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The degree-one Bockstein homomorphism makes modulo-$n$ cohomology into a cochain complex. Its cohomology is the Bockstein cohomology. For prime $n$ this is a graded vector space; for composite $n$ it is a graded $\mathbb{Z}/n$-module. For ${\mathbb{RP}}^3$ at modulus two, its only nonzero groups are ${\mathbb{F}}_2$ in degrees zero and three. This is the cohomological counterpart of Bockstein homology.

The category $D^{+}(X)$ is obtained from bounded below cochain complexes of sheaves by passing to cochain-homotopy classes of maps and then inverting quasi-isomorphisms. It can equally be modeled by injective resolutions. The cohomology-sheaf functors descend to this localization, and an object is zero exactly when all its cohomology sheaves vanish.

A finite filtration of a cochain complex produces a convergent spectral sequence. The induced abutment filtration is the image of $H^n(F^{p}C)\to H^n(C)$. Finiteness in each degree suffices; a bounded range of cochain degrees is not required.

An element in the image of the preceding differential of a cochain complex. Every coboundary is a cocycle, since consecutive differentials compose to zero.

A coboundary map is the degree-one differential $d^n:C^n\to C^{n+1}$ of a cochain complex.

A cochain differential is a degree-one map in a cochain complex:The square-zero identity makes every coboundary a cocycle, so the quotient $H^n(C)=\ker d^n/\mathrm{im}{d}^{n-1}$ defining cohomology is meaningful. The maps are homomorphisms of the abelian groups, or module-linear maps, used in the complex.

For $C^0=\mathbb{Z}$, $C^1=\mathbb{Z}$, $d^0(n)=2n$, and all other groups zero, the identity is automatic. The degree-zero cohomology is zero, while the degree-one cohomology is $\mathbb{Z}/2\mathbb{Z}$: every degree-one element is closed, but only the even ones are exact.

The exterior derivative on differential forms is another cochain differential, with $d^2=0$. In contrast, a chain differential lowers degree. The word differential records this graded square-zero structure, not necessarily differentiation of scalar functions.

A sequence of submodules preserved by the differential of a cochain complex. Its inclusion is a map of complexes.

An element killed by the next differential of a cochain complex. Its class modulo coboundaries is a cohomology class.

A cohomology theory assigns abelian groups $H^q(X;R)$ to a topological space $X$. For singular cohomology with coefficients in a commutative ring $R$, form the cochain complex $C^q(X;R)$ with coboundary $\delta$ and takeA cocycle represents a cohomology class; adding a coboundary leaves that class unchanged.

A cohomology class evaluates on a homology class by applying its cochain to a representing cycle. Replacing either representative changes the result by zero, explaining the relation to homology. A continuous map $f:X\to Y$ induces cochain pullback and hence $f^{\ast }:H^q(Y;R)\to H^q(X;R)$: the direction reverses.

The cup product combines degree-$p$ and degree-$q$ classes into degree $p+q$, giving a graded cohomology ring. The circle has one integral generator in degrees zero and one. A two-dimensional torus has two independent degree-one classes whose cup product generates degree two. That multiplication distinguishes it from $S^1\vee S^1\vee S^2$, which has the same additive cohomology groups but zero products between its positive-degree generators.

The equivalence class of a cocycle in a cochain complex, where adding a coboundary leaves its class unchanged.

A degree-$q$ cohomology group is the quotient of degree-$q$ cocycles by coboundaries in a cochain complex. Topological cohomology groups arise from singular or cellular cochains. Their direct sum carries the cohomology ring structure when equipped with the cup product.

For a cochain complex of sheaves $E$, its $i$th cohomology sheaf is ${\mathcal{H}}^i(E)=\ker (d^i)/\mathrm{im}(d^{i-1})$, where the image and quotient are taken in sheaves. Its stalk at $x$ is the cohomology of the stalk complex, because taking stalks is exact.

A short exact sequence of cochain complexes of sheaves $0\to E\to F\to G\to 0$ induces $\cdots \to {\mathcal{H}}^i(E)\to {\mathcal{H}}^i(F)\to {\mathcal{H}}^i(G)\to {\mathcal{H}}^{i+1}(E)\to \cdots$. Taking stalks is exact, so the construction and exactness follow from the ordinary long exact sequence of cohomology of complexes at every stalk. The stalkwise connecting maps come from the lift-and-differentiate construction and form a sheaf morphism.

For an additive contravariant left-exact functor, apply it to a projective resolution of $M$. Reversal of the arrows gives a cochain complex. Comparison maps lift using projectivity, and any two lifts are chain homotopic, so its cohomology is independent of the resolution. The functors are contravariant, vanish on projective modules in positive degrees, and send $0\to M^{\prime }\to M\to M^{\prime \prime }\to 0$ to $0\to F(M^{\prime \prime })\to F(M)\to F(M^{\prime })\to R^{1}F(M^{\prime \prime })\to \cdots$. Degreewise split resolutions from the horseshoe lemma yield this sequence.

