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étale Homotopy Type
In mathematics, especially in algebraic geometry, the étale homotopy type is an analogue of the homotopy type of topological spaces for algebraic varieties. Roughly speaking, for a variety or scheme ''X'', the idea is to consider étale coverings U \rightarrow X and to replace each connected component of ''U'' and the higher "intersections", i.e., fiber products, U_n := U \times_X U \times_X \dots \times_X U (''n''+1 copies of ''U'', n \geq 0) by a single point. This gives a simplicial set which captures some information related to ''X'' and the étale topology of it. Slightly more precisely, it is in general necessary to work with étale hypercovers (U_n)_ instead of the above simplicial scheme determined by a usual étale cover. Taking finer and finer hypercoverings (which is technically accomplished by working with the pro-object in simplicial sets determined by taking all hypercoverings), the resulting object is the étale homotopy type of ''X''. Similarly to classica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Scheme
In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s. Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site. Examples Example: Consider the étale site of a scheme ''S''. Each ''U'' in the site represents the presheaf \operatorname(-, U). Thus, a simplicial scheme, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf). Example: Let ''G'' be a presheaf of groupoids. Then taking nerves section-wise, one obtains a simplicial presheaf BG. For example, one might set B\operatorname = \varinjlim B\operatorname. These types of examples appe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Scholze
Peter Scholze (; born 11 December 1987) is a German mathematician known for his work in arithmetic geometry. He has been a professor at the University of Bonn since 2012 and co-director at the Max Planck Institute for Mathematics since 2018. He has been called one of the leading mathematicians in the world. In 2018, he won the Fields Medal, an award regarded as the highest professional honor in mathematics. Early life and education Scholze was born in Dresden and grew up in Berlin. His father is a physicist, his mother a computer scientist, and his sister studied chemistry. He attended the in Berlin-Friedrichshain, a gymnasium devoted to mathematics and science. As a student, Scholze participated in the International Mathematical Olympiad, winning three gold medals and one silver medal. He studied at the University of Bonn and completed his bachelor's degree in three semesters and his master's degree in two further semesters. He obtained his Ph.D. in 2012 under the superv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Category
In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any model category and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra. The naive homotopy category The category of topological spaces Top has topological spaces as objects and as morphisms the continuous maps between them. The older definition of the homotopy category hTop, called the naive homotopy category for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps ''f'' : ''X'' → ''Y'' are considered the same in the na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forgetful Functor
In mathematics, more specifically in the area of category theory, a forgetful functor (also known as a stripping functor) "forgets" or drops some or all of the input's structure or properties mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case. Overview As an example, there are several forgetful functors from the category of commutative rings. A ( unital) ring, described in the language of universal algebra, is an ordered tuple (R,+,\times,a,0,1) satisfying certain axioms, where + and \times are binary functions on the set R, a is a unary operation corresponding to additive inverse, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf (mathematics)
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data are well-behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every datum is the sum of its constituent data). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their precise definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type. History Étale cohomology was introduced by , using some suggestions by Jean-Pierre Serre, and was motivated by the attempt to construct a Weil cohomology theory in order to prove the Weil conjectures. The foundations were soon after worked out by Grothendieck together with Michael Artin, and published as and SGA 4. Grothendieck used étale cohomology to prove some of the Weil conjectures (Bernard Dwork had already managed to prove the rationality par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
étale Fundamental Group
The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces. Topological analogue/informal discussion In algebraic topology, the fundamental group \pi_1(X,x) of a pointed topological space (X, x) is defined as the group of homotopy classes of loops based at x. This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the Zariski topology. In the classification of covering spaces, it is shown that the fundamental group is exactly the group of deck transformations of the universal covering space. This is more promising: finite étale morphisms of algebraic varieties are the appropriate analogue of covering spaces of topological spaces. Unfortunately, an algebraic variety X often fails to have a "universal cover" that is finite over ''X'', so one must consider the entire category of finite étale coverings of ''X''. One ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pro-object
In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category ''C''. The objects in this ind-completed category, denoted Ind(''C''), are known as direct systems, they are functors from a small filtered category ''I'' to ''C''. The dual concept is the pro-completion, Pro(''C''). Definitions Filtered categories Direct systems depend on the notion of ''filtered categories''. For example, the category N, whose objects are natural numbers, and with exactly one morphism from ''n'' to ''m'' whenever n \le m, is a filtered category. Direct systems A ''direct system'' or an ''ind-object'' in a category ''C'' is defined to be a functor :F : I \to C from a small filtered category ''I'' to ''C''. For example, if ''I'' is the category N mentioned above, this datum is equivalent to a sequence :X_0 \to X_1 \to \cdots of objects in ''C'' together with morphisms as displayed. The ind-completion Ind-objects in ''C'' form a cate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercover
In mathematics, and in particular homotopy theory, a hypercovering (or hypercover) is a simplicial object that generalises the Čech nerve of a cover. For the Čech nerve of an open cover one can show that if the space X is compact and if every intersection of open sets in the cover is contractible, then one can contract these sets and get a simplicial set that is weakly equivalent to X in a natural way. For the étale topology and other sites, these conditions fail. The idea of a hypercover is to instead of only working with n-fold intersections of the sets of the given open cover \mathcal U, to allow the pairwise intersections of the sets in \mathcal U=\mathcal U_0 to be covered by an open cover \mathcal U_1, and to let the triple intersections of this cover to be covered by yet another open cover \mathcal U_2, and so on, iteratively. Hypercoverings have a central role in étale homotopy and other areas where homotopy theory is applied to algebraic geometry, such as motivic homoto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Type
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second parameter of ''H'' as time then ''H'' describes a ''continuous def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
