Inmathematics, specifically inhomology theory andalgebraic topology,cohomology is a general term for a sequence ofabelian groups, usually one associated with atopological space, often defined from acochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains arefunctions on the group ofchains in homology theory.

From its start intopology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughoutgeometry andalgebra. The terminology tends to hide the fact that cohomology, acontravariant theory, is more natural than homology in many applications. At a basic level, this has to do with functions andpullbacks in geometric situations: given spaces${\displaystyle X}$ and${\displaystyle Y}$, and some function${\displaystyle F}$ on${\displaystyle Y}$, for anymapping${\displaystyle f:X\to Y}$, composition with${\displaystyle f}$ gives rise to a function${\displaystyle F\circ f}$ on${\displaystyle X}$. The most important cohomology theories have a product, thecup product, which gives them aring structure. Because of this feature, cohomology is usually a stronger invariant than homology.

Singular cohomology is a powerful invariant in topology, associating agraded-commutative ring with any topological space. Everycontinuous map${\displaystyle f:X\to Y}$ determines ahomomorphism from the cohomology ring of${\displaystyle Y}$ to that of${\displaystyle X}$; this puts strong restrictions on the possible maps from${\displaystyle X}$ to${\displaystyle Y}$. Unlike more subtle invariants such ashomotopy groups, the cohomology ring tends to be computable in practice for spaces of interest.

For a topological space${\displaystyle X}$, the definition of singular cohomology starts with thesingular chain complex:^[1]$${\displaystyle \cdots \to C_{i+1}\stackrel{{\mathrm{\partial }}_{i+1}}{\to }{C}_i\stackrel{{\mathrm{\partial }}_i}{\to }{C}_{i-1}\to \cdots }$$By definition, thesingular homology of${\displaystyle X}$ is the homology of this chain complex (the kernel of one homomorphism modulo the image of the previous one). In more detail,${\displaystyle C_i}$ is thefree abelian group on the set of continuous maps from the standard${\displaystyle i}$-simplex to${\displaystyle X}$ (called "singular${\displaystyle i}$-simplices in${\displaystyle X}$"), and${\displaystyle {\mathrm{\partial }}_i}$ is the${\displaystyle i}$-th boundary homomorphism. The groups${\displaystyle C_i}$ are zero for${\displaystyle i}$ negative.

Now fix an abelian group${\displaystyle A}$, and replace each group${\displaystyle C_i}$ by itsdual group${\displaystyle C_i^{\ast }=\mathrm{H}\mathrm{o}\mathrm{m}(C_i,A),}$ and${\displaystyle {\mathrm{\partial }}_i}$ by itsdual homomorphism$${\displaystyle d_{i-1}:C_{i-1}^{\ast }\to C_i^{\ast }.}$$

This has the effect of "reversing all the arrows" of the original complex, leaving acochain complex$${\displaystyle \cdots \leftarrow C_{i+1}^{\ast }\stackrel{d_i}{\leftarrow }{C}_i^{\ast }\stackrel{d_{i-1}}{\leftarrow }{C}_{i-1}^{\ast }\leftarrow \cdots }$$

For an integer${\displaystyle i}$, the${\displaystyle i}$^thcohomology group of${\displaystyle X}$ with coefficients in${\displaystyle A}$ is defined to be${\displaystyle \ker (d_i)/\mathrm{im}(d_{i-1})}$ and denoted by${\displaystyle H^i(X,A)}$. The group${\displaystyle H^i(X,A)}$ is zero for${\displaystyle i}$ negative. The elements of${\displaystyle C_i^{\ast }}$ are calledsingular${\displaystyle i}$-cochains with coefficients in${\displaystyle A}$. (Equivalently, an${\displaystyle i}$-cochain on${\displaystyle X}$ can be identified with a function from the set of singular${\displaystyle i}$-simplices in${\displaystyle X}$ to${\displaystyle A}$.) Elements of${\displaystyle \ker (d)}$ and${\displaystyle \text{im}(d)}$ are calledcocycles andcoboundaries, respectively, while elements of${\displaystyle \ker (d_i)/\mathrm{im}(d_{i-1})=H^i(X,A)}$ are calledcohomology classes (because they areequivalence classes of cocycles).

In what follows, the coefficient group${\displaystyle A}$ is sometimes not written. It is common to take${\displaystyle A}$ to be acommutative ring${\displaystyle R}$; then the cohomology groups are${\displaystyle R}$-modules. A standard choice is the ring${\displaystyle \mathbb{Z}}$ ofintegers.

