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Serre's Inequality On Height
In algebra, specifically in the theory of commutative rings, Serre's inequality on height states: given a (Noetherian) regular ring ''A'' and a pair of prime ideals \mathfrak, \mathfrak in it, for each prime ideal \mathfrak r that is a minimal prime ideal over the sum \mathfrak p + \mathfrak q, the following inequality on heights holds: :\operatorname(\mathfrak r) \le \operatorname(\mathfrak p) + \operatorname(\mathfrak q). Without the assumption on regularity, the inequality can fail; see scheme-theoretic intersection#Proper intersection. Sketch of Proof Serre gives the following proof of the inequality, based on the validity of Serre's multiplicity conjectures for formal power series ring over a complete discrete valuation ring. By replacing A by the localization at \mathfrak r, we assume (A, \mathfrak r) is a local ring. Then the inequality is equivalent to the following inequality: for finite A-modules M, N such that M \otimes_A N has finite length, :\dim_A M + \dim_A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
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 algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Definition and first examples Definition A ''ring'' is a set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Ring
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ideal m, and suppose ''a''1, ..., ''a''''n'' is a minimal set of generators of m. Then by Krull's principal ideal theorem ''n'' ≥ dim ''A'', and ''A'' is defined to be regular if ''n'' = dim ''A''. The appellation ''regular'' is justified by the geometric meaning. A point ''x'' on an algebraic variety ''X'' is nonsingular if and only if the local ring \mathcal_ of germs at ''x'' is regular. (See also: regular scheme.) Regular local rings are ''not'' related to von Neumann regular rings. For Noetherian local rings, there is the following chain of inclusions: Characterizations There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if A is a Noetherian local ring with maximal idea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary and semiprime. Prime ideals for commutative rings An ideal of a commutative ring is prime if it has the following two properties: * If and are two elements of such that their product is an element of , then is in or is in , * is not the whole ring . This generalizes the following property of prime numbers, known as Euclid's lemma: if is a prime number and if divides a product of two integers, then divides or divides . We can therefore say :A positive integer is a prime number if and only if n\Z is a prime ideal in \Z. Examples * A simple example: In the ring R=\Z, the subset of even numbers is a prime ideal. * Given an integral domain R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Prime Ideal
In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes. Definition A prime ideal ''P'' is said to be a minimal prime ideal over an ideal ''I'' if it is minimal among all prime ideals containing ''I''. (Note: if ''I'' is a prime ideal, then ''I'' is the only minimal prime over it.) A prime ideal is said to be a minimal prime ideal if it is a minimal prime ideal over the zero ideal. A minimal prime ideal over an ideal ''I'' in a Noetherian ring ''R'' is precisely a minimal associated prime (also called isolated prime) of R/I; this follows for instance from the primary decomposition of ''I''. Examples * In a commutative artinian ring, every maximal ideal is a minimal prime ideal. * In an integral domain, the only minimal prime ideal is the zero ideal. * In the ring Z of integers, the minimal prime ideals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Height Of An Ideal
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules. The Krull dimension was introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal ''I'' in a polynomial ring ''R'' is the Krull dimension of ''R''/''I''. A field ''k'' has Krull dimension 0; more generally, ''k'' 'x''1, ..., ''x''''n''has Krull dimension ''n''. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent. There are several other ways that have been used to define the dimension of a ring. Most of them ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme-theoretic Intersection
In algebraic geometry, the scheme-theoretic intersection of closed subschemes ''X'', ''Y'' of a scheme ''W'' is X \times_W Y, the fiber product of the closed immersions X \hookrightarrow W, Y \hookrightarrow W. It is denoted by X \cap Y. Locally, ''W'' is given as \operatorname R for some ring ''R'' and ''X'', ''Y'' as \operatorname(R/I), \operatorname(R/J) for some ideals ''I'', ''J''. Thus, locally, the intersection X \cap Y is given as :\operatorname(R/(I+J)). Here, we used R/I \otimes_R R/J \simeq R/(I + J) (for this identity, see tensor product of modules#Examples.) Example: Let X \subset \mathbb^n be a projective variety with the homogeneous coordinate ring ''S/I'', where ''S'' is a polynomial ring. If H = \ \subset \mathbb^n is a hypersurface defined by some homogeneous polynomial ''f'' in ''S'', then : X \cap H = \operatorname(S/(I, f)). If ''f'' is linear (deg = 1), it is called a hyperplane section. See also: Bertini's theorem. Now, a scheme-theoretic intersection may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serre's Multiplicity Conjectures
