duality (mathematics)


In mathematics, duality has numerous meanings, and although it is “a very pervasive and important concept in (modern) mathematics”(1) and “an important general theme that has manifestations in almost every area of mathematics”,(2) there is no single universally agreed definition that unifies all concepts of duality.(3)

Order-reversing dualities

A particularly simple form of duality comes from order theory. The dual of a poset P = (X, ≤) is the poset P^d = (X, ≥) comprising the same ground set but the converse relation. Familiar examples of dual partial orders include

• the subset and superset relations
  &su(;
  and
  su(set
  on any collection of sets,

• the divides and multiple-of relations on the integers, and

• the descendant-of and ancestor-of relations on the set of humans.

A concept defined for a partial order P will correspond to a dual concept on the dual poset P^d. For instance, a minimal element of P will be a maximal element of P^d: minimality and maximality are dual concepts in order theory. Other pairs of dual concepts are upper and lower bounds, lower sets and upper sets, and ideals and filters.A particular order reversal of this type occurs in the family of all subsets of some set S: if denotes the complement set, then if and only if . In topology, open sets and closed sets are dual concepts: the complement of an open set is closed, and vice versa. In matroid theory, the family of sets complementary to the independent sets of a given matroid themselves form another matroid, called the dual matroid. In logic, one may represent a truth assignment to the variables of an unquantified formula as a set, the variables that are true for the assignment. A truth assignment satisfies the formula if and only if the complementary truth assignment satisfies the De Morgan dual of its formula. The existential and universal quantifiers in logic are similarly dual.A partial order may be interpreted as a category in which there is an arrow from x to y in the category if and only if x ≤ y in the partial order. The order-reversing duality of partial orders can be extended to the concept of a dual category, the category formed by reversing all the arrows in a given category. Many of the specific dualities described later are dualities of categories in this sense.According to Artstein-Avidan and Milman,(4)(5) a duality transform is just an involutive antiautomorphism of a partially ordered set S, that is, an order reversing involution Surprisingly, in several important cases these simple properties determine the transform uniquely up to some simple symmetries. If are two duality transforms then their composition is an order automorphism of S; thus, any two duality transforms differ only by an order automorphism. For example, all order automorphisms of a power set S=2^R are induced by permutations of R. The papers cited above treat only sets S of functions on Rⁿ satisfying some condition of convexity and prove that all order automorphisms are induced by linear or affine transformations of Rⁿ.