Some of the formal properties of cohomology are only minor variants of the properties of homology:

• A continuous map${\displaystyle f:X\to Y}$ determines apushforward homomorphism${\displaystyle f_{\ast }:H_i(X)\to H_i(Y)}$ on homology and apullback homomorphism${\displaystyle f^{\ast }:H^i(Y)\to H^i(X)}$ on cohomology. This makes cohomology into acontravariant functor from topological spaces to abelian groups (or${\displaystyle R}$-modules).
• Twohomotopic maps from${\displaystyle X}$ to${\displaystyle Y}$ induce the same homomorphism on cohomology (just as on homology).
• TheMayer–Vietoris sequence is an important computational tool in cohomology, as in homology. Note that the boundary homomorphism increases (rather than decreases) degree in cohomology. That is, if a space${\displaystyle X}$ is the union ofopen subsets${\displaystyle U}$ and${\displaystyle V}$, then there is along exact sequence:$${\displaystyle \cdots \to H^i(X)\to H^i(U)\oplus H^i(V)\to H^i(U\cap V)\to H^{i+1}(X)\to \cdots }$$
• There arerelative cohomology groups${\displaystyle H^i(X,Y;A)}$ for anysubspace${\displaystyle Y}$ of a space${\displaystyle X}$. They are related to the usual cohomology groups by a long exact sequence:$${\displaystyle \cdots \to H^i(X,Y)\to H^i(X)\to H^i(Y)\to H^{i+1}(X,Y)\to \cdots }$$
• Theuniversal coefficient theorem describes cohomology in terms of homology, usingExt groups. Namely, there is ashort exact sequence$${\displaystyle 0\to {\mathrm{Ext}}_{\mathbb{Z}}^1({\mathrm{H}}_{i-1}(X,\mathbb{Z}),A)\to H^i(X,A)\to {\mathrm{Hom}}_{\mathbb{Z}}(H_i(X,\mathbb{Z}),A)\to 0.}$$ A related statement is that for afield${\displaystyle F}$,${\displaystyle H^i(X,F)}$ is precisely thedual space of thevector space${\displaystyle H_i(X,F)}$.
• If${\displaystyle X}$ is a topologicalmanifold or aCW complex, then the cohomology groups${\displaystyle H^i(X,A)}$ are zero for${\displaystyle i}$ greater than thedimension of${\displaystyle X}$.^[2] If${\displaystyle X}$ is acompact manifold (possibly with boundary), or a CW complex with finitely many cells in each dimension, and${\displaystyle R}$ is a commutativeNoetherian ring, then the${\displaystyle R}$-module${\displaystyle H^i(X,R)}$ isfinitely generated for each${\displaystyle i}$.^[3]

On the other hand, cohomology has a crucial structure that homology does not: for any topological space${\displaystyle X}$ and commutative ring${\displaystyle R}$, there is abilinear map, called thecup product:$${\displaystyle H^i(X,R)\times H^j(X,R)\to H^{i+j}(X,R),}$$defined by an explicit formula on singular cochains. The product of cohomology classes${\displaystyle u}$ and${\displaystyle v}$ is written as${\displaystyle u\cup v}$ or simply as${\displaystyle uv}$. This product makes thedirect sum$${\displaystyle H^{\ast }(X,R)=\underset{i}{⨁}{H}^i(X,R)}$$into agraded ring, called thecohomology ring of${\displaystyle X}$. It isgraded-commutative in the sense that:^[4]$${\displaystyle uv=(-1{)}^{ij}vu,u\in H^i(X,R),v\in H^j(X,R).}$$

For any continuous map${\displaystyle f:X\to Y,}$ the pullback${\displaystyle f^{\ast }:H^{\ast }(Y,R)\to H^{\ast }(X,R)}$ is a homomorphism of graded${\displaystyle R}$-algebras. It follows that if two spaces arehomotopy equivalent, then their cohomology rings are isomorphic.

Here are some of the geometric interpretations of the cup product. In what follows,manifolds are understood to be without boundary, unless stated otherwise. Aclosed manifold means a compact manifold (without boundary), whereas a closedsubmanifoldN of a manifoldM means a submanifold that is aclosed subset ofM, not necessarily compact (althoughN is automatically compact ifM is).

