Chain complex ($(C_{\bullet },d)$)

A chain complex is a sequence of abelian groups or modules and homomorphisms $d_i:C_i\to C_{i-1}$ satisfying $d_{i-1}{d}_i=0$. Its homology is $H_i(C)=\ker d_i/\mathrm{im}{d}_{i+1}$.

For an endomorphism of a finite-dimensional bounded chain complex over a field, the alternating chain trace equals the alternating trace on homology. Split each chain space into boundaries, representatives for homology, and a complement mapping isomorphically to the preceding boundaries. The chain-map relation makes the traces on the first and third summands cancel in successive degrees, leaving the homology traces. Over a field the induced cohomology maps are dual and have the same traces. This identity turns the absence of diagonal simplex contributions into a vanishing Lefschetz number.

The chain differential of a homological chain complex is a family of group or module homomorphisms ${\partial }_n:C_n\to C_{n-1}$ satisfying ${\partial }_{n-1}{\partial }_n=0$. The identity implies $\mathrm{im}{\partial }_{n+1}\subset \ker {\partial }_n$, so the quotient $H_n=\ker {\partial }_n/\mathrm{im}{\partial }_{n+1}$ is defined. In a graded complex over $\mathbb{Z}[U]$, the chain differential can have degree $-1$ while $U$ has degree $-2$, with $\partial (Ux)=U\partial x$.

The reversed dual module of a chain complex becomes a chain complex with the unsigned precomposition differential above. A functional on $C_{-i}$ is sent to a functional on $C_{1-i}$, lowering the reversed degree by one.

A graded Hom functor construction consists of maps shifting degree by $j$. One homological convention is $d_{M}f=f{d}_C+(-1{)}^{j-1}{d}_{C^{\prime }}f$. Its degree-zero cycles are chain maps, and its degree-zero boundaries are null-homotopic chain maps, so its zeroth homology is the module of chain homotopy classes. This convention differs by degree-dependent signs from the alternative convention $d^{\prime }f-(-1{)}^{j}fd$.

For finite free total chain complexes, evaluation identifies the underlying graded tensor product $X\otimes C^{\prime }$ with the graded Hom complex of chain complexes, where $X$ is the reversed dual chain complex. Under the precomposition-first Hom differential, conjugation by the sign $(-1{)}^{j(j-1)/2}$ makes this a chain isomorphism. The identity $\rho (j)-\rho (j-1)=j-1$ works for every integer degree. If the grading is unbounded and only degreewise finite, products on the Hom side need not equal direct sums on the tensor side.

The tensor product of homologically graded chain complexes has differential $D(x_p\otimes y)=d{x}_p\otimes y+(-1{)}^p{x}_p\otimes dy$. The sign makes the two mixed terms in $D^2$ cancel. Total degree is the sum of the two degrees.

Interchanging homogeneous objects of degrees $p$ and $q$ introduces $(-1{)}^{pq}$. In particular, moving a degree-minus-one differential past a degree-$p$ factor introduces $(-1{)}^p$. This convention produces the differential on the tensor product of chain complexes.

For $n\ge 1$, the chain complex with copies of $R$ in degrees $n,n-1$ and identity differential. It is acyclic; a map from it chooses an arbitrary degree-$n$ element and its boundary.

The chain complex with $R$ in a single degree $n$ and zero differential. A map from it into a complex chooses a cycle in degree $n$.

A double complex has modules $C_{p,q}$ and differentials lowering either index, each squaring to zero and anticommuting with the other. The total chain complex has term ${⨁}_{p+q=i}{C}_{p,q}$ and differential the sum of the two differentials. For the tensor product of two chain complexes, a sign $(-1{)}^p$ on the second differential ensures anticommutation. If the double complex is in the first quadrant and one direction has homology only in degree zero, its total homology is computed by the surviving degree-zero complex. Applying this twice to two free resolutions proves the balanced calculation of the Tor functor.