Dimension-reversing dualities

(File:Dual Cube-Octahedron.svg|thumb|200px|The features of the cube and its dual octahedron correspond one-for-one with dimensions reversed.)There are many distinct but interrelated dualities in which geometric or topological objects correspond to other objects of the same type, but with a reversal of the dimensions of the features of the objects. A classical example of this is the duality of the platonic solids, in which the cube and the octahedron form a dual pair, the dodecahedron and the icosahedron form a dual pair, and the tetrahedron is self-dual. The dual polyhedron of any of these polyhedra may be formed as the convex hull of the center points of each face of the primal polyhedron, so the vertices of the dual correspond one-for-one with the faces of the primal. Similarly, each edge of the dual corresponds to an edge of the primal, and each face of the dual corresponds to a vertex of the primal. These correspondences are incidence-preserving: if two parts of the primal polyhedron touch each other, so do the corresponding two parts of the dual polyhedron. More generally, using the concept of polar reciprocation, any convex polyhedron, or more generally any convex polytope, corresponds to a dual polyhedron or dual polytope, with an i-dimensional feature of an n-dimensional polytope corresponding to an (n − i − 1)-dimensional feature of the dual polytope. The incidence-preserving nature of the duality is reflected in the fact that the face lattices of the primal and dual polyhedra or polytopes are themselves order-theoretic duals. Duality of polytopes and order-theoretic duality are both involutions: the dual polytope of the dual polytope of any polytope is the original polytope, and reversing all order-relations twice returns to the original order. Choosing a different center of polarity leads to geometrically different dual polytopes, but all have the same combinatorial structure. From any three-dimensional polyhedron, one can form a planar graph, the graph of its vertices and edges. The dual polyhedron has a dual graph, a graph with one vertex for each face of the polyhedron and with one edge for every two adjacent faces. The same concept of planar graph duality may be generalized to graphs that are drawn in the plane but that do not come from a three-dimensional polyhedron, or more generally to graph embeddings on surfaces of higher genus: one may draw a dual graph by placing one vertex within each region bounded by a cycle of edges in the embedding, and drawing an edge connecting any two regions that share a boundary edge. An important example of this type comes from computational geometry: the duality for any finite set S of points in the plane between the Delaunay triangulation of S and the Voronoi diagram of S. As with dual polyhedra and dual polytopes, the duality of graphs on surfaces is a dimension-reversing involution: each vertex in the primal embedded graph corresponds to a region of the dual embedding, each edge in the primal is crossed by an edge in the dual, and each region of the primal corresponds to a vertex of the dual. The dual graph depends on how the primal graph is embedded: different planar embeddings of a single graph may lead to different dual graphs. Matroid duality is an algebraic extension of planar graph duality, in the sense that the dual matroid of the graphic matroid of a planar graph is isomorphic to the graphic matroid of the dual graph.In topology, Poincaré duality also reverses dimensions; it corresponds to the fact that, if a topological manifold is respresented as a cell complex, then the dual of the complex (a higher dimensional generalization of the planar graph dual) represents the same manifold. In Poincaré duality, this homeomorphism is reflected in an isomorphism of the kth homology group and the (n − k)th cohomology group.File:Complete-quads.svg|thumb|240px|The complete quadranglecomplete quadrangleAnother example of a dimension-reversing duality arises in projective geometry.(6) In the projective plane, it is possible to find geometric transformations that map each point of the projective plane to a line, and each line of the projective plane to a point, in an incidence-preserving way: in terms of the incidence matrix of the points and lines in the plane, this operation is just that of forming the transpose. Transformations of this type exist also in any higher dimension; one way to construct them is to use the same polar transformations that generate polyhedron and polytope duality. Due to this ability to replace any configuration of points and lines with a corresponding configuration of lines and points, there arises a general principle of duality in projective geometry: given any theorem in plane projective geometry, exchanging the terms "point" and "line" everywhere results in a new, equally valid theorem.(7)The points, lines, and higher dimensional subspaces n-dimensional projective space may be interpreted as describing the linear subspaces of an (n + 1)-dimensional vector space; if this vector space is supplied with an inner product the transformation from any linear subspace to its perpendicular subspace is an example of a projective duality. The Hodge dual extends this duality within an inner product space by providing a canonical correspondence between the elements of the exterior algebra.A kind of geometric duality also occurs in optimization theory, but not one that reverses dimensions. A linear program may be specified by a system of real variables (the coordinates for a point in Euclidean space Rⁿ), a system of linear constraints (specifying that the point lie in a halfspace; the intersection of these halfspaces is a convex polytope, the feasible region of the program), and a linear function (what to optimize). Every linear program has a dual problem with the same optimal solution, but the variables in the dual problem correspond to constraints in the primal problem and vice versa.{{-}}

Duality in logic and set theory

In logic, functions or relations A and B are considered dual if A(¬x) = ¬B(x), where ¬ is logical negation. The basic duality of this type is the duality of the ∃ and ∀ quantifiers. These are dual because ∃x.¬P(x) and ¬∀x.P(x) are equivalent for all predicates P: if there exists an x for which P fails to hold, then it is false that P holds for all x. From this fundamental logical duality follow several others:

• A formula is said to be satisfiable in a certain model if there are assignments to its free variables that render it true; it is valid if every assignment to its free variables makes it true. Satisfiability and validity are dual because the invalid formulas are precisely those whose negations are satisfiable, and the unsatisfiable formulas are those whose negations are valid. This can be viewed as a special case of the previous item, with the quantifiers ranging over interpretations.

• In classical logic, the ∧ and ∨ operators are dual in this sense, because (¬x ∧ ¬y) and ¬(x ∨ y) are equivalent. This means that for every theorem of classical logic there is an equivalent dual theorem. De Morgan's laws are examples. More generally,
  big∧ (≠g xi) = ≠gbig∨ xi
  . The left side is true if and only if ∀i.¬x_i, and the right side if and only if ¬∃i.x_i.

