea works for modules, sheaves, or any setting where kernels, quotients, and exact sequences make sense.

The abelian-category version is the common language behind derived functors, sheaf cohomology, Ext, Tor, and derived categories.

Visual Beginner

The complex is useful because the failure of exactness at each object becomes a new invariant.

Worked example Beginner

Take a triangle with its edges and vertices. One boundary map sends the filled triangle to the sum of its three oriented edges. The next boundary map sends those edges to their endpoints.

Going from the filled triangle all the way to vertices gives zero, because each vertex appears once with a plus sign and once with a minus sign.

What this tells us: the boundary of a boundary cancels. Chain complexes turn that cancellation into algebra.

Check your understanding Beginner

Formal definition Intermediate+

Let $\mathcal{A}$ be an abelian category. A cochain complex in $\mathcal{A}$ is a family of objects $$ {C^n}_{n\in\mathbb Z} $$ together with differentials $$ d^n^n\to C^{n+1} $$ such that $$ d^{n+1}\circ d^n=0 $$ for every $n$.

The category of cochain complexes is denoted $$ \mathrm{Ch}(\mathcal A). $$ A morphism of complexes $f:C^{\bullet }\to D^{\bullet }$ is a family of maps $f^n:C^n\to D^n$ satisfying $$ d_D^n f^n=f^{n+1}d_C^n. $$ This commuting-square condition says that $f$ respects the differentials.

The $n$-th cohomology object is $$ H^n(C^\bullet)=\ker(d^n)/\operatorname{im}(d^{n-1}). $$ In an abelian category, this quotient is made as the cokernel of the image map into the kernel. The condition $d^n{d}^{n-1}=0$ ensures that $\mathrm{im}(d^{n-1})$ lies inside $\ker (d^n)$.

Bounded variants are named as follows: $$ \mathrm{Ch}^+(\mathcal A),\quad \mathrm{Ch}^-(\mathcal A),\quad \mathrm{Ch}^b(\mathcal A), $$ for complexes bounded below, bounded above, and bounded on both sides. Chain-complex conventions use subscripts and lower degree, with differentials $d_n:C_n\to C_{n-1}$. Cochain conventions use superscripts and raise degree.

Counterexamples to common slips

• A graded family of objects is not a complex until it has differentials with consecutive composite zero.
• A complex need not be exact; its failure of exactness is its cohomology.
• Chain and cochain conventions carry opposite degree directions, so signs and shifts must be stated.

Key theorem with proof Intermediate+

Theorem (long exact sequence in cohomology). If $$ 0\to A^\bullet\xrightarrow{i}B^\bullet\xrightarrow{p}C^\bullet\to 0 $$ is a short exact sequence of cochain complexes in an abelian category, then there is a natural long exact sequence $$ \cdots\to H^n(A)\to H^n(B)\to H^n(C) \xrightarrow{\delta}H^{n+1}(A)\to H^{n+1}(B)\to\cdots. $$

Proof. In a module category, the connecting map is the standard snake-lemma diagram chase. Start with a cohomology class $[c]\in H^n(C)$. Lift $c$ to some $b\in B^n$. Since $c$ is closed, $d_{B}b$ maps to zero in $C^{n+1}$. Exactness then places $d_{B}b$ in the image of $A^{n+1}$, so $d_{B}b=i(a)$ for a unique $a$ after identifying $A^{n+1}$ with its image.

The equation $d_{A}a=0$ follows from $d_B^2=0$ and injectivity of $i$. Define $\delta ([c])=[a]\in H^{n+1}(A)$. Changing the lift $b$ or the representative $c$ changes $a$ by a coboundary, so the map is well-defined.

Exactness of the resulting sequence is checked by the same diagram chase at each cohomology object. In a general abelian category, the snake lemma supplies the same construction without choosing elements, applied degreewise to the diagram of kernels and cokernels. $\square$

Bridge. This unit upgrades exact sequences and the snake lemma from 01.02.11 into the language of complexes. It prepares chain homotopy 01.02.31, mapping cones 01.02.32, and derived categories 04.03.11.

Exercises Intermediate+

Advanced results Master

The category $\mathrm{Ch}(\mathcal{A})$ is itself additive, and when $\mathcal{A}$ is abelian, $\mathrm{Ch}(\mathcal{A})$ is abelian with kernels and cokernels computed degreewise. This makes complexes a stable environment for exact-sequence arguments.

The cohomology functor $$ H^n:\mathrm{Ch}(\mathcal A)\to\mathcal A $$ is additive but not exact. A short exact sequence of complexes does not usually split into short exact sequences on cohomology. Instead, it produces the long exact sequence with connecting morphism. This is the first signal that cohomology is naturally derived rather than exact.

Boundedness conventions matter in later derived categories. The categories $D^{+}(\mathcal{A})$, $D^{-}(\mathcal{A})$, and $D^b(\mathcal{A})$ are obtained from bounded-below, bounded-above, and bounded complexes after passing through homotopy and then inverting quasi-isomorphisms. The present unit supplies the raw category before homotopies and localization enter.