Two degree-raising differentials square to zero and anticommute. If an initial convention uses commuting differentials, introduce a sign in one of them before forming the total cochain complex.

Filtering the total cochain complex by each index yields two spectral sequences. One first takes vertical cohomology and the other horizontal cohomology; boundedness ensures both converge to total cohomology.

The total differential is $d_h+d_v$ for anticommuting differentials. For a double cochain complex bounded in both indices, both index filtrations are finite and have the same total cohomology as abutment.

The Koszul complex on $f_1,\dots ,f_r$ has term $K_i={\bigwedge }^i{R}^r$ and differential $\partial (e_{j_1}\wedge \cdots \wedge e_{j_i})={\sum }_a(-1{)}^{a-1}{f}_{j_a}{e}_{j_1}\wedge \cdots \wedge \widehat{e_{j_a}}\wedge \cdots \wedge e_{j_i}$. Its degree-zero homology is $R/(f_1,\dots ,f_r)$. For a regular sequence it is exact in positive degrees, hence a Koszul resolution of that quotient. The variables of a polynomial ring form a regular sequence, so this resolves the coefficient field in length equal to the number of variables.

The perfect exterior pairing in complementary degrees identifies the dual of a Koszul complex with its degree reversal, with suitable differential signs. For a regular sequence, its Koszul resolution consequently gives ${\mathrm{Ext}}_R^i(R/(f),R)=0$ except in degree $r$, where it is $R/(f)$.

Let $(R,\mathfrak{m})$ be a Noetherian local ring, let $M$ be nonzero and finitely generated, and let every $f_i$ lie in $\mathfrak{m}$. Then positive Koszul homology vanishes exactly when $f$ is a regular sequence on a module $M$. The mapping cone exact sequence proves the forward implication by induction; surjectivity of the last generator on earlier homology and the Nakayama lemma prove the converse.

Tensor the finite free Koszul complex on $f_1,\dots ,f_r$ with the module $M$. For a regular sequence on a module, its positive Koszul homology vanishes. If the sequence is regular on $R$, this complex instead computes ${\mathrm{Tor}}^R(R/(f),M)$ for arbitrary $M$.

Koszul homology is the homology of the Koszul complex with module coefficients. Its degree-zero term is $M/(f)M$, and its positive terms record failures of regularity. Each generator acts trivially on homology by the Koszul homotopy for multiplication by a generator.

Exterior multiplication $h_j(u)=e_j\wedge u$ is a chain homotopy between multiplication by $f_j$ and zero on a Koszul complex. A linear combination of these homotopies is a contracting homotopy when the generators span the unit ideal.

For a possibly noncommutative ring and central $t_i$, the formal exterior basis gives free bimodules with differential $d(e_{i_1}\wedge \cdots \wedge e_{i_k})={\sum }_a(-1{)}^{a-1}{t}_{i_a}{e}_{i_1}\wedge \cdots \widehat{e_{i_a}}\cdots \wedge e_{i_k}$. Centrality makes this a bimodule differential. Pairwise cancellation proves $d^2=0$.

An augmented chain complex includes an augmentation $\epsilon :C_0\to \mathbb{Z}$ with $\epsilon d_1=0$. For a nonempty simplicial complex, the augmentation sends every vertex to $1$ and the augmented homology in degree zero is its reduced homology.

The chain group $C_i$ is the group in degree $i$ of a chain complex. Its elements are $i$-chains.

A chain subcomplex $D_{\bullet }\subseteq C_{\bullet }$ consists of subgroups $D_i\le C_i$ preserved by the boundary operator: $\partial (D_i)\subseteq D_{i-1}$.

The boundary operator of a chain complex is its degree-minus-one differential $\partial :C_i\to C_{i-1}$. The identity ${\partial }^2=0$ says that every chain boundary is a chain cycle.

For a CW complex, $C_n^{\mathrm{cell}}(X)=H_n(X^n,X^{n-1})$ is freely generated by the oriented $n$-cells. Connecting maps of skeleton pairs define its differential.