• In modal logic,
  Box (
  means that the proposition p is "necessarily" true, and
  Diamond (
  that p is "possibly" true. Most interpretations of modal logic assign dual meanings to these two operators. For example in Kripke semantics, "p is possibly true" means "there exists some world W in which p is true", while "p is necessarily true" means "for all worlds W, p is true". The duality of
  Box
  and
  Diamond
  then follows from the analogous duality of ∀ and ∃. Other dual modal operators behave similarly. For example, temporal logic has operators denoting "will be true at some time in the future" and "will be true at all times in the future" which are similarly dual.

Other analogous dualities follow from these:

• Set-theoretic union and intersection are dual under the set complement operator ^C. That is,
  (AC&ca(; BC) = (A &cu(; B)C
  , and more generally,
  big&ca(; A&al(ha;C = ((big&cu(; A&al(ha;&nbs(;))C
  . This follows from the duality of ∀ and ∃: an element x is a member of
  big&ca(; A&al(ha;C
  if and only if ∀α.¬x∈A_α, and is a member of
  ((big&cu(; A&al(ha;&nbs(;))C
  if and only if ¬∃α.x∈A_α.

Topology inherits a duality between open and closed subsets of some fixed topological space X: a subset U of X is closed if and only if its complement in X is open. Because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of open sets is open, so dually, any intersection of closed sets is closed. The interior of a set is the largest open set contained in it, and the closure of the set is the smallest closed set that contains it. Because of the duality, the complement of the interior of any set U is equal to the closure of the complement of U.The collection of all open subsets of a topological space X forms a complete Heyting algebra. There is a duality, known as Stone duality, connecting sober spaces and spatial locales.

• Birkhoff's representation theorem relating distributive lattices and partial orders

Dual objects

A group of dualities can be described by endowing, for any mathematical object X, the set of morphisms Hom(X, D) into some fixed object D, with a structure similar to the one of X. This is sometimes called internal Hom. In general, this yields a true duality only for specific choices of D, in which case X^∗=Hom(X, D) is referred to as the dual of X. It may or may not be true that the bidual, that is to say, the dual of the dual, X^∗∗ = (X^∗)^∗ is isomorphic to X, as the following example, which is underlying many other dualities, shows: the dual vector space V^∗ of a K-vector space V is defined as The set of morphisms, i.e., linear maps, is a vector space in its own right. There is always a natural, injective map V → V^∗∗ given by v ↦ (f ↦ f(v)), where f is an element of the dual space. That map is an isomorphism if and only if the dimension of V is finite.In the realm of topological vector spaces, a similar construction exists, replacing the dual by the topological dual vector space. Topological vector spaces that are isomorphic to its bidual are called reflexive spaces.The dual lattice of a lattice L is given by which is used in the construction of toric varieties.(8) The Pontryagin dual of locally compact topological groups G is given by continuous group homomorphisms with values in the circle (with multiplication of complex numbers as group operation).

Dual categories

Opposite category and adjoint functors

In another group of dualities, the objects of one theory are translated into objects of another theory and the maps between objects in the first theory are translated into morphisms in the second theory, but with direction reversed. Using the parlance of category theory, this amounts to a contravariant functor between two categories C and D: which for any two objects X and Y of C gives a map That functor may or may not be an equivalence of categories. There are various situations, where such a functor is an equivalence between the opposite category C^op of C, and D. Using a duality of this type, every statement in the first theory can be translated into a "dual" statement in the second theory, where the direction of all arrows has to be reversed.(9) Therefore, any duality between categories C and D is formally the same as an equivalence between C and D^op (C^op and D). However, in many circumstances the opposite categories have no inherent meaning, which makes duality an additional, separate concept.(10)Many category-theoretic notions come in pairs in the sense that they correspond to each other while considering the opposite category. For example, Cartesian products Y₁ × Y₂ and disjoint unions Y₁ ⊔ Y₂ of sets are dual to each other in the sense that and for any set X. This is a particular case of a more general duality phenomenon, under which limits in a category C correspond to colimits in the opposite category C^op; further concrete examples of this are epimorphisms vs. monomorphism, in particular factor modules (or groups etc.) vs. submodules, direct products vs. direct sums (also called coproducts to emphasize the duality aspect). Therefore, in some cases, proofs of certain statements can be halved, using such a duality phenomenon. Further notions displaying related by such a categorical duality are projective and injective modules in homological algebra,(11) fibrations and cofibrations in topology and more generally model categories.(12)Two functors F: C → D and G: D → C are adjoint if for all objects c in C and d in D in a natural way. Actually, the correspondence of limits and colimits is an example of adjoints, since there is an adjunction between the colimit functor that assigns to any diagram in C indexed by some category I its colimit and the diagonal functor that maps any object c of C to the constant diagramm which has c at all places. Dually,