• LetX be a closedoriented manifold of dimensionn. ThenPoincaré duality gives an isomorphismHⁱX ≅H_n−iX. As a result, a closed oriented submanifoldS ofcodimensioni inX determines a cohomology class inHⁱX, called [S]. In these terms, the cup product describes the intersection of submanifolds. Namely, ifS andT are submanifolds of codimensioni andj that intersecttransversely, then$${\displaystyle [S][T]=[S\cap T]\in H^{i+j}(X),}$$ where the intersectionS ∩T is a submanifold of codimensioni +j, with an orientation determined by the orientations ofS,T, andX. In the case ofsmooth manifolds, ifS andT do not intersect transversely, this formula can still be used to compute the cup product [S][T], by perturbingS orT to make the intersection transverse.
  More generally, without assuming thatX has an orientation, a closed submanifold ofX with an orientation on itsnormal bundle determines a cohomology class onX. IfX is a noncompact manifold, then a closed submanifold (not necessarily compact) determines a cohomology class onX. In both cases, the cup product can again be described in terms of intersections of submanifolds.
  Note thatThom constructed an integral cohomology class of degree 7 on a smooth 14-manifold that is not the class of any smooth submanifold.^[5] On the other hand, he showed that every integral cohomology class of positive degree on a smooth manifold has a positive multiple that is the class of a smooth submanifold.^[6] Also, every integral cohomology class on a manifold can be represented by a "pseudomanifold", that is, a simplicial complex that is a manifold outside a closed subset of codimension at least 2.
• For a smooth manifoldX,de Rham's theorem says that the singular cohomology ofX withreal coefficients is isomorphic to the de Rham cohomology ofX, defined usingdifferential forms. The cup product corresponds to the product of differential forms. This interpretation has the advantage that the product on differential forms is graded-commutative, whereas the product on singular cochains is only graded-commutative up tochain homotopy. In fact, it is impossible to modify the definition of singular cochains with coefficients in the integers${\displaystyle \mathbb{Z}}$ or in${\displaystyle \mathbb{Z}/p}$ for a prime numberp to make the product graded-commutative on the nose. The failure of graded-commutativity at the cochain level leads to theSteenrod operations on modp cohomology.

Very informally, for any topological spaceX, elements of${\displaystyle H^i(X)}$ can be thought of as represented by codimension-i subspaces ofX that can move freely onX. For example, one way to define an element of${\displaystyle H^i(X)}$ is to give a continuous mapf fromX to a manifoldM and a closed codimension-i submanifoldN ofM with an orientation on the normal bundle. Informally, one thinks of the resulting class${\displaystyle f^{\ast }([N])\in H^i(X)}$ as lying on the subspace${\displaystyle f^{-1}(N)}$ ofX; this is justified in that the class${\displaystyle f^{\ast }([N])}$ restricts to zero in the cohomology of the open subset${\displaystyle X-f^{-1}(N).}$ The cohomology class${\displaystyle f^{\ast }([N])}$ can move freely onX in the sense thatN could be replaced by any continuous deformation ofN insideM.

In what follows, cohomology is taken with coefficients in the integersZ, unless stated otherwise.

• The cohomology ring of a point is the ringZ in degree 0. By homotopy invariance, this is also the cohomology ring of anycontractible space, such as Euclidean spaceRⁿ.