For a CW complex, the qth cellular chain group is the relative homology group of successive skeleta, a free abelian group with one generator for each q-cell. The attaching maps determine the cellular boundary, and the resulting cellular chain complex computes singular homology.

A finite linear combination, with coefficients in $R$, of the oriented $n$-cells of a CW complex.

A cellular chain whose cellular boundary is zero.

A cellular chain that is the boundary of a chain one dimension higher. The chain-complex identity $d_n{d}_{n+1}=0$ makes every cellular boundary a cellular cycle; their quotient is cellular homology.

The product cells of finite CW complexes give the tensor product of their cellular chain complexes. The boundary formula is the displayed signed product rule. The sign comes from moving the outward normal of the second factor past the $p$ oriented tangent directions of the first factor. This is the cellular counterpart of the Eilenberg–Zilber theorem.

Collapse two distinct parallel circles of a torus to two separate points. Cutting between the circles gives two annuli, each of which becomes a two-sphere with the two collapsed points as its poles. The resulting CW complex has two vertices, two edges connecting them, and two two-cells with zero cellular boundary. Its homology is as displayed, with all higher groups zero; it is homotopy equivalent to $S^1\vee S^2\vee S^2$.

For a CW complex, the homology of its cellular chain complex naturally equals its singular homology. The relative groups of successive skeleton pairs, free on the cells in the corresponding dimension and zero in other dimensions, yield this identification through their connecting maps.

For a CW complex, applying $\mathrm{Hom}(-,A)$ to its integral cellular chain complex gives a cochain complex. The differential is precomposition with the cellular boundary. For finitely many cells it is represented by the transpose of the boundary matrix, reduced in the coefficient group when appropriate.

The cohomology of the cellular cochain complex naturally agrees with singular cohomology. A CW complex with no odd cells has zero cellular differentials and free integral cohomology on its even cells, although its cup product still requires a separate computation.

If an $n$-dimensional CW complex is extended by exactly one cell in dimension $n+1$, then its cellular cochain group in that degree is the coefficient field $k$. Regardless of how many higher cells are attached, the cohomology is a quotient of a subspace of that one-dimensional group. This gives the displayed bound. Finiteness of the higher cell sets is unnecessary for this particular conclusion.

The cellular boundary ${\mathbb{Z}}^2\to \mathbb{Z}$ is $(m,n)$, with kernel $\mathbb{Z}$ and zero cokernel when the degrees are coprime. The van Kampen theorem gives trivial fundamental group. The Hurewicz theorem supplies a map from the sphere representing a homology generator, and the homological Whitehead theorem makes it a homotopy equivalence. For degrees $2,3$ the space cannot be homeomorphic to a sphere: an embedded circle in a sphere bounds its two complementary regions, so both characteristic boundary maps would have degree of absolute value one.

The coefficient of an $(n-1)$-cell in the cellular boundary of an $n$-cell is the degree of the attaching map after collapsing the complement of that $(n-1)$-cell to obtain a map $S^{n-1}\to S^{n-1}$.

An element of a chain group is a cycle when its boundary operator kills it. A cycle need not itself be a chain boundary. The homology group $H_n(C)$ identifies cycles whose difference is a chain boundary.

An element of a chain group is a boundary when it is the image of a higher-degree element under the boundary operator. Since ${\partial }^2=0$, every boundary is a chain cycle. Thus $H_n(C)=Z_n(C)/B_n(C)$, and a chain cycle represents zero in homology exactly when it is a boundary.

The coefficients of a chain are the scalars multiplying its basis simplices.

A chain map $f:C\to C^{\prime }$ is a family $f_i:C_i\to C_i^{\prime }$ satisfying $d_i^{\prime }{f}_i=f_{i-1}{d}_i$.

A chain map inducing isomorphisms on all homology groups. The same definition applies to a map of cochain complexes using cohomology groups. A quasi-isomorphism need not be an isomorphism in each degree.