A chain map $f:C^{\bullet }\to D^{\bullet }$ is a quasi-isomorphism when it induces isomorphisms $$ H^n(f)^n(C^\bullet)\to H^n(D^\bullet) $$ for every $n$. Derived categories are built by formally treating quasi-isomorphisms as equivalences. This is why cohomology objects are not only invariants of complexes; they define the class of maps that derived categories invert.

The topological origin remains visible. Poincare's chains were attached to cells and simplices; Cartan-Eilenberg and Grothendieck abstracted the same exactness mechanisms into modules, sheaves, and abelian categories. Gelfand-Manin's treatment begins from this categorical level because modern geometry and representation theory need complexes of sheaves and functors, not only complexes of abelian groups.

Synthesis. A chain complex packages many exactness tests into one graded object. Its cohomology records the controlled failure of exactness degree by degree. In an abelian category, this construction is abstract enough to include modules, sheaves, and later derived functors, while still retaining the diagram-chase machinery of elementary exact sequences.

Full proof set Master

Proposition 1 (cohomology is functorial). A morphism of complexes $f:C^{\bullet }\to D^{\bullet }$ induces morphisms $H^n(f):H^n(C^{\bullet })\to H^n(D^{\bullet })$ for all $n$.

Proof. The chain-map identity $d_{D}f=f{d}_C$ sends closed elements of $C^n$ to closed elements of $D^n$. It also sends boundaries in $C^n$ to boundaries in $D^n$. Therefore it induces a map from closed objects modulo boundaries in $C$ to closed objects modulo boundaries in $D$. In abelian-category language, this is the induced morphism between the corresponding kernel-over-image quotients. $\square$

Proposition 2 (degreewise kernels make complex kernels). If $f:C^{\bullet }\to D^{\bullet }$ is a morphism of complexes, the degreewise kernels $\ker (f^n)$ form a subcomplex of $C^{\bullet }$.

Proof. Let $x$ lie in $\ker (f^n)$. Then $$ f^{n+1}(d_C^n x)=d_D^n(f^n x)=d_D^n(0)=0. $$ Thus $d_C^{n}x$ lies in $\ker (f^{n+1})$. Hence the differentials of $C^{\bullet }$ restrict to the degreewise kernels. The cokernel argument is dual, so kernels and cokernels in $\mathrm{Ch}(\mathcal{A})$ are computed degreewise. $\square$

Proposition 3 (exact complexes have zero cohomology). If a complex is exact at every degree, then $H^n(C^{\bullet })=0$ for all $n$.

Proof. Exactness at degree $n$ means $\mathrm{im}(d^{n-1})=\ker (d^n)$. Therefore $$ H^n(C^\bullet)=\ker(d^n)/\operatorname{im}(d^{n-1})=0. $$ Conversely, in an abelian category, zero cohomology at every degree says the same image-to-kernel quotient vanishes, which recovers exactness. $\square$

Connections Master

• Exact sequence and snake lemma 01.02.11. The long exact sequence in cohomology is the snake lemma applied degree by degree.

• Chain homotopy 01.02.31. Homotopy is the next equivalence relation on maps of complexes.

• Mapping cone 01.02.32. The cone packages a chain map into a new complex whose cohomology measures whether the map is a quasi-isomorphism.

• Abelian categories and Grothendieck axioms 01.02.33. Abelian categories provide the kernel/cokernel setting in which cohomology objects exist.

• Derived categories 04.03.11. Derived categories begin with complexes and then invert quasi-isomorphisms.

Historical & philosophical context Master

Poincare's 1895 topology introduced chains and boundaries as a way to compute holes in spaces ^[Poincare]. The identity "boundary of a boundary is zero" is the ancestral form of the complex condition.

Cartan and Eilenberg's 1956 book gave homological algebra its modern axiomatic language ^{[Cartan-Eilenberg]}. Complexes were no longer only topological gadgets; they became the central algebraic objects used to define derived functors.

Gelfand and Manin place complexes at the start of the derived-category story ^{[Gelfand-Manin]}. That choice reflects the modern viewpoint: many geometric objects are best studied through complexes, and ordinary objects often appear as complexes concentrated in one degree.

Bibliography Master

@book{GelfandManinHomologicalAlgebraComplexes,
  author = {Gelfand, Sergei I. and Manin, Yuri I.},
  title = {Methods of Homological Algebra},
  publisher = {Springer},
  year = {1996}
}

@book{Weibel1994Complexes,
  author = {Weibel, Charles A.},
  title = {An Introduction to Homological Algebra},
  publisher = {Cambridge University Press},
  year = {1994}
}

@book{CartanEilenberg1956Complexes,
  author = {Cartan, Henri and Eilenberg, Samuel},
  title = {Homological Algebra},
  publisher = {Princeton University Press},
  year = {1956}
}

@book{MacLane1963HomologyComplexes,
  author = {Mac Lane, Saunders},
  title = {Homology},
  publisher = {Springer},
  year = {1963}
}

@article{Poincare1895AnalysisSitusComplexes,
  author = {Poincare, Henri},
  title = {Analysis Situs},
  journal = {Journal de l'Ecole Polytechnique},
  volume = {1},
  pages = {1--121},
  year = {1895}
}