Scheme (mathematics)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).
Group schemeIn mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance.
Hilbert schemeIn algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by . Hironaka's example shows that non-projective varieties need not have Hilbert schemes.
Fiber product of schemesIn mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determines a variety over a bigger field, or the pullback of a family of varieties, or a fiber of a family of varieties. Base change is a closely related notion. The of schemes is a broad setting for algebraic geometry.
Formal schemeIn mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme. For this reason, formal schemes frequently appear in topics such as deformation theory. But the concept is also used to prove a theorem such as the theorem on formal functions, which is used to deduce theorems of interest for usual schemes.
Derived schemeIn algebraic geometry, a derived scheme is a pair consisting of a topological space X and a sheaf either of simplicial commutative rings or of commutative ring spectra on X such that (1) the pair is a scheme and (2) is a quasi-coherent -module. The notion gives a homotopy-theoretic generalization of a scheme. A derived stack is a stacky generalization of a derived scheme. Over a field of characteristic zero, the theory is closely related to that of a differential graded scheme.
1,3-Dipolar cycloadditionThe 1,3-dipolar cycloaddition is a chemical reaction between a 1,3-dipole and a dipolarophile to form a five-membered ring. The earliest 1,3-dipolar cycloadditions were described in the late 19th century to the early 20th century, following the discovery of 1,3-dipoles. Mechanistic investigation and synthetic application were established in the 1960s, primarily through the work of Rolf Huisgen. Hence, the reaction is sometimes referred to as the Huisgen cycloaddition (this term is often used to specifically describe the 1,3-dipolar cycloaddition between an organic azide and an alkyne to generate 1,2,3-triazole).
Descriptor (chemistry)In chemical nomenclature, a descriptor is a notational prefix placed before the systematic substance name, which describes the configuration or the stereochemistry of the molecule. Some listed descriptors are only of historical interest and should not be used in publications anymore as they do not correspond with the modern recommendations of the IUPAC. Stereodescriptors are often used in combination with locants to clearly identify a chemical structure unambiguously.
Click chemistryIn chemical synthesis, click chemistry is a class of simple, atom-economy reactions commonly used for joining two molecular entities of choice. Click chemistry is not a single specific reaction, but describes a way of generating products that follow examples in nature, which also generates substances by joining small modular units. In many applications, click reactions join a biomolecule and a reporter molecule. Click chemistry is not limited to biological conditions: the concept of a "click" reaction has been used in chemoproteomic, pharmacological, biomimetic and molecular machinery applications.
Octane ratingAn octane rating, or octane number, is a standard measure of a fuel's ability to withstand compression in an internal combustion engine without detonating. The higher the octane number, the more compression the fuel can withstand before detonating. Octane rating does not relate directly to the power output or the energy content of the fuel per unit mass or volume, but simply indicates gasoline's capability against compression. Whether or not a higher octane fuel improves or impairs an engine's performance depends on the design of the engine.