Let $f$ be a chain map between degreewise finitely generated free abelian groups. If its reduction modulo every prime number is a quasi-isomorphism, the mapping cone acyclicity criterion makes $M(f)\otimes {\mathbb{F}}_p$ acyclic. The short exact sequence of chain complexes $0\to M\stackrel{p}{\to }M\to M\otimes {\mathbb{F}}_p\to 0$ implies that multiplication by $p$ is an isomorphism on every $H_i(M)$. These are finitely generated abelian groups. Their free summands would make multiplication by $p$ fail to be surjective, and any nonzero finite summand would make it fail to be injective for a prime dividing its order. Thus $H_i(M)=0$ and $f$ is a quasi-isomorphism. Degreewise finite generation is essential to this argument.

A chain map sends cycles to cycles and boundaries to boundaries, and therefore induces $f_{\ast }:H_i(C)\to H_i(C^{\prime })$ by $[x]\mapsto [f_i(x)]$.

Chain maps $f,g:C\to C^{\prime }$ are chain homotopic when there are maps $h_i:C_i\to C_{i+1}^{\prime }$ such thatChain-homotopic maps induce the same map on homology.

A contracting homotopy on a chain complex is a degree-one map $h$ whose displayed identity makes the identity chain-homotopic to zero. It implies that all homology vanishes. The converse is not valid for arbitrary complexes of modules.

For cochain maps $f,g:C^{\bullet }\to D^{\bullet }$, a cochain homotopy is a family $K:C^q\to D^{q-1}$ satisfyingCochain-homotopic maps induce the same map on cohomology.

For an open cover $\mathcal{U}$ of $X$, singular chains generated by simplices lying inside one member of $\mathcal{U}$ form a subcomplex chain-homotopy equivalent to the full singular chain complex. Repeated barycentric subdivision makes chains small.

For a chain map $f:C\to C^{\prime }$, one mapping-cone convention takes $M(f{)}_i=C_{i-1}\oplus C_i^{\prime }$ with a differential combining $d,d^{\prime }$ and $f$. Its short exact sequence with $C^{\prime }$ and a shift of $C$ produces a long exact sequence in homology.

For a chain map $f:C\to D$, use $M_i=C_{i-1}\oplus D_i$ and $d(x,y)=(d_{C}x,(-1{)}^{i}f(x)+d_{D}y)$. The chain map identity makes $d^2=0$. The short exact sequence of chain complexes $0\to D\to M\to C_{\ast -1}\to 0$ has connecting homomorphism $(-1{)}^i{f}_{\ast }$ in degree $i$. Its long exact sequence in homology shows that the homology of $M$ vanishes precisely when every $f_{\ast }$ is an isomorphism.

A degreewise short exact sequence $0\to A\to B\to C\to 0$ of chain complexes induces a long exact sequence of homology groups through the connecting homomorphisms.

The connecting map sends a homology class in the quotient complex to the class obtained by lifting a representative, applying the middle differential, and identifying the result in the subcomplex.

The connecting homomorphism is the degree-shifting map produced by the lift-and-boundary construction in a long exact sequence.

A relative cycle is represented by a singular chain $b$ whose boundary lies in $A$. Define $\delta [\bar{b}]=[\partial b]$. If a representative changes by $\partial c+a$ with $a\in C_n(A)$, its boundary changes by $\partial a$, an $A$-boundary. Thus the map is well-defined and additive. Its image is exactly the kernel of $H_{n-1}(A)\to H_{n-1}(B)$: a cycle in $A$ dies in $B$ precisely when it is the boundary of a relative cycle lifted to $B$. This proves the corresponding exactness in the long exact sequence in relative homology directly.

The Bockstein homomorphism is the connecting homomorphism associated with a short exact sequence of coefficient groups. For $0\to \mathbb{Z}/n\to \mathbb{Z}/n^2\to \mathbb{Z}/n\to 0$, it has degree $-1$ on homology and degree $+1$ on cohomology.

