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In mathematics, Artin–Verdier duality is a
duality Duality may refer to: Mathematics * Duality (mathematics), a mathematical concept ** Dual (category theory), a formalization of mathematical duality ** Duality (optimization) ** Duality (order theory), a concept regarding binary relations ** Dual ...
theorem for constructible abelian sheaves over the
spectrum of a ring In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with ...
of
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the p ...
s, introduced by , that generalizes
Tate duality In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by and . Local Tate duality For a ''p''-adic local ...
. It shows that, as far as etale (or
flat Flat or flats may refer to: Architecture * Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), a ...
)
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
is concerned, the
ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...
in a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
behaves like a
3-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical ...
.


Statement

Let ''X'' be the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...
of the
ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...
in a totally imaginary
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
''K'', and ''F'' a constructible étale
abelian sheaf In mathematics, a sheaf of ''O''-modules or simply an ''O''-module over a ringed space (''X'', ''O'') is a sheaf ''F'' such that, for any open subset ''U'' of ''X'', ''F''(''U'') is an ''O''(''U'')-module and the restriction maps ''F''(''U'')  ...
on ''X''. Then the Yoneda pairing :H^r(X,F)\times \operatorname^(F,\mathbb_m)\to H^3(X,\mathbb_m)=\Q/\Z is a non-degenerate pairing of finite abelian groups, for every integer ''r''. Here, ''Hr''(''X,F'') is the ''r''-th
étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conject ...
group of the
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
''X'' with values in ''F,'' and Ext''r''(''F,G'') is the group of ''r''-
extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Ext ...
of the étale sheaf ''G'' by the étale sheaf ''F'' in the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
of étale abelian sheaves on ''X.'' Moreover, ''Gm'' denotes the étale sheaf of
units Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...
in the
structure sheaf In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
of ''X.'' proved Artin–Verdier duality for constructible, but not necessarily torsion sheaves. For such a sheaf ''F'', the above pairing induces isomorphisms :\begin H^r(X, F)^* &\cong \operatorname^(F, \mathbb_m) && r = 0, 1 \\ H^r(X, F) &\cong \operatorname^(F, \mathbb_m)^* && r = 2, 3 \end where :(-)^* = \operatorname(-, \Q /\Z).


Finite flat group schemes

Let ''U'' be an open subscheme of the spectrum of the ring of integers in a number field ''K'', and ''F'' a finite flat commutative
group scheme In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups ha ...
over ''U''. Then the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutat ...
defines a non-degenerate pairing :H^r(U,F^D)\times H_c^(U,F)\to H_c^3(U,_m)=\Q/\Z of finite abelian groups, for all integers ''r''. Here ''FD'' denotes the Cartier dual of ''F'', which is another finite flat commutative group scheme over ''U''. Moreover, H^r(U,F) is the ''r''-th
flat cohomology In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry. It is used to define the theory of flat cohomology; it also plays a fundamental role in the theory of descent (faithfully flat descent). The term ''flat'' he ...
group of the scheme ''U'' with values in the flat abelian sheaf ''F'', and H_c^r(U,F) is the ''r''-th ''flat cohomology with compact supports'' of ''U'' with values in the flat abelian sheaf ''F.'' The ''flat cohomology with compact supports'' is defined to give rise to a long exact sequence :\cdots\to H^r_c(U,F)\to H^r(U,F)\to \bigoplus\nolimits_ H^r(K_v,F)\to H^_c(U,F) \to\cdots The sum is taken over all
places Place may refer to: Geography * Place (United States Census Bureau), defined as any concentration of population ** Census-designated place, a populated area lacking its own municipal government * "Place", a type of street or road name ** Often ...
of ''K'', which are not in ''U'', including the archimedean ones. The local contribution ''Hr''(''Kv'', ''F'') is the
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a nat ...
of the
Henselization In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now res ...
''Kv'' of ''K'' at the place ''v'', modified a la
Tate Tate is an institution that houses, in a network of four art galleries, the United Kingdom's national collection of British art, and international modern and contemporary art. It is not a government institution, but its main sponsor is the U ...
: :H^r(K_v,F)=H^r_T(\mathrm(K_v^s/K_v),F(K_v^s)). Here K_v^s is a separable closure of K_v.


References

* * * * {{DEFAULTSORT:Artin-Verdier duality Theorems in number theory Duality theories