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Concept# Scheme (mathematics)

Summary

In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem.
Formally, a scheme is a topological space together with commutative rings for

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MATH-510: Modern algebraic geometry

The aim of this course is to learn the basics of the modern scheme theoretic language of algebraic geometry.

MATH-436: Homotopical algebra

This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous examples of model categories and their applications in algebra and topology.

MATH-679: Group schemes

This is a course about group schemes, with an emphasis on structural theorems for algebraic groups (e.g. Barsotti--Chevalley's theorem). All the basics will be covered towards the proof of such theorem, with an estress on the modern presentation using scheme theory and modern algebraic geometry.

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Algebraic geometry is a branch of mathematics which classically studies zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly fro

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In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebr

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Nonempirical hybrid functionals are investigated for band-gap predictions of inorganic metal-halide perovskites belonging to the class CsBX3, with B = Ge, Sn, Pb and X = Cl, Br, I. We consider both global and range-separated hybrid functionals and determine the parameters through two different schemes. The first scheme is based on the static screening response of the material and thus yields dielectric-dependent hybrid functionals. The second scheme defines the hybrid functionals through the enforcement of Koopmans' condition for localized defect states. We also carry out quasiparticle self-consistent GW calculations with vertex corrections to establish state-of-the-art references. For the investigated class of materials, dielectric-dependent functionals and those fulfilling Koopmans' condition yield band gaps of comparable accuracy (similar to 0.2 eV), but the former only require calculations for the primitive unit cell and are less subject to the specifics of the material.

2019Let R be a semilocal Dedekind domain. Under certain assumptions, we show that two (not necessarily unimodular) hermitian forms over an R-algebra with involution, which are rationally isomorphic and have isomorphic semisimple coradicals, are in fact isomorphic. The same result is also obtained for quadratic forms equipped with an action of a finite group. The results have cohomological restatements that resemble the Grothendieck-Serre conjecture, except the group schemes involved are not reductive. We show that these group schemes are closely related to group schemes arising in Bruhat-Tits theory. (C) 2017 Elsevier Inc. All rights reserved.