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Perfectoid Space
In mathematics, perfectoid spaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic", such as local fields of characteristic zero which have residue fields of characteristic prime ''p''. A perfectoid field is a complete topological field ''K'' whose topology is induced by a nondiscrete valuation of rank 1, such that the Frobenius endomorphism Φ is surjective on ''K''°/''p'' where ''K''° denotes the ring of power-bounded elements. Perfectoid spaces may be used to (and were invented in order to) compare mixed characteristic situations with purely finite characteristic ones. Technical tools for making this precise are the tilting equivalence and the almost purity theorem. The notions were introduced in 2012 by Peter Scholze. Tilting equivalence For any perfectoid field ''K'' there is a tilt ''K''♭, which is a perfectoid field of finite characteristic ''p''. As a set, it may be defined as :K^\flat = \varprojlim_ K. Explicitly, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra (structure)
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over the field ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a field ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Purity (algebraic Geometry)
In the mathematical field of algebraic geometry, purity is a theme covering a number of results and conjectures, which collectively address the question of proving that "when something happens, it happens in a particular codimension". Purity of the branch locus For example, ramification is a phenomenon of codimension 1 (in the geometry of complex manifolds, reflecting as for Riemann surfaces that ramify at single points that it happens in real codimension two). A classical result, Zariski–Nagata purity of Masayoshi Nagata and Oscar Zariski, called also purity of the branch locus, proves that on a non-singular algebraic variety a ''branch locus'', namely the set of points at which a morphism ramifies, must be made up purely of codimension 1 subvarieties (a Weil divisor). There have been numerous extensions of this result into theorems of commutative algebra and scheme theory, establishing purity of the branch locus in the sense of description of the restrictions on the possi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Mathematics
In mathematics, almost modules and almost rings are certain objects interpolating between rings and their fields of fractions. They were introduced by in his study of ''p''-adic Hodge theory. Almost modules Let ''V'' be a local integral domain with the maximal ideal ''m'', and ''K'' a fraction field of ''V''. The category of ''K''-modules, ''K''-Mod, may be obtained as a quotient of ''V''-Mod by the Serre subcategory of torsion modules, i.e. those ''N'' such that any element ''n'' in ''N'' is annihilated by some nonzero element in the maximal ideal. If the category of torsion modules is replaced by a smaller subcategory, we obtain an intermediate step between ''V''-modules and ''K''-modules. Faltings proposed to use the subcategory of almost zero modules, i.e. ''N'' ∈ ''V''-Mod such that any element ''n'' in ''N'' is annihilated by ''all'' elements of the maximal ideal. For this idea to work, ''m'' and ''V'' must satisfy certain technical conditions. Let ''V'' be a ring (n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Hodge Theory
In mathematics, ''p''-adic Hodge theory is a theory that provides a way to classify and study ''p''-adic Galois representations of characteristic 0 local fields with residual characteristic ''p'' (such as Q''p''). The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge–Tate representation. Hodge–Tate representations are related to certain decompositions of ''p''-adic cohomology theories analogous to the Hodge decomposition, hence the name ''p''-adic Hodge theory. Further developments were inspired by properties of ''p''-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field. General classification of ''p''-adic representations Let ''K'' be a local field with residue field ''k'' of characteristic ''p''. In this article, a ''p-adic representation'' of ''K'' (or of ''GK'', the absolute Galois group of ''K' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerd Faltings
Gerd Faltings (; born 28 July 1954) is a German mathematician known for his work in arithmetic geometry. Education From 1972 to 1978, Faltings studied mathematics and physics at the University of Münster. In 1978 he received his PhD in mathematics. Career and research In 1981 he obtained the ''venia legendi'' (Habilitation) in mathematics, from the University of Münster. During this time he was an assistant professor at the University of Münster. From 1982 to 1984, he was professor at the University of Wuppertal. From 1985 to 1994, he was professor at Princeton University. In the fall of 1988 and in the academic year 1992–1993 he was a visiting scholar at the Institute for Advanced Study. In 1986 he was awarded the Fields Medal at the ICM at Berkeley for proving the Tate conjecture for abelian varieties over number fields, the Shafarevich conjecture for abelian varieties over number fields and the Mordell conjecture, which states that any non-singular projective curve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
étale Morphism
In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology. The word ''étale'' is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle. Definition Let \phi : R \to S be a ring homomorphism. This makes S an R-algebra. Choose a monic polynomial f in R /math> and a polynomial g in R /math> such that the derivative f' of f is a unit in (R fR _g. We say that \phi is ''standard étale'' if f and g can be chos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Morphism
In algebraic geometry, a finite morphism between two affine varieties X, Y is a dense regular map which induces isomorphic inclusion k\left \righthookrightarrow k\left \right/math> between their coordinate rings, such that k\left \right/math> is integral over k\left \right/math>. This definition can be extended to the quasi-projective varieties, such that a regular map f\colon X\to Y between quasiprojective varieties is finite if any point like y\in Y has an affine neighbourhood V such that U=f^(V) is affine and f\colon U\to V is a finite map (in view of the previous definition, because it is between affine varieties). Definition by Schemes A morphism ''f'': ''X'' → ''Y'' of schemes is a finite morphism if ''Y'' has an open cover by affine schemes :V_i = \mbox \; B_i such that for each ''i'', :f^(V_i) = U_i is an open affine subscheme Spec ''A''''i'', and the restriction of ''f'' to ''U''''i'', which induces a ring homomorphism :B_i \rightarrow A_i, makes ''A''''i'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism Of Schemes
In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. A morphism of algebraic stacks generalizes a morphism of schemes. Definition By definition, a morphism of schemes is just a morphism of locally ringed spaces. A scheme, by definition, has open affine charts and thus a morphism of schemes can also be described in terms of such charts (compare the definition of morphism of varieties). Let ƒ:''X''→''Y'' be a morphism of schemes. If ''x'' is a point of ''X'', since ƒ is continuous, there are open affine subsets ''U'' = Spec ''A'' of ''X'' containing ''x'' and ''V'' = Spec ''B'' of ''Y'' such that ƒ(''U'') ⊆ ''V''. Then ƒ: ''U'' → ''V'' is a morphism of affine schemes and thus is induced by some ring homomorphism ''B'' → ''A'' (cf. #Affine case.) In fact, one can use this description to "define" a morphism of schemes; on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Of Categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation. If a category is equivalent to the opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent. An equivalence of categories consists of a functor between the involved categories, which is required ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