The de Rham complex of a smooth manifold $M$ is the cochain complex of smooth differential forms with exterior derivative:Here $n=\dim M$ and ${\Omega }^0(M)=C^{\infty }(M)$. The identity $d^2=0$ makes every exact differential form closed. The quotient of closed $k$-forms by exact $k$-forms is the de Rham cohomology group $H_{\mathrm{dR}}^k(M)$.^[ref]

In Euclidean space the Poincaré lemma makes this complex exact in positive degrees. On a general manifold it is locally exact in small coordinate balls, but global exactness can fail. For the unit circle, the restriction of $\omega =-ydx+xdy$ has integral $2\pi$ around one positive turn. It is closed, since the circle has no nonzero two-forms. It cannot equal $df$ for a globally defined smooth function on the circle: the integral of a derivative around a closed loop is zero.

The complex therefore separates local differential identities from global obstructions. In degree zero, $df=0$ means that $f$ is locally constant; in degree one, periods around loops can detect a failure to possess a global potential.

An elliptic complex on a smooth manifold is a finite cochain complex of differential operators between sections of vector bundles,with $D_{i+1}{D}_i=0$, whose principal symbol sequence is exact at every nonzero real cotangent vector. Thus for $\xi \in T_x^{\ast }M\setminus \{0\}$, the kernel of each symbol map equals the image of the preceding one, including injectivity and surjectivity at the ends.^[ref]

The de Rham complex is the basic example. Up to an inessential nonzero scalar convention, the symbol of $d$ is $\alpha \mapsto \xi \wedge \alpha$. Choose a tangent vector $v$ with $\xi (v)=1$. The interior product identitygives a contracting homotopy for that symbol sequence and proves exactness.

The condition belongs to the whole sequence: the individual maps need not be elliptic operators between equal-rank bundles. On a compact manifold without boundary, ellipticity gives finite-dimensional cohomology. Boundary problems require additional conditions. This finiteness is what makes an alternating cohomological trace meaningful in the Atiyah-Bott fixed-point theorem.

A cochain complex with a decreasing filtration by cochain subcomplexes. A filtration finite in each degree is bounded in the degreewise sense; uniform bounds $F^{a}C=C$ and $F^{b}C=0$ are sufficient. The associated spectral sequence starts with the graded quotient complexes.

A flasque resolution is an exact sequence of sheaves of abelian groups whose terms ${\mathcal{I}}^j$ are flasque sheaves. Every such sheaf embeds by its section germs into the sheaf $U\mapsto {\prod }_{P\in U}{\mathcal{F}}_P$. Its restriction maps are projections, so it is flasque. Apply this construction successively to the cokernels to obtain a resolution. The sheaf cohomology of $\mathcal{F}$ is the cohomology of the cochain complex $\Gamma (X,{\mathcal{I}}^{\bullet })$. A quotient of two flasque sheaves in a short exact sequence of sheaves is flasque: the flasque kernel allows local lifts to be glued into a section of the middle sheaf over the smaller open set; extend that lift using flasqueness of the middle sheaf, and project. Consequently the resolution's successive cokernels are flasque when $\mathcal{F}$ is flasque, and exactness of sections shows that all its positive-degree cohomology vanishes.

For a group $G$ acting on an abelian group $M$, group cohomology consists of the groups $H^n(G,M)$ obtained from the standard cochain complex of maps $G^n\to M$. In degree zero it gives the invariant subgroup $M^G$. In degree one it gives one-cocycles modulo group coboundaries.

If the action is trivial, the one-cocycle identity becomes $f(gh)=f(g)+f(h)$ and all one-coboundaries vanish, so $H^1(G,M)=\mathrm{Hom}(G,M)$. Degree two encodes group extensions with the prescribed action on the kernel. This explains why cohomology naturally measures whether local multiplication data can be assembled into a globally associative extension.

Homological algebra studies chain complexes and cochain complexes, their homology and cohomology, and the functors that compare them. The identity $d^2=0$ ensures that every boundary is a cycle, so the quotient measures cycles not explained by boundaries.

A basic motivation is failure of exactness. Tensoring the inclusion $2\mathbb{Z}\hookrightarrow \mathbb{Z}$ with $\mathbb{Z}/2\mathbb{Z}$ gives a zero map between two nonzero groups, so tensoring need not preserve injections. Tor functor measures this failure, while Ext functor measures related failures for Hom functors. Projective resolutions replace a module by a complex of easier modules and make these invariants computable.