Examples

For example, there is a duality between commutative rings and affine schemes: to every commutative ring A there is an affine spectrum, Spec A, conversely, given an affine scheme S, one gets back a ring by taking global sections of the structure sheaf O_S. In addition, ring homomorphisms are in one-to-one correspondence with morphisms of affine schemes, thereby there is an equivalence Compare with noncommutative geometry and Gelfand duality.In a number of situations, the objects of two categories linked by a duality are partially ordered, i.e., there is some notion of an object "being smaller" than another one. In such a situation, a duality that respects the orderings in question is known as a Galois connection. An example is the standard duality in Galois theory (fundamental theorem of Galois theory) between field extensions and subgroups of the Galois group: a bigger field extension corresponds—under the mapping that assigns to any extension L ⊃ K (inside some fixed bigger field Ω) the Galois group Gal(Ω / L)—to a smaller group.(14) Pontryagin duality gives a duality on the category of locally compact abelian groups: given any such group G, the character group given by continuous group homomorphisms from G to the circle group S¹ can be endowed with the compact-open topology. Pontryagin duality states that the character group is again locally compact abelian and that Moreover, discrete groups correspond to compact abelian groups; finite groups correspond to finite groups. Pontryagin is the background to Fourier analysis, see below.

• Tannaka-Krein duality, a non-commutative analogue of Pontryagin duality(16)
• Gelfand duality relating commutative C-algebras and compact Hausdorff spaces

Both Gelfand and Pontryagin duality can be deduced in a largely formal. category-theoretic way.(17)

Analytic dualities

In analysis, problems are frequently solved by passing to the dual description of functions and operators.Fourier transform switches between functions on a vector space and its dual: and conversely If f is an L²-function on R or R^N, say, then so is and . Moreover, the transform interchanges operations of multiplication and convolution on the corresponding function spaces. A conceptual explanation of the Fourier transform is obtained by the aforementioned Pontryagin duality, applied to the locally compact groups R (or R^N etc.): any character of R is given by . The dualizing character of Fourier transform has many other manifestations, for example, in alternative descriptions of quantum mechanical systems in terms of coordinate and momentum representations.

• Laplace transform is similar to Fourier transform and interchanges operators of multiplication by polynomials with constant coefficient linear differential operators.
• Legendre transformation is an important analytic duality which switches between velocities in Lagrangian mechanics and momenta in Hamiltonian mechanics.

Poincaré-style dualities

Theorems showing that certain objects of interest are the dual spaces (in the sense of linear algebra) of other objects of interest are often called dualities. Many of these dualities are given by a bilinear pairing of two K-vector spaces For perfect pairings, there is, therefore, an isomorphism of A to the dual of B.For example, Poincaré duality of a smooth compact complex manifold X is given by a pairing of singular cohomology with C-coefficients (equivalently, sheaf cohomology of the constant sheaf C) where n is the (complex) dimension of X.(18) Poincaré duality can also be expressed as a relation of singular homology and de Rham cohomology, by asserting that the map (integrating a differential k-form over an 2n−k-(real)-dimensional cycle) is a perfect pairing. The same duality pattern holds for a smooth projective variety over a separably closed field, using l-adic cohomology with Q_ℓ-coefficients instead.(19) This is further generalized to possibly singular varieties, using intersection cohomology instead, a duality called Verdier duality.(20) With increasing level of generality, it turns out, an increasing amount of technical background is helpful or necessary to understand these theorems: the modern formulation of both these dualities can be done using derived categories and certain direct and inverse image functors of sheaves, applied to locally constant sheaves (with respect to the classical analytical topology in the first case, and with respect to the étale topology in the second case).Yet another group of similar duality statements is encountered in arithmetics: étale cohomology of finite, local and global fields (also known as Galois cohomology, since étale cohomology over a field is equivalent to group cohomology of the (absolute) Galois group of the field) admit similar pairings. The absolute Galois group G(F_q) of a finite field, for example, is isomorphic to , the profinite completion of Z, the integers. Therefore, the perfect pairing (for any G-module M) is a direct consequence of Pontryagin duality of finite groups. For local and global fields, similar statements exist (local duality and global or Poitou–Tate duality).(22)Serre duality or coherent duality are similar to the statements above, but applies to cohomology of coherent sheaves instead.(23)