• 

  For a positive integern, the cohomology ring of thesphere${\displaystyle S^n}$ isZ[x]/(x²) (thequotient ring of apolynomial ring by the givenideal), withx in degreen. In terms of Poincaré duality as above,x is the class of a point on the sphere.
• The cohomology ring of thetorus${\displaystyle (S^1{)}^n}$ is theexterior algebra overZ onn generators in degree 1.^[7] For example, letP denote a point in the circle${\displaystyle S^1}$, andQ the point (P,P) in the 2-dimensional torus${\displaystyle (S^1{)}^2}$. Then the cohomology of (S¹)² has a basis as afreeZ-module of the form: the element 1 in degree 0,x := [P ×S¹] andy := [S¹ ×P] in degree 1, andxy = [Q] in degree 2. (Implicitly, orientations of the torus and of the two circles have been fixed here.) Note thatyx = −xy = −[Q], by graded-commutativity.
• More generally, letR be a commutative ring, and letX andY be any topological spaces such thatH^*(X,R) is a finitely generated freeR-module in each degree. (No assumption is needed onY.) Then theKünneth formula gives that the cohomology ring of theproduct spaceX ×Y is atensor product ofR-algebras:^[8]$${\displaystyle H^{\ast }(X\times Y,R)\cong H^{\ast }(X,R){\otimes }_R{H}^{\ast }(Y,R).}$$
• The cohomology ring ofreal projective spaceRPⁿ withZ/2 coefficients isZ/2[x]/(xⁿ⁺¹), withx in degree 1.^[9] Herex is the class of ahyperplaneRPⁿ⁻¹ inRPⁿ; this makes sense even thoughRP^j is not orientable forj even and positive, because Poincaré duality withZ/2 coefficients works for arbitrary manifolds.
  With integer coefficients, the answer is a bit more complicated. TheZ-cohomology ofRP^2a has an elementy of degree 2 such that the whole cohomology is the direct sum of a copy ofZ spanned by the element 1 in degree 0 together with copies ofZ/2 spanned by the elementsyⁱ fori=1,...,a. TheZ-cohomology ofRP^2a+1 is the same together with an extra copy ofZ in degree 2a+1.^[10]
• The cohomology ring ofcomplex projective spaceCPⁿ isZ[x]/(xⁿ⁺¹), withx in degree 2.^[9] Herex is the class of a hyperplaneCPⁿ⁻¹ inCPⁿ. More generally,x^j is the class of a linear subspaceCP^n−j inCPⁿ.
• The cohomology ring of the closed oriented surfaceX ofgenusg ≥ 0 has a basis as a freeZ-module of the form: the element 1 in degree 0,A₁,...,A_g andB₁,...,B_g in degree 1, and the classP of a point in degree 2. The product is given by:A_iA_j =B_iB_j = 0 for alli andj,A_iB_j = 0 ifi ≠j, andA_iB_i =P for alli.^[11] By graded-commutativity, it follows thatB_iA_i = −P.
• On any topological space, graded-commutativity of the cohomology ring implies that 2x² = 0 for all odd-degree cohomology classesx. It follows that for a ringR containing 1/2, all odd-degree elements ofH^*(X,R) have square zero. On the other hand, odd-degree elements need not have square zero ifR isZ/2 orZ, as one sees in the example ofRP² (withZ/2 coefficients) orRP⁴ ×RP² (withZ coefficients).

The cup product on cohomology can be viewed as coming from thediagonal map${\displaystyle \mathrm{\Delta }:X\to X\times X}$,${\displaystyle x\mapsto (x,x)}$. Namely, for any spaces${\displaystyle X}$ and${\displaystyle Y}$ with cohomology classes${\displaystyle u\in H^i(X,R)}$ and${\displaystyle v\in H^j(Y,R)}$, there is anexternal product (orcross product) cohomology class${\displaystyle u\times v\in H^{i+j}(X\times Y,R)}$. The cup product of classes${\displaystyle u\in H^i(X,R)}$ and${\displaystyle v\in H^j(X,R)}$ can be defined as the pullback of the external product by the diagonal:^[12]$${\displaystyle uv={\mathrm{\Delta }}^{\ast }(u\times v)\in H^{i+j}(X,R).}$$

Alternatively, the external product can be defined in terms of the cup product. For spaces${\displaystyle X}$ and${\displaystyle Y}$, write${\displaystyle f:X\times Y\to X}$ and${\displaystyle g:X\times Y\to Y}$ for the two projections. Then the external product of classes${\displaystyle u\in H^i(X,R)}$ and${\displaystyle v\in H^j(Y,R)}$ is:$${\displaystyle u\times v=(f^{\ast }(u))(g^{\ast }(v))\in H^{i+j}(X\times Y,R).}$$

Another interpretation of Poincaré duality is that the cohomology ring of a closed oriented manifold is self-dual in a strong sense. Namely, let${\displaystyle X}$ be a closedconnected oriented manifold of dimension${\displaystyle n}$, and let${\displaystyle F}$ be a field. Then${\displaystyle H^n(X,F)}$ is isomorphic to${\displaystyle F}$, and the product

${\displaystyle H^i(X,F)\times H^{n-i}(X,F)\to H^n(X,F)\cong F}$

is aperfect pairing for each integer${\displaystyle i}$.^[13] In particular, the vector spaces${\displaystyle H^i(X,F)}$ and${\displaystyle H^{n-i}(X,F)}$ have the same (finite) dimension. Likewise, the product on integral cohomology modulotorsion with values in${\displaystyle H^n(X,\mathbb{Z})\cong \mathbb{Z}}$ is a perfect pairing over${\displaystyle \mathbb{Z}}$.

An oriented realvector bundleE of rankr over a topological spaceX determines a cohomology class onX, theEuler class χ(E) ∈H^r(X,Z). Informally, the Euler class is the class of the zero set of a generalsection ofE. That interpretation can be made more explicit whenE is a smooth vector bundle over a smooth manifoldX, since then a general smooth section ofX vanishes on a codimension-r submanifold ofX.