For the coefficient sequence $0\to {\mathbb{Z}}_2\to {\mathbb{Z}}_4\to {\mathbb{Z}}_2\to 0$, the Bockstein homomorphism on infinite-dimensional real projective space is zero from an even degree and an isomorphism from an odd degree. In cellular cohomology, lift the mod-two generator to $1\in {\mathbb{Z}}_4$. The coboundary is zero in even degree and two in odd degree; identifying two with the image of $1\in {\mathbb{Z}}_2$ gives the formula.

The Bockstein homomorphism for $0\to \mathbb{Z}/n\to \mathbb{Z}/n^2\to \mathbb{Z}/n\to 0$, with injection $[a]\mapsto [na]$, is coefficient reduction $\rho$ after the integral Bockstein homomorphism. The commuting maps of coefficient sequences prove this by naturality of the connecting homomorphism.

Reduction $\rho$ of the integral Bockstein homomorphism associated to $0\to \mathbb{Z}\stackrel{2}{\to }\mathbb{Z}\to {\mathbb{F}}_2\to 0$ equals the cup product square on degree-one classes. One can see this on an ordered two-simplex: lift the values of a mod-two one-cocycle to zero or one. The half-coboundary is one precisely when both consecutive edge values are one, giving the cup-square cocycle.

Exactness of the integral coefficient sequence gives $\hat{\beta }\rho =0$. The Bockstein factorization through integral cohomology therefore gives ${\beta }^2=\rho \hat{\beta }\rho \hat{\beta }=0$, for every modulus $n\ge 2$, including composite moduli.

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 integral Bockstein is the connecting homomorphism of $0\to \mathbb{Z}\stackrel{n}{\to }\mathbb{Z}\to \mathbb{Z}/n\to 0$. Lift a modulo-$n$ cocycle to an integral cochain $a$; write its coboundary as $\delta a=nb$. Then $\hat{\beta }[a]=[b]$. Its image is the subgroup of integral cohomology killed by $n$.

For the three-dimensional lens space $L(p)$, the integral coefficient sequence shows that the integral connecting map $H^1(L(p);{\mathbb{F}}_p)\to H^2(L(p);\mathbb{Z})$ is an isomorphism between groups of order $p$. Reduction modulo $p$ is also an isomorphism in degree two, so their composite Bockstein is an isomorphism.

For a generator $a\in H^1(L(p);{\mathbb{F}}_p)$, Poincare duality and the Bockstein isomorphism for a three-dimensional lens space make $t(a)$ nonzero. Replacing $a$ by $na$ multiplies $t(a)$ by $n^2$. An orientation-reversing homotopy equivalence multiplies the evaluation by $-1$, so its existence forces $n^2=-1$ in ${\mathbb{F}}_p$ for some $n$.

When the homological Bockstein satisfies ${\beta }^2=0$, it makes $H_{\ast }(C\otimes \mathbb{Z}/n)$ into a chain complex. Its homology is the Bockstein homology $\beta H_{\ast }(C;n)$.

A short exact sequence of coefficient groups $0\to A\to B\to C\to 0$ induces a natural long exact sequenceThe connecting map is a Bockstein homomorphism.

For the modulo-$m$ cohomological Bockstein homomorphism,This follows by lifting cocycles to integral cochains and applying the graded Leibniz rule for the coboundary before dividing by $m$.

Over the principal ideal domain $\mathbb{Z}$, Smith normal form decomposes a chain complex of finitely generated free modules into direct sums of one-term complexes $\mathbb{Z}$ and two-term complexesThe one-term summands record free homology, the summands with $m>1$ record cyclic torsion, and those with $m=1$ are acyclic.

If a chain complex of finitely generated free abelian groups has zero homology after tensoring with ${\mathbb{F}}_p$ for every prime $p$, then its integral homology vanishes. The universal coefficient theorem for homology detects a free summand modulo every prime and detects a nonzero finite summand modulo any prime dividing its order.