• Alexander duality

See also

• Dual (category theory)
• Dual numbers, a certain associative algebra; the term "dual" here is synonymous with double, and is unrelated to the notions given above.
• Duality (electrical engineering)
• Lagrange duality
• Dual code
• Dual lattice
• Dual basis
• dual abelian variety
• Adjoint functors

Notes

1. {{harvnb|Kostrikin|2001}}



2. {{harvnb|Gowers|2008|loc=p. 187, col. 1}}



3. Generally speaking, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have fixed points, the dual of A is sometimes A itself. For example, Desargues' theorem in projective geometry is self-dual in this sense. Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.{{harvnb|Gowers|2008|loc=p. 189, col. 2}}



4. {{harvnb|Artstein-Avidan|Milman|2007}}



5. {{harvnb|Artstein-Avidan|Milman|2008}}



6. {{harvnb|Veblen|Young|1965}}.



7. {{Harvard citations|last1=Veblen|last2=Young|year=1965|loc = Ch. I, Theorem 11}}



8. {{Harvard citations| last1=Fulton | year=1993|nb=yes}}



9. {{Harvnb|Mac Lane|1998|loc=Ch. II.1}}.



10. {{Harvard citations | last1=Lam|year=1999|loc=§19C}}



11. {{Harvard citations | last1=Weibel |year=1994|txt}}



12. {{Harvard citations | last1=Dwyer | last2=Spaliński | year=1995|txt}}



13. {{Harvard citations | last1=Hartshorne | year=1966 | loc=Ch. II.2, esp. Prop. II.2.3|nb=yes}}



14. See {{Harvard citations | last1=Lang|year=2002|loc=Theorem VI.1.1}} for finite Galois extensions.



15. {{Harvard citations | last1=Loomis|year=1953 | loc=p. 151, section 37D}}



16. {{Harvard citations | last1=Joyal | last2=Street|year=1991 |txt}}



17. {{Harvnb|Negrepontis|1971}}.



18. {{Harvard citations | last1=Griffiths | last2=Harris | year=1994|nb=yes|loc=p. 56}}



19. {{Harvard citations | last1=Milne |year=1980|loc=Ch. VI.11|nb=yes}}



20. {{Harvard citations | last1=Iversen | year=1986|nb=yes|loc=Ch. VII.3, VII.5}}



21. {{Harvard citations|last=Milne|year=2006|loc=Example I.1.10|txt}}



22. {{Harvard citations|last1=Mazur|year=1973|txt}}; {{Harvard citations|last=Milne|year=2006|txt}}



23. {{Harvard citations | last1=Hartshorne | year=1966 | loc=Ch. III.7|nb=yes}}

References

Duality in general

• {{Springer|title=Duality|id=D/d034120|first=A. I.|last=Kostrikin}}.
• {{citation|contribution=III.19 Duality|title=The Princeton Companion to Mathematics|pages=187–190|first=Timothy|last=Gowers|publisher=Princeton University Press|year=2008}}.
• {{Citation | last1=Cartier | first1=Pierre | title=A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry | url=http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/ | id={{MathSciNet | id = 1848254}} | year=2001 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=38 | issue=4 | pages=389–408}} (a non-technical overview about several aspects of geometry, including dualities)