There are several other types ofcharacteristic classes for vector bundles that take values in cohomology, includingChern classes,Stiefel–Whitney classes, andPontryagin classes.

For each abelian groupA and natural numberj, there is a space${\displaystyle K(A,j)}$ whosej-th homotopy group is isomorphic toA and whose other homotopy groups are zero. Such a space is called anEilenberg–MacLane space. This space has the remarkable property that it is aclassifying space for cohomology: there is a natural elementu of${\displaystyle H^j(K(A,j),A)}$, and every cohomology class of degreej on every spaceX is the pullback ofu by some continuous map${\displaystyle X\to K(A,j)}$. More precisely, pulling back the classu gives a bijection

${\displaystyle [X,K(A,j)]\stackrel{\cong }{\to }{H}^j(X,A)}$

for every spaceX with the homotopy type of a CW complex.^[14] Here${\displaystyle [X,Y]}$ denotes the set of homotopy classes of continuous maps fromX toY.

For example, the space${\displaystyle K(\mathbb{Z},1)}$ (defined up to homotopy equivalence) can be taken to be the circle${\displaystyle S^1}$. So the description above says that every element of${\displaystyle H^1(X,\mathbb{Z})}$ is pulled back from the classu of a point on${\displaystyle S^1}$ by some map${\displaystyle X\to S^1}$.

There is a related description of the first cohomology with coefficients in any abelian groupA, say for a CW complexX. Namely,${\displaystyle H^1(X,A)}$ is in one-to-one correspondence with the set of isomorphism classes of Galoiscovering spaces ofX with groupA, also calledprincipalA-bundles overX. ForX connected, it follows that${\displaystyle H^1(X,A)}$ is isomorphic to${\displaystyle \mathrm{Hom}({\pi }_1(X),A)}$, where${\displaystyle {\pi }_1(X)}$ is thefundamental group ofX. For example,${\displaystyle H^1(X,\mathbb{Z}/2)}$ classifies the double covering spaces ofX, with the element${\displaystyle 0\in H^1(X,\mathbb{Z}/2)}$ corresponding to the trivial double covering, the disjoint union of two copies ofX.

For any topological spaceX, thecap product is a bilinear map

${\displaystyle \cap :H^i(X,R)\times H_j(X,R)\to H_{j-i}(X,R)}$

for any integersi andj and any commutative ringR. The resulting map

${\displaystyle H^{\ast }(X,R)\times H_{\ast }(X,R)\to H_{\ast }(X,R)}$

makes the singular homology ofX into a module over the singular cohomology ring ofX.

Fori =j, the cap product gives the natural homomorphism

${\displaystyle H^i(X,R)\to {\mathrm{Hom}}_R(H_i(X,R),R),}$

which is an isomorphism forR a field.

For example, letX be an oriented manifold, not necessarily compact. Then a closed oriented codimension-i submanifoldY ofX (not necessarily compact) determines an element ofHⁱ(X,R), and a compact orientedj-dimensional submanifoldZ ofX determines an element ofH_j(X,R). The cap product [Y] ∩ [Z] ∈H_j−i(X,R) can be computed by perturbingY andZ to make them intersect transversely and then taking the class of their intersection, which is a compact oriented submanifold of dimensionj −i.

A closed oriented manifoldX of dimensionn has afundamental class [X] inH_n(X,R). The Poincaré duality isomorphism$${\displaystyle H^i(X,R)\stackrel{\cong }{\to }{H}_{n-i}(X,R)}$$is defined by cap product with the fundamental class ofX.

Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology. The concept ofdual cell structure, whichHenri Poincaré used in his proof of his Poincaré duality theorem, contained the beginning of the idea of cohomology, but this was not seen until later.

There were various precursors to cohomology.^[15] In the mid-1920s,J. W. Alexander andSolomon Lefschetz foundedintersection theory of cycles on manifolds. On a closed orientedn-dimensional manifoldM ani-cycle and aj-cycle with nonempty intersection will, if in thegeneral position, have as their intersection a (i + j − n)-cycle. This leads to a multiplication of homology classes

${\displaystyle H_i(M)\times H_j(M)\to H_{i+j-n}(M),}$

which (in retrospect) can be identified with thecup product on the cohomology ofM.

Alexander had by 1930 defined a first notion of a cochain, by thinking of ani-cochain on a spaceX as a function on small neighborhoods of the diagonal inXⁱ⁺¹.

In 1931,Georges de Rham related homology and differential forms, provingde Rham's theorem. This result can be stated more simply in terms of cohomology.