In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial definition of intersection numbers, around 1949, there had been a question of how to provide a more flexible and computable theory. Let ''R'' be a (Noetherian, commutative) regular local ring and ''P'' and ''Q'' be prime ideals of ''R''. In 1958, Serre realized that classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra. Serre defined the intersection multiplicity of ''R''/''P'' and ''R''/''Q'' by means of the Tor functors of homological algebra, as : \chi (R/P,R/Q):=\sum _^\infty (-1)^i\ell_R (\operatorname^R_i(R/P,R/Q)). This requires the concept of the length of a module, denoted here by \ell_R, and the assumption that : \ell _R((R/P)\otimes(R/Q)) < \infty. If this idea were to work, howe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Power Series Ring
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, whose terms are of the form a x^n where x^n is the nth power of a variable x (n is a non-negative integer), and a is called the coefficient. Hence, power series can be viewed as a generalization of polynomials, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the x^n are used only as position-holders for the coefficients, so that the coefficient of x^5 is the fifth term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completion (ring Theory)
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure than general ones, and Hensel's lemma applies to them. In algebraic geometry, a completion of a ring of functions ''R'' on a space ''X'' concentrates on a formal neighborhood of a point of ''X'': heuristically, this is a neighborhood so small that ''all'' Taylor series centered at the point are convergent. An algebraic completion is constructed in a manner analogous to completion of a metric space with Cauchy sequences, and agrees with it in the case when ''R'' has a metric given by a non-Archimedean absolute value. General construction Suppose that ''E'' is an abelian group with a descending filtration : E = F^0 E \supset F^1 E \supset F^2 E \supset \cdots \, of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Valuation Ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' is a local principal ideal domain, and not a field. # ''R'' is a valuation ring with a value group isomorphic to the integers under addition. # ''R'' is a local Dedekind domain and not a field. # ''R'' is a Noetherian local domain whose maximal ideal is principal, and not a field.https://mathoverflow.net/a/155639/114772 # ''R'' is an integrally closed Noetherian local ring with Krull dimension one. # ''R'' is a principal ideal domain with a unique non-zero prime ideal. # ''R'' is a principal ideal domain with a unique irreducible element ( up to multiplication by units). # ''R'' is a unique factorization domain with a unique irreducible element (up to multiplication by units). # ''R'' is Noetherian, not a field, and every nonzero fractio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohen's Structure Theorem
In mathematics, the Cohen structure theorem, introduced by , describes the structure of complete Noetherian local rings. Some consequences of Cohen's structure theorem include three conjectures of Krull: *Any complete regular equicharacteristic Noetherian local ring is a ring of formal power series over a field. (Equicharacteristic means that the local ring and its residue field have the same characteristic, and is equivalent to the local ring containing a field.) *Any complete regular Noetherian local ring that is not equicharacteristic but is unramified is uniquely determined by its residue field and its dimension. *Any complete Noetherian local ring is the image of a complete regular Noetherian local ring. Statement The most commonly used case of Cohen's theorem is when the complete Noetherian local ring contains some field. In this case Cohen's structure theorem states that the ring is of the form ''k'' ''x''1,...,''x''''n''/(''I'') for some ideal ''I'', where ''k'' is it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tor Spectral Sequence
Tor, TOR or ToR may refer to: Places * Tor, Pallars, a village in Spain * Tor, former name of Sloviansk, Ukraine, a city * Mount Tor, Tasmania, Australia, an extinct volcano * Tor Bay, Devon, England * Tor River, Western New Guinea, Indonesia Science and technology * ''Tor'' (fish), a genus of fish commonly known as mahseers * Target of rapamycin, a regulatory enzyme * Tor functor, in mathematics * Tor (network), an Internet communication method for enabling online anonymity ** The Tor Project, a software organization that maintains the Tor network and the related Tor Browser People * Tor (given name), a Nordic masculine given name * Tor (surname) * Tor Johnson, stage name of Swedish professional wrestler and actor Karl Erik Tore Johansson (1902 or 1903–1971) * Tor (musician), Canadian electronic musician Tor Sjogren Arts, entertainment, and media Fictional characters * Tor (comics), a prehistoric human character * Tor, a character in the book ''The Hero and the Crown ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