Specific dualities

• {{Citation | last1=Artstein-Avidan | first1=Shiri | last2=Milman | first2=Vitali | author2-link=Vitali Milman | year=2008 | title=The concept of duality for measure projections of convex bodies | journal=Journal of functional analysis | volume=254 | issue=10 | pages=2648–2666 | doi=10.1016/j.jfa.2007.11.008}}. Also author's site.
• {{Citation | last1=Artstein-Avidan | first1=Shiri | last2=Milman | first2=Vitali | author2-link=Vitali Milman | year=2007 | title=A characterization of the concept of duality | journal=Electronic research announcements in mathematical sciences | volume=14 | pages=42–59 | url=http://www.aimsciences.org/journals/pdfs.jsp?paperID=2887&mode=full}}. Also author's site.
• {{Citation | last1=Dwyer | first1=William G. | last2=Spaliński | first2=J. | title=Handbook of algebraic topology | url=http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Spalinski/theories | publisher=North-Holland | location=Amsterdam | id={{MathSciNet | id = 1361887}} | year=1995 | chapter=Homotopy theories and model categories | pages=73–126}}
• {{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician)| title=Introduction to toric varieties | publisher=Princeton University Press | isbn=978-0-691-00049-7 | year=1993}}
• {{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=John Wiley & Sons | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | id={{MathSciNet | id = 1288523}} | year=1994}}
• {{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Residues and Duality | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics 20 | year=1966 | pages=20–48}}
• {{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Algebraic Geometry | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90244-9 | oclc=13348052 | id={{MathSciNet | id = 0463157}} | year=1977}}
• {{Citation | last1=Iversen | first1=Birger | title=Cohomology of sheaves | publisher=Springer-Verlag | location=Berlin, New York | series=Universitext | isbn=978-3-540-16389-3 | id={{MathSciNet | id = 842190}} | year=1986}}
• {{Citation | last1=Joyal | first1=André | author1-link=André Joyal | last2=Street | first2=Ross | author2-link=Ross Street | title=Category theory (Como, 1990) | url=http://www.maths.mq.edu.au/~street/CT90Como.pdf | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture notes in mathematics | id={{MathSciNet | id = 1173027}} | year=1991 | volume=1488 | chapter=An introduction to Tannaka duality and quantum groups | pages=413–492}}
• {{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | id={{MathSciNet|id=1653294}} | year=1999}}
• {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Algebra | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-95385-4 | id={{MathSciNet | id = 1878556}} | year=2002 | volume=211}}
• {{Citation | last1=Loomis | first1=Lynn H. | title=An introduction to abstract harmonic analysis | publisher=D. Van Nostrand Company, Inc. | location=Toronto-New York-London | year=1953 | pages=pp. x+190}}
• {{Citation | last1=Mac Lane | first1=Saunders | author1-link=Saunders Mac Lane | title=Categories for the Working Mathematician | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | isbn=978-0-387-98403-2 | year=1998}}
• {{Citation | last1=Mazur | first1=Barry | author1-link=Barry Mazur | title=Notes on étale cohomology of number fields | id={{MathSciNet | id = 0344254}} | year=1973 | journal=Annales Scientifiques de l'École Normale Supérieure. Quatrième Série | issn=0012-9593 | volume=6 | pages=521–552}}
• {{Citation | last1=Milne | first1=James S. | title=Étale cohomology | publisher=Princeton University Press | isbn=978-0-691-08238-7 | year=1980}}
• {{Citation | last1=Milne | first1=James S. | title=Arithmetic duality theorems | url=http://www.jmilne.org/math/Books/adt.html | publisher=BookSurge, LLC | location=Charleston, SC | edition=2nd | isbn=978-1-4196-4274-6 | id={{MathSciNet | id = 2261462}} | year=2006}}
• {{Citation | last1=Negrepontis | first1=Joan W. | title=Duality in analysis from the point of view of triples | id={{MathSciNet | id = 0280571}} | year=1971 | journal=Journal of Algebra | issn=0021-8693 | volume=19 | pages=228–253}}
• {{Citation | last1=Veblen | first1=Oswald | author1-link=Oswald Veblen | last2=Young | first2=John Wesley | title=Projective geometry. Vols. 1, 2 | publisher=Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London | id={{MathSciNet | id = 0179666}} | year=1965}}
• {{Citation | last1=Weibel | first1=Charles A. | title=An introduction to homological algebra | publisher=Cambridge University Press | isbn=978-0-521-55987-4 | id={{MathSciNet | id = 1269324}} | year=1994}}

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