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Kähler Differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available. Definition Let and be commutative rings and be a ring homomorphism. An important example is for a field and a unital algebra over (such as the coordinate ring of an affine variety). Kähler differentials formalize the observation that the derivatives of polynomials are again polynomial. In this sense, differentiation is a notion which can be expressed in purely algebraic terms. This observation can be turned into a definition of the module :\Omega_ of differentials in different, but equivalent ways. Definition using derivations An -linear ''derivation'' on is an -modu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modulo (jargon)
In mathematics, the term ''modulo'' ("with respect to a modulus of", the Latin ablative of '' modulus'' which itself means "a small measure") is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their difference is accounted for by an additional factor. It was initially introduced into mathematics in the context of modular arithmetic by Carl Friedrich Gauss in 1801. Since then, the term has gained many meanings—some exact and some imprecise (such as equating "modulo" with "except for"). For the most part, the term often occurs in statements of the form: :''A'' is the same as ''B'' modulo ''C'' which means :''A'' and ''B'' are the same—except for differences accounted for or explained by ''C''. History ''Modulo'' is a mathematical jargon that was introduced into mathematics in the book ''Disquisitiones Arithmeticae'' by Carl Friedrich Gauss in 1801. Given the integers ''a'', ''b'' and ''n'', the expression "''a'' ≡ ''b'' (mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Product Of Schemes
In 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. Definition The category of schemes is a broad setting for algebraic geometry. A fruitful philosophy (known as Grothendieck's relative point of view) is that much of algebraic geometry should be developed for a morphism of schemes ''X'' → ''Y'' (called a scheme ''X'' over ''Y''), rather than for a single scheme ''X''. For example, rather than simply studying algebraic curves, one can study families of curves over any base scheme ''Y''. Indeed, the two approaches enrich each other. In particular, a scheme over a commutative ring ''R'' means a scheme ''X'' together with a morphism ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek διαγώνιος ''diagonios'', "from angle to angle" (from διά- ''dia-'', "through", "across" and γωνία ''gonia'', "angle", related to ''gony'' "knee"); it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as ''diagonus'' ("slanting line"). In matrix algebra, the diagonal of a square matrix consists of the entries on the line from the top left corner to the bottom right corner. There are also other, non-mathematical uses. Non-mathematical uses In engineering, a diagonal brace is a beam used to brace a rectangular structure (such as scaffolding) to withstand strong forces pushing into it; although called a diagonal, due to practical consideration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cotangent Complex
In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X \to Y is a morphism of geometric or algebraic objects, the corresponding cotangent complex \mathbf_^\bullet can be thought of as a universal "linearization" of it, which serves to control the deformation theory of f. It is constructed as an object in a certain derived category of sheaves on X using the methods of homotopical algebra. Restricted versions of cotangent complexes were first defined in various cases by a number of authors in the early 1960s. In the late 1960s, Michel André and Daniel Quillen independently came up with the correct definition for a morphism of commutative rings, using simplicial methods to make precise the idea of the cotangent complex as given by taking the (non-abelian) left derived functor of Kähler differentials. Luc Illusie then globalized this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short Exact Sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i |
Multiplicative Set
In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold: * 1 \in S, * xy \in S for all x, y \in S. In other words, ''S'' is closed under taking finite products, including the empty product 1.Eisenbud, p. 59. Equivalently, a multiplicative set is a submonoid of the multiplicative monoid of a ring. Multiplicative sets are important especially in commutative algebra, where they are used to build localizations of commutative rings. A subset ''S'' of a ring ''R'' is called saturated if it is closed under taking divisors: i.e., whenever a product ''xy'' is in ''S'', the elements ''x'' and ''y'' are in ''S'' too. Examples Examples of multiplicative sets include: * the set-theoretic complement of a prime ideal in a commutative ring; * the set , where ''x'' is an element of a ring; * the set of units of a ring; * the set of non-zero-divisors in a ring; * for an ideal ''I''. * t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Of A Ring
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractions \frac, such that the denominator ''s'' belongs to a given subset ''S'' of ''R''. If ''S'' is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field \Q of rational numbers from the ring \Z of integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term ''localization'' originated in algebraic geometry: if ''R'' is a ring of functions defined on some geometric object (algebraic variety) ''V'', and one wants to study this variety "locally" near a point ''p'', then one considers the set ''S'' of all functions that are not zero at ''p'' and localizes ''R'' wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extension Of Scalars
In algebra, given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a left ''R''-module ''M'' and a left ''S''-module ''N'', *f_! M = S\otimes_R M, the induced module. *f_* M = \operatorname_R(S, M), the coinduced module. *f^* N = N_R, the restriction of scalars. They are related as adjoint functors: :f_! : \text_R \leftrightarrows \text_S : f^* and :f^* : \text_S \leftrightarrows \text_R : f_*. This is related to Shapiro's lemma. Operations Restriction of scalars Throughout this section, let R and S be two rings (they may or may not be commutative, or contain an identity), and let f:R \to S be a homomorphism. Restriction of scalars changes ''S''-modules into ''R''-modules. In algebraic geometry, the term "restriction of scalars" is often used as a synonym for Weil restriction. Definition Suppose that M is a module over S. Then it can be regarded as a module over R where the action of R is given via : \begin M\ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is that of the ring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (algebra)
In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the kernel of a linear map. The kernel of a matrix, also called the ''null space'', is the kernel of the linear map defined by the matrix. The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective.See and . For some types of structure, such as abelian groups and vector spaces, the possible kernels are exactly the substructures of the same type. This is not always the case, and, sometimes, the possible kernels have received a special name, such as normal subgroup for groups and two-sided ideals for r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