In 1934,Lev Pontryagin proved thePontryagin duality theorem; a result ontopological groups. This (in rather special cases) provided an interpretation of Poincaré duality andAlexander duality in terms of groupcharacters.

At a 1935 conference inMoscow,Andrey Kolmogorov and Alexander both introduced cohomology and tried to construct a cohomology product structure.

In 1936,Norman Steenrod constructedČech cohomology by dualizing Čech homology.

From 1936 to 1938,Hassler Whitney andEduard Čech developed thecup product (making cohomology into a graded ring) andcap product, and realized that Poincaré duality can be stated in terms of the cap product. Their theory was still limited to finite cell complexes.

In 1944,Samuel Eilenberg overcame the technical limitations, and gave the modern definition of singular homology and cohomology.

In 1945, Eilenberg and Steenrod stated theaxioms defining a homology or cohomology theory, discussed below. In their 1952 book,Foundations of Algebraic Topology, they proved that the existing homology and cohomology theories did indeed satisfy their axioms.

In 1946,Jean Leray defined sheaf cohomology.

In 1948Edwin Spanier, building on work of Alexander and Kolmogorov, developedAlexander–Spanier cohomology.

Sheaf cohomology is a rich generalization of singular cohomology, allowing more general "coefficients" than simply an abelian group. For everysheaf of abelian groupsE on a topological spaceX, one has cohomology groupsHⁱ(X,E) for integersi. In particular, in the case of theconstant sheaf onX associated with an abelian groupA, the resulting groupsHⁱ(X,A) coincide with singular cohomology forX a manifold or CW complex (though not for arbitrary spacesX). Starting in the 1950s, sheaf cohomology has become a central part ofalgebraic geometry andcomplex analysis, partly because of the importance of the sheaf ofregular functions or the sheaf ofholomorphic functions.

Grothendieck elegantly defined and characterized sheaf cohomology in the language ofhomological algebra. The essential point is to fix the spaceX and think of sheaf cohomology as a functor from theabelian category of sheaves onX to abelian groups. Start with the functor taking a sheafE onX to its abelian group of global sections overX,E(X). This functor isleft exact, but not necessarily right exact. Grothendieck defined sheaf cohomology groups to be the rightderived functors of the left exact functorE ↦E(X).^[16]

That definition suggests various generalizations. For example, one can define the cohomology of a topological spaceX with coefficients in any complex of sheaves, earlier calledhypercohomology (but usually now just "cohomology"). From that point of view, sheaf cohomology becomes a sequence of functors from thederived category of sheaves onX to abelian groups.

In a broad sense of the word, "cohomology" is often used for the right derived functors of a left exact functor on an abelian category, while "homology" is used for the left derived functors of a right exact functor. For example, for a ringR, theTor groups Tor_i^R(M,N) form a "homology theory" in each variable, the left derived functors of the tensor productM⊗_RN ofR-modules. Likewise, theExt groups Extⁱ_R(M,N) can be viewed as a "cohomology theory" in each variable, the right derived functors of the Hom functor Hom_R(M,N).

Sheaf cohomology can be identified with a type of Ext group. Namely, for a sheafE on a topological spaceX,Hⁱ(X,E) is isomorphic to Extⁱ(Z_X,E), whereZ_X denotes the constant sheaf associated with the integersZ, and Ext is taken in the abelian category of sheaves onX.

There are numerous machines built for computing the cohomology ofalgebraic varieties. The simplest case being the determination of cohomology forsmoothprojective varieties over a field ofcharacteristic${\displaystyle 0}$. Tools fromHodge theory, calledHodge structures, help give computations of cohomology of these types of varieties (with the addition of more refined information). In the simplest case the cohomology of a smoothhypersurface in${\displaystyle {\mathbb{P}}^n}$ can be determined from the degree of the polynomial alone.

When considering varieties over afinite field, or a field of characteristic${\displaystyle p}$, more powerful tools are required because the classical definitions of homology/cohomology break down. This is because varieties over finite fields will only be a finite set of points. Grothendieck came up with the idea for aGrothendieck topology and used sheaf cohomology over theétale topology to define the cohomology theory for varieties over a finite field. Using the étale topology for a variety over a field of characteristic${\displaystyle p}$ one can construct${\displaystyle \ell }$-adic cohomology for${\displaystyle \ell \ne p}$. This is defined as theprojective limit

