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Witt Vector Cohomology
In mathematics, Witt vector cohomology was an early ''p''-adic cohomology theory for algebraic varieties introduced by . Serre constructed it by defining a sheaf of truncated Witt rings ''W''''n'' over a variety ''V'' and then taking the inverse limit of the sheaf cohomology groups ''H''''i''(''V'', ''W''''n'') of these sheaves. Serre observed that though it gives cohomology groups over a field of characteristic 0, it cannot be a Weil cohomology 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 ... theory because the cohomology groups vanish when ''i'' > dim(''V''). For Abelian varieties showed that one could obtain a reasonable first cohomology group by taking the direct sum of the Witt vector cohomology and the Tate module of the Picard variety. References * *{{citation, mr=0098100 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Cohomology
In mathematics, p-adic cohomology means a cohomology theory for varieties of characteristic ''p'' whose values are modules over a ring of ''p''-adic integers. Examples (in roughly historical order) include: * Serre's Witt vector cohomology * Monsky–Washnitzer cohomology * Infinitesimal cohomology * Crystalline cohomology * Rigid cohomology See also *p-adic Hodge theory *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 conjectu ..., taking values over a ring of ''l''-adic integers for ''l''≠''p'' Arithmetic geometry Cohomology theories {{SIA, mathematics ... [...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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf Cohomology
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper. Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria. From 1940 to 1945, Leray and other prisoners organized a "université en captivité" in the camp. Leray's definitions were simplified and clarified in the 1950s. It became clear that sheaf cohomology was not only a new approach to cohomology in algebraic topology, but also a powerful method in complex analytic geometry and algebraic geometry. These subjects often involve constructing global functions with specified local properties, and sheaf cohomology is ideally suited to such probl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Cohomology
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] |
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 topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
