Nisnevich Topology
In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. It was originally introduced by Yevsey Nisnevich, who was motivated by the theory of adeles. Definition A morphism of schemes f:Y \to X is called a Nisnevich morphism if it is an étale morphism such that for every (possibly non-closed) point ''x'' ∈ ''X'', there exists a point ''y'' ∈ ''Y'' in the fiber such that the induced map of residue fields ''k''(''x'') → ''k''(''y'') is an isomorphism. Equivalently, ''f'' must be flat, unramified, locally of finite presentation, and for every point ''x'' ∈ ''X'', there must exist a point ''y'' in the fiber such that ''k''(''x'') → ''k''(''y'') is an isomorphism. A family of morphisms is a Nisnevich cover if each morphism in the family is étale and for every (possibly non-clo ...
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Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ...
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Cdh Topology
In algebraic geometry, the ''h'' topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes. It combines several good properties possessed by its related "sub"topologies, such as the ''qfh'' and ''cdh'' topologies. It has subsequently been used by Beilinson to study p-adic Hodge theory, in Bhatt and Scholze's work on projectivity of the affine Grassmanian, Huber and Jörder's study of differential forms, etc. Definition Voevodsky defined the ''h'' topology to be the topology associated to finite families \ of morphisms of finite type such that \amalg U_i \to X is a universal topological epimorphism (i.e., a set of points in the target is an open subset if and only if its preimage is open, and any base change also has this property). Voevodsky worked with this topology exclusively on categories Sch^_ of schemes of finite type over a ''Noetherian'' base scheme S. Bhatt-Scholze define the ''h'' topology on the category Sch^_ of schemes ...
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Henselian Ring
In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now restrict them to be commutative. Some standard references for Hensel rings are , , and . Definitions In this article rings will be assumed to be commutative, though there is also a theory of non-commutative Henselian rings. * A local ring ''R'' with maximal ideal ''m'' is called Henselian if Hensel's lemma holds. This means that if ''P'' is a monic polynomial in ''R'' 'x'' then any factorization of its image ''P'' in (''R''/''m'') 'x''into a product of coprime monic polynomials can be lifted to a factorization in ''R'' 'x'' * A local ring is Henselian if and only if every finite ring extension is a product of local rings. * A Henselian local ring is called strictly Henselian if its residue field is separably closed. * By abuse of term ...
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Mixed Motives (math)
Mixed is the past tense of ''mix''. Mixed may refer to: * Mixed (United Kingdom ethnicity category), an ethnicity category that has been used by the United Kingdom's Office for National Statistics since the 2001 Census Music * ''Mixed'' (album), a compilation album of two avant-garde jazz sessions featuring performances by the Cecil Taylor Unit and the Roswell Rudd Sextet See also * Mix (other) * Mixed breed, an animal whose family are from different breeds or species * Mixed ethnicity The term multiracial people refers to people who are mixed with two or more races (human categorization), races and the term multi-ethnic people refers to people who are of more than one ethnicity, ethnicities. A variety of terms have been used ...

Presheaf With Transfers
In algebraic geometry, a presheaf with transfers is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps. Precisely, it is, by definition, a contravariant additive functor from the category of finite correspondences (defined below) to the category of abelian groups (in category theory, “presheaf” is another term for a contravariant functor). When a presheaf ''F'' with transfers is restricted to the subcategory of smooth separated schemes, it can be viewed as a presheaf on the category with ''extra'' maps F(Y) \to F(X), not coming from morphisms of schemes but also from finite correspondences from ''X'' to ''Y'' A presheaf ''F'' with transfers is said to be \mathbb^1-homotopy invariant if F(X) \simeq F(X \times \mathbb^1) for every ''X''. For example, Chow groups as well as motivic cohomology groups form presheaves with transfers. Finite correspondence Let X, Y be algebraic schemes (i.e., separated and of finite type over a fiel ...
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