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Local Cohomology
In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by , and in 1961-2 at IHES written up as SGA2 - , republished as . Given a function (more generally, a section of a quasicoherent sheaf) defined on an open subset of an algebraic variety (or scheme), local cohomology measures the obstruction to extending that function to a larger domain. The rational function 1/x, for example, is defined only on the complement of 0 on the affine line \mathbb^1_K over a field K, and cannot be extended to a function on the entire space. The local cohomology module H^1_(K (where K /math> is the coordinate ring of \mathbb^1_K) detects this in the nonvanishing of a cohomology class /x/math>. In a similar manner, 1/xy is defined away from the x and y axes in the affine plane, but cannot be extended to either the complement of the x-axis or the complement of the y-axis alone (nor can it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Functors
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in various quite different settings that a short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations. Suppose we are given a covariant left exact functor ''F'' : A → B between two abelian categories A and B. If 0 → ''A'' → ''B'' → ''C'' → 0 is a short exact sequence in A, then applying ''F'' yields the exact sequence 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if A is "nice" enough) th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annihilator (ring Theory)
In mathematics, the annihilator of a subset of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by an element of . Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator. The above definition applies also in the case noncommutative rings, where the left annihilator of a left module is a left ideal, and the right-annihilator, of a right module is a right ideal. Definitions Let ''R'' be a ring, and let ''M'' be a left ''R''-module. Choose a non-empty subset ''S'' of ''M''. The annihilator of ''S'', denoted Ann''R''(''S''), is the set of all elements ''r'' in ''R'' such that, for all ''s'' in ''S'', . In set notation, :\mathrm_R(S)=\ It is the set of all elements of ''R'' that "annihilate" ''S'' (the elements for which ''S'' is a torsion set). Subsets of right modules may be used as well, after the modification of "" in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Subscheme
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme ''S'' and a morphism an ''S''-morphism. !$@ A B C D E F G H I J K L M N O P ... [...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] |
Noetherian Ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence I_1\subseteq I_2 \subseteq I_3 \subseteq \cdots of left (or right) ideals has a largest element; that is, there exists an such that: I_=I_=\cdots. Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on Noetherian property (for example, the Laskerâ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum Of A Ring
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings \mathcal. Zariski topology For any ideal ''I'' of ''R'', define V_I to be the set of prime ideals containing ''I''. We can put a topology on \operatorname(R) by defining the collection of closed sets to be :\. This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows. For ''f'' ∈ ''R'', define ''D''''f'' to be the set of prime ideals of ''R'' not containing ''f''. Then each ''D''''f'' is an open subset of \operatorname(R), and \ is a basis for the Zariski topology. \operatorname(R) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in ''R'' are precisely the closed points in this topology. By the same reasoning, it is not, in general, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in various quite different settings that a short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations. Suppose we are given a covariant left exact functor ''F'' : A → B between two abelian categories A and B. If 0 → ''A'' → ''B'' → ''C'' → 0 is a short exact sequence in A, then applying ''F'' yields the exact sequence 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if A is "nice" enough) the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Subset
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold. Equivalent definitions By definition, a subset A of a topological space (X, \tau) is called if its complement X \setminus A is an open subset of (X, \tau); that is, if X \setminus A \in \tau. A set is closed in X if and only if it is equal to its closure in X. Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset A \subseteq X is always contained in its (topological) closure in X, which is denoted by \operatorname_X A; that is, if A \subseteq X then A \subseteq \operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support (mathematics)
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis. Formulation Suppose that f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname(f), is the set of points in X where f is non-zero: \operatorname(f) = \. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then f is said to have . If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f vanishes in an appropriate sense on its complement. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
