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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] |
Weil Cohomology Theory
In algebraic geometry, a Weil cohomology or Weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups. The name is in honor of André Weil. Any Weil cohomology theory factors uniquely through the category of Chow motives, but the category of Chow motives itself is not a Weil cohomology theory, since it is not an abelian category. Definition Fix a base field ''k'' of arbitrary characteristic and a "coefficient field" ''K'' of characteristic zero. A ''Weil cohomology theory'' is a contravariant functor :H^*: \ \longrightarrow \ satisfying the axioms below. For each smooth projective algebraic variety ''X'' of dimension ''n'' over ''k'', then the graded ''K''-algebra :H^*(X) = \bigoplus\nolimits_i H^i(X) is required to satisfy the following: * H^i(X) is a finite-dimensional ''K''-vector space for each integer ''i''. * H^i(X) = 0 for each ''i'' 2''n''. * H^(X) is isomorphic to ''K'' (the so-called or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramification Theory
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping. In complex analysis In complex analysis, the basic model can be taken as the ''z'' → ''z''''n'' mapping in the complex plane, near ''z'' = 0. This is the standard local picture in Riemann surface theory, of ramification of order ''n''. It occurs for example in the Riemann–Hurwitz formula for the effect of mappings on the genus. See also branch point. In algebraic topology In a covering map the Euler–Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The ''z'' → ''z''''n'' mapping shows this as a local p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unramified Extension
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping. In complex analysis In complex analysis, the basic model can be taken as the ''z'' → ''z''''n'' mapping in the complex plane, near ''z'' = 0. This is the standard local picture in Riemann surface theory, of ramification of order ''n''. It occurs for example in the Riemann–Hurwitz formula for the effect of mappings on the genus. See also branch point. In algebraic topology In a covering map the Euler–Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The ''z'' → ''z''''n'' mapping shows this as a local p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1972) pp.74-76 or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field ''K'' is unique up to an isomorphism that fixes every member of ''K''. Because of this essential uniqueness, we often speak of ''the'' algebraic closure of ''K'', rather than ''an'' algebraic closure of ''K''. The algebraic closure of a field ''K'' can be thought of as the largest algebraic extension of ''K''. To see this, note that if ''L'' is any algebraic extension of ''K'', then the algebraic closure of ''L'' is also an algebraic closure of ''K'', and so ''L'' is contained within the algebraic closure of ''K''. The algebraic closure of ''K'' is also the smallest algebraically closed fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ground Field
In mathematics, a ground field is a field ''K'' fixed at the beginning of the discussion. Use It is used in various areas of algebra: In linear algebra In linear algebra, the concept of a vector space may be developed over any field. In algebraic geometry In algebraic geometry, in the foundational developments of André Weil the use of fields other than the complex numbers was essential to expand the definitions to include the idea of abstract algebraic variety over ''K'', and generic point relative to ''K''. In Lie theory Reference to a ground field may be common in the theory of Lie algebras (''qua'' vector spaces) and algebraic groups (''qua'' algebraic varieties). In Galois theory In Galois theory, given a field extension ''L''/''K'', the field ''K'' that is being extended may be considered the ground field for an argument or discussion. Within algebraic geometry, from the point of view of scheme theory, the spectrum ''Spec''(''K'') of the ground field ''K'' plays the role ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion Algebra
In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a matrix algebra by '' extending scalars'' (equivalently, tensoring with a field extension), i.e. for a suitable field extension ''K'' of ''F'', A \otimes_F K is isomorphic to the 2 × 2 matrix algebra over ''K''. The notion of a quaternion algebra can be seen as a generalization of Hamilton's quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over F = \mathbb, and indeed the only one over \mathbb apart from the 2 × 2 real matrix algebra, up to isomorphism. When F = \mathbb, then the biquaternions form the quaternion algebra over ''F''. Structure ''Quaternion algebra'' here means something more general than the algebra of Hamilton's quaternions. When th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Order
In mathematics, an order in the sense of ring theory is a subring \mathcal of a ring A, such that #''A'' is a finite-dimensional algebra over the field \mathbb of rational numbers #\mathcal spans ''A'' over \mathbb, and #\mathcal is a \mathbb-lattice in ''A''. The last two conditions can be stated in less formal terms: Additively, \mathcal is a free abelian group generated by a basis for ''A'' over \mathbb. More generally for ''R'' an integral domain contained in a field ''K'', we define \mathcal to be an ''R''-order in a ''K''-algebra ''A'' if it is a subring of ''A'' which is a full ''R''-lattice. When ''A'' is not a commutative ring, the idea of order is still important, but the phenomena are different. For example, the Hurwitz quaternions form a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates in the most obvious sense. Maximal orders exist in general, but need not be unique: there is in general no largest or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endomorphism Ring
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map 0: x \mapsto 0 as additive identity and the identity map 1: x \mapsto x as multiplicative identity. The functions involved are restricted to what is defined as a homomorphism in the context, which depends upon the category of the object under consideration. The endomorphism ring consequently encodes several internal properties of the object. As the resulting object is often an algebra over some ring ''R,'' this may also be called the endomorphism algebra. An abelian group is the same thing as a module over the ring of integers, which is the initial object in the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersingular Elliptic Curve
In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic ''p'' > 0 with unusually large endomorphism rings. Elliptic curves over such fields which are not supersingular are called ''ordinary'' and these two classes of elliptic curves behave fundamentally differently in many aspects. discovered supersingular elliptic curves during his work on the Riemann hypothesis for elliptic curves by observing that positive characteristic elliptic curves could have endomorphism rings of unusually large rank 4, and developed their basic theory. The term "supersingular" has nothing to do with singular points of curves, and all supersingular elliptic curves are non-singular. It comes from the phrase "singular values of the j-invariant" used for values of the j-invariant for which a complex elliptic curve has complex multiplication. The complex elliptic curves with complex multiplication are those for which the endomorp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Integer
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic P
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's 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 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 requirements for a ring (see Multiplicative identity and the te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