${\displaystyle H^k(X;{\mathbb{Q}}_{\ell }):=\underset{n\in \mathbb{N}}{\underleftarrow{\lim }}{H}_{et}^k(X;\mathbb{Z}/({\ell }^n)){\otimes }_{{\mathbb{Z}}_{\ell }}{\mathbb{Q}}_{\ell }.}$

If we have a scheme of finite type

${\displaystyle X=\mathrm{Proj}\left(\frac{\mathbb{Z}\left[x_0,\dots ,x_n\right]}{\left(f_1,\dots ,f_k\right)}\right)}$

then there is an equality of dimensions for the Betti cohomology of${\displaystyle X(\mathbb{C})}$ and the${\displaystyle \ell }$-adic cohomology of${\displaystyle X({\mathbb{F}}_q)}$ whenever the variety is smooth over both fields. In addition to these cohomology theories there are other cohomology theories calledWeil cohomology theories which behave similarly to singular cohomology. There is a conjectured theory of motives which underlie all of the Weil cohomology theories.

Another useful computational tool is the blowup sequence. Given a codimension${\displaystyle \ge 2}$ subscheme${\displaystyle Z\subset X}$ there is aCartesian square

From this there is an associated long exact sequence

${\displaystyle \cdots \to H^n(X)\to H^n(Z)\oplus H^n(B{l}_Z(X))\to H^n(E)\to H^{n+1}(X)\to \cdots }$

If the subvariety${\displaystyle Z}$ is smooth, then the connecting morphisms are all trivial, hence

${\displaystyle H^n(B{l}_Z(X))\oplus H^n(Z)\cong H^n(X)\oplus H^n(E)}$

There are various ways to define cohomology for topological spaces (such as singular cohomology,Čech cohomology,Alexander–Spanier cohomology orsheaf cohomology). (Here sheaf cohomology is considered only with coefficients in a constant sheaf.) These theories give different answers for some spaces, but there is a large class of spaces on which they all agree. This is most easily understood axiomatically: there is a list of properties known as theEilenberg–Steenrod axioms, and any two constructions that share those properties will agree at least on all CW complexes.^[17] There are versions of the axioms for a homology theory as well as for a cohomology theory. Some theories can be viewed as tools for computing singular cohomology for special topological spaces, such assimplicial cohomology forsimplicial complexes,cellular cohomology for CW complexes, andde Rham cohomology for smooth manifolds.

One of the Eilenberg–Steenrod axioms for a cohomology theory is thedimension axiom: ifP is a single point, thenHⁱ(P) = 0 for alli ≠ 0. Around 1960,George W. Whitehead observed that it is fruitful to omit the dimension axiom completely: this gives the notion of a generalized homology theory or a generalized cohomology theory, defined below. There are generalized cohomology theories such as K-theory or complex cobordism that give rich information about a topological space, not directly accessible from singular cohomology. (In this context, singular cohomology is often called "ordinary cohomology".)

By definition, ageneralized homology theory is a sequence offunctorsh_i (for integersi) from thecategory of CW-pairs (X, A) (soX is a CW complex andA is a subcomplex) to the category of abelian groups, together with anatural transformation∂_i:h_i(X,A) →h_i−1(A) called theboundary homomorphism (hereh_i−1(A) is a shorthand forh_i−1(A,∅)). The axioms are:

1. Homotopy: If${\displaystyle f:(X,A)\to (Y,B)}$ is homotopic to${\displaystyle g:(X,A)\to (Y,B)}$, then the induced homomorphisms on homology are the same.
2. Exactness: Each pair (X,A) induces a long exact sequence in homology, via the inclusionsf:A →X andg: (X,∅) → (X,A):$${\displaystyle \cdots \to h_i(A)\stackrel{f_{\ast }}{\to }{h}_i(X)\stackrel{g_{\ast }}{\to }{h}_i(X,A)\stackrel{\mathrm{\partial }}{\to }{h}_{i-1}(A)\to \cdots .}$$
3. Excision: IfX is the union of subcomplexesA andB, then the inclusionf: (A,A∩B) → (X,B) induces an isomorphism$${\displaystyle h_i(A,A\cap B)\stackrel{f_{\ast }}{\to }{h}_i(X,B)}$$ for everyi.
4. Additivity: If (X,A) is the disjoint union of a set of pairs (X_α,A_α), then the inclusions (X_α,A_α) → (X,A) induce an isomorphism from thedirect sum:$${\displaystyle \underset{\alpha }{⨁}{h}_i(X_{\alpha },A_{\alpha })\to h_i(X,A)}$$ for everyi.

