arithmetic geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...
, a Frobenioid is a
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) ...
with some extra structure that generalizes the theory of
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organisi ...
s on models of finite extensions of
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
*Algebraic number field: A finite extension of \mathbb
*Global function f ...
s. Frobenioids were introduced by . The word "Frobenioid" is a
portmanteau
A portmanteau word, or portmanteau (, ) is a blend of wordsFrobenius and
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
, as certain Frobenius morphisms between Frobenioids are analogues of the usual
Frobenius morphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphi ...
, and some of the simplest examples of Frobenioids are essentially monoids.
The Frobenioid of a monoid
If ''M'' is a
commutative monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ar ...
, it is acted on naturally by the monoid ''N'' of positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s under multiplication, with an element ''n'' of ''N'' multiplying an element of ''M'' by ''n''. The Frobenioid of ''M'' is the semidirect product of ''M'' and ''N''. The underlying category of this Frobenioid is category of the monoid, with one object and a morphism for each element of the monoid. The standard Frobenioid is the special case of this construction when ''M'' is the additive monoid of non-negative integers.
Elementary Frobenioids
An elementary Frobenioid is a generalization of the Frobenioid of a commutative monoid, given by a sort of semidirect product of the monoid of positive integers by a family Φ of commutative monoids over a base category ''D''. In applications the category ''D'' is sometimes the category of models of finite separable extensions of a global field, and Φ corresponds to the line bundles on these models, and the action of a positive integers ''n'' in ''N'' is given by taking the ''n''th power of a line bundle.
Frobenioids and poly-Frobenioids
A Frobenioid consists of a category ''C'' together with a functor to an elementary Frobenioid, satisfying some complicated conditions related to the behavior of line bundles and divisors on models of global fields. One of Mochizuki's fundamental theorems states that under various conditions a Frobenioid can be reconstructed from the category ''C''. A poly-Frobenioid is an extension of a Frobenioid.
Category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
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Anabelian geometry
Anabelian geometry is a theory in number theory which describes the way in which the algebraic fundamental group ''G'' of a certain arithmetic variety ''X'', or some related geometric object, can help to restore ''X''. The first results for n ...