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F-crystal
In algebraic geometry, F-crystals are objects introduced by that capture some of the structure of crystalline cohomology groups. The letter ''F'' stands for Frobenius, indicating that ''F''-crystals have an action of Frobenius on them. F-isocrystals are crystals "up to isogeny". F-crystals and F-isocrystals over perfect fields Suppose that ''k'' is a perfect field, with ring of Witt vectors ''W'' and let ''K'' be the quotient field of ''W'', with Frobenius automorphism σ. Over the field ''k'', an ''F''-crystal is a free module ''M'' of finite rank over the ring ''W'' of Witt vectors of ''k'', together with a σ-linear injective endomorphism of ''M''. An ''F''-isocrystal is defined in the same way, except that ''M'' is a module for the quotient field ''K'' of ''W'' rather than ''W''. Dieudonné–Manin classification theorem The Dieudonné–Manin classification theorem was proved by and . It describes the structure of ''F''-isocrystals over an algebraically closed fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystalline Cohomology
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes ''X'' over a base field ''k''. Its values ''H''''n''(''X''/''W'') are modules over the ring ''W'' of Witt vectors over ''k''. It was introduced by and developed by . Crystalline cohomology is partly inspired by the ''p''-adic proof in of part of the Weil conjectures and is closely related to the algebraic version of de Rham cohomology that was introduced by Grothendieck (1963). Roughly speaking, crystalline cohomology of a variety ''X'' in characteristic ''p'' is the de Rham cohomology of a smooth lift of ''X'' to characteristic 0, while de Rham cohomology of ''X'' is the crystalline cohomology reduced mod ''p'' (after taking into account higher ''Tor''s). The idea of crystalline cohomology, roughly, is to replace the Zariski open sets of a scheme by infinitesimal thickenings of Zariski open sets with divided power structures. The motivation for this is that it can then be calculated by taking a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic And Geometric Frobenius
In mathematics, the Frobenius endomorphism is defined in any commutative ring ''R'' that has characteristic ''p'', where ''p'' is a prime number. Namely, the mapping φ that takes ''r'' in ''R'' to ''r''''p'' is a ring endomorphism of ''R''. The image of φ is then ''R''''p'', the subring of ''R'' consisting of ''p''-th powers. In some important cases, for example finite fields, φ is surjective. Otherwise φ is an endomorphism but not a ring ''automorphism''. The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to φ. This gives a mapping :φ*: Spec(''R''''p'') → Spec(''R'') of affine schemes. Even in cases where ''R''''p'' = ''R'' this is not the identity, unless ''R'' is the prime field. Mappings created by fibre product with φ*, i.e. base changes, tend in scheme theory to be called ''geometric Frobenius''. The reason for a careful terminology is that the Frobenius automorphism in Galois groups, or defined by transport of stru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Field
In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k'' is separable. * Every algebraic extension of ''k'' is separable. * Either ''k'' has characteristic 0, or, when ''k'' has characteristic , every element of ''k'' is a ''p''th power. * Either ''k'' has characteristic 0, or, when ''k'' has characteristic , the Frobenius endomorphism is an automorphism of ''k''. * The separable closure of ''k'' is algebraically closed. * Every reduced commutative ''k''-algebra ''A'' is a separable algebra; i.e., A \otimes_k F is reduced for every field extension ''F''/''k''. (see below) Otherwise, ''k'' is called imperfect. In particular, all fields of characteristic zero and all finite fields are perfect. Perfect fields are significant because Galois theory over these fields becomes simpler, since the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witt Vector
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field of order p is the ring of p-adic integers. They have a highly non-intuitive structure upon first glance because their additive and multiplicative structure depends on an infinite set of recursive formulas which do not behave like addition and multiplication formulas for standard p-adic integers. The main idea behind Witt vectors is instead of using the standard p-adic expansiona = a_0+a_1p+a_2p^2 + \cdotsto represent an element in \mathbb_p, we can instead consider an expansion using the Teichmüller character\omega: \mathbb_p^* \to \mathbb_p^*which sends each element in the solution set of x^-1 in \mathbb_p to an element in the solution set of x^-1 in \mathbb_p. That is, we expand out elements in \mathbb_p in terms of roots of unity instead of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Valuation Ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' is a local principal ideal domain, and not a field. # ''R'' is a valuation ring with a value group isomorphic to the integers under addition. # ''R'' is a local Dedekind domain and not a field. # ''R'' is a Noetherian local domain whose maximal ideal is principal, and not a field.https://mathoverflow.net/a/155639/114772 # ''R'' is an integrally closed Noetherian local ring with Krull dimension one. # ''R'' is a principal ideal domain with a unique non-zero prime ideal. # ''R'' is a principal ideal domain with a unique irreducible element ( up to multiplication by units). # ''R'' is a unique factorization domain with a unique irreducible element (up to multiplication by units). # ''R'' is Noetherian, not a field, and every nonzero fractio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (algebra)
In mathematics, the characteristic of a ring (mathematics), ring , often denoted , is defined to be the smallest number of times one must use the ring's identity element, multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, is the smallest positive number such that: :\underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent (group theory), exponent of the ring's additive group, that is, the smallest positive integer such that: :\underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). Some authors do not include the multiplicative identity element in their r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Field
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements. The field of fractions of R is sometimes denoted by \operatorname(R) or \operatorname(R), and the construction is sometimes also called the fraction field, field of quotients, or quotient field of R. All four are in common usage, but are not to be confused with the quotient of a ring by an ideal, which is a quite different concept. For a commutative ring which is not an integral domain, the analogous construction is called the localization or ring of quotients. Definition Given an integral domain and letting R^* = R \setminus \, we define an equivalence relation on R \times R^* by letting (n,d) \sim (m,b) whenever nb = md. We denote the equivale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