The axioms for a generalized cohomology theory are obtained by reversing the arrows, roughly speaking. In more detail, ageneralized cohomology theory is a sequence of contravariant functorshⁱ (for integersi) from the category of CW-pairs to the category of abelian groups, together with a natural transformationd:hⁱ(A) →hⁱ⁺¹(X,A) called theboundary homomorphism (writinghⁱ(A) forhⁱ(A,∅)). The axioms are:

1. Homotopy: Homotopic maps induce the same homomorphism on cohomology.
2. Exactness: Each pair (X,A) induces a long exact sequence in cohomology, via the inclusionsf:A →X andg: (X,∅) → (X,A):$${\displaystyle \cdots \to h^i(X,A)\stackrel{g_{\ast }}{\to }{h}^i(X)\stackrel{f_{\ast }}{\to }{h}^i(A)\stackrel{d}{\to }{h}^{i+1}(X,A)\to \cdots .}$$
3. Excision: IfX is the union of subcomplexesA andB, then the inclusionf: (A,A∩B) → (X,B) induces an isomorphism$${\displaystyle h^i(X,B)\stackrel{f_{\ast }}{\to }{h}^i(A,A\cap B)}$$ for everyi.
4. Additivity: If (X,A) is the disjoint union of a set of pairs (X_α,A_α), then the inclusions (X_α,A_α) → (X,A) induce an isomorphism to theproduct group:$${\displaystyle h^i(X,A)\to \prod _{\alpha }{h}^i(X_{\alpha },A_{\alpha })}$$ for everyi.

Aspectrum determines both a generalized homology theory and a generalized cohomology theory. A fundamental result by Brown, Whitehead, andAdams says that every generalized homology theory comes from a spectrum, and likewise every generalized cohomology theory comes from a spectrum.^[18] This generalizes the representability of ordinary cohomology by Eilenberg–MacLane spaces.

A subtle point is that the functor from the stable homotopy category (the homotopy category of spectra) to generalized homology theories on CW-pairs is not an equivalence, although it gives a bijection on isomorphism classes; there are nonzero maps in the stable homotopy category (calledphantom maps) that induce the zero map between homology theories on CW-pairs. Likewise, the functor from the stable homotopy category to generalized cohomology theories on CW-pairs is not an equivalence.^[19] It is the stable homotopy category, not these other categories, that has good properties such as beingtriangulated.

If one prefers homology or cohomology theories to be defined on all topological spaces rather than on CW complexes, one standard approach is to include the axiom that everyweak homotopy equivalence induces an isomorphism on homology or cohomology. (That is true for singular homology or singular cohomology, but not for sheaf cohomology, for example.) Since every space admits a weak homotopy equivalence from a CW complex, this axiom reduces homology or cohomology theories on all spaces to the corresponding theory on CW complexes.^[20]

Some examples of generalized cohomology theories are:

• Stablecohomotopy groups${\displaystyle {\pi }_S^{\ast }(X).}$ The corresponding homology theory is used more often:stable homotopy groups${\displaystyle {\pi }_{\ast }^S(X).}$
• Various different flavors ofcobordism groups, based on studying a space by considering all maps from it to manifolds: unoriented cobordism${\displaystyle M{O}^{\ast }(X)}$ oriented cobordism${\displaystyle MS{O}^{\ast }(X),}$complex cobordism${\displaystyle M{U}^{\ast }(X),}$ and so on. Complex cobordism has turned out to be especially powerful in homotopy theory. It is closely related toformal groups, via a theorem ofDaniel Quillen.
• Various different flavors of topologicalK-theory, based on studying a space by considering all vector bundles over it:${\displaystyle K{O}^{\ast }(X)}$ (real periodic K-theory),${\displaystyle k{o}^{\ast }(X)}$ (real connective K-theory),${\displaystyle K^{\ast }(X)}$ (complex periodic K-theory),${\displaystyle k{u}^{\ast }(X)}$ (complex connective K-theory), and so on.
• Brown–Peterson cohomology,Morava K-theory, Morava E-theory, and other theories built from complex cobordism.
• Various flavors ofelliptic cohomology.

Many of these theories carry richer information than ordinary cohomology, but are harder to compute.

A cohomology theoryE is said to bemultiplicative if${\displaystyle E^{\ast }(X)}$ has the structure of a graded ring for each spaceX. In the language of spectra, there are several more precise notions of aring spectrum, such as anE_∞ ring spectrum, where the product is commutative and associative in a strong sense.

Cohomology theories in a broader sense (invariants of other algebraic or geometric structures, rather than of topological spaces) include:

• complex-oriented cohomology theory

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• Cohomology theories

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