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Popescu's Theorem
In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu, states: :Let ''A'' be a Noetherian ring and ''B'' a Noetherian algebra over it. Then, the structure map ''A'' → ''B'' is a regular homomorphism if and only if ''B'' is a direct limit of smooth ''A''-algebras. For example, if ''A'' is a local G-ring (e.g., a local excellent ring) and ''B'' its completion, then the map ''A'' → ''B'' is regular by definition and the theorem applies. Another proof of Popescu's theorem was given by Tetsushi Ogoma, while an exposition of the result was provided by Richard Swan Richard Gordon Swan (; born 1933) is an American mathematician who is known for the Serre–Swan theorem relating the geometric notion of vector bundles to the algebraic concept of projective modules, and for the Swan representation, an ''l''-a .... The usual proof of the Artin approximation theorem relies crucially on Popescu's theorem. Popescu's result was proved by an alte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb; and p-adic number, ''p''-adic integers. Commutative algebra is the main technical tool of algebraic geometry, and many results and concepts of commutative algebra are strongly related with geometrical concepts. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. Several concepts of commutative algebras have been developed in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
G-ring
In commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular (defined below). Almost all Noetherian rings that occur naturally in algebraic geometry or number theory are G-rings, and it is quite hard to construct examples of Noetherian rings that are not G-rings. The concept is named after Alexander Grothendieck. A ring that is both a G-ring and a J-2 ring is called a quasi-excellent ring, and if in addition it is universally catenary it is called an excellent ring. Definitions *A (Noetherian) ring ''R'' containing a field ''k'' is called geometrically regular over ''k'' if for any finite extension ''K'' of ''k'' the ring ''R'' ⊗''k'' ''K'' is a regular ring. *A homomorphism of rings from ''R'' to ''S'' is called regular if it is flat and for every ''p'' ∈ Spec(''R'') the fiber ''S'' ⊗''R'' ''k''(''p'') is geometrically regular over the residue field ''k' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Property (ring Theory)
In algebra, a commutative Noetherian ring ''A'' is said to have the approximation property with respect to an ideal ''I'' if each finite system of polynomial equations with coefficients in ''A'' has a solution in ''A'' if and only if it has a solution in the ''I''-adic completion of ''A''. The notion of the approximation property is due to Michael Artin. See also * Artin approximation theorem *Popescu's theorem In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu, states: :Let ''A'' be a Noetherian ring and ''B'' a Noetherian algebra over it. Then, the structure map ''A'' → ''B'' is a regular homomorphism if and ... Notes References * * * * Ring theory {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Mathematical Society
The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstracted and indexed in: 2011. American Mathematical Society. * Mathematical Reviews * Zentralblatt MATH * Science Citation Index * ISI Alerting Services * CompuMath Citation Index * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Approximation Theorem
In mathematics, the Artin approximation theorem is a fundamental result of in deformation theory which implies that formal power series with coefficients in a field ''k'' are well-approximated by the algebraic functions on ''k''. More precisely, Artin proved two such theorems: one, in 1968, on approximation of complex analytic solutions by formal solutions (in the case k = \Complex); and an algebraic version of this theorem in 1969. Statement of the theorem Let \mathbf = x_1, \dots, x_n denote a collection of ''n'' indeterminates, k \mathbf the ring of formal power series with indeterminates \mathbf over a field ''k'', and \mathbf = y_1, \dots, y_n a different set of indeterminates. Let :f(\mathbf, \mathbf) = 0 be a system of polynomial equations in k mathbf, \mathbf/math>, and ''c'' a positive integer. Then given a formal power series solution \hat(\mathbf) \in k \mathbf, there is an algebraic solution \mathbf(\mathbf) consisting of algebraic functions (more precisely, alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Swan
Richard Gordon Swan (; born 1933) is an American mathematician who is known for the Serre–Swan theorem relating the geometric notion of vector bundles to the algebraic concept of projective modules, and for the Swan representation, an ''l''-adic projective representation of a Galois group. His work has mainly been in the area of algebraic K-theory. Education and career As an undergraduate at Princeton University, Swan was one of five winners in the William Lowell Putnam Mathematical Competition in 1952. He earned his Ph.D. in 1957 from Princeton University under the supervision of John Coleman Moore. In 1969 he proved in full generality what is now known as the Stallings–Swan theorem. He is the Louis Block Professor Emeritus of Mathematics at the University of Chicago.. His doctoral students at Chicago include Charles Weibel, also known for his work in K-theory. Together with Otto Forster he proved the Forster–Swan theorem. Awards and honors In 1970 Swan was awarded t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical evidence, empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completion Of A Ring
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 \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excellent Ring
In commutative algebra, a quasi-excellent ring is a Noetherian ring, Noetherian commutative ring that behaves well with respect to the operation of completion of a ring, completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" ring (mathematics), rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Ring
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The concept of local rings was introduced by Wolfgang Krull in 1938 under the name ''Stellenringe''. The English term ''local ring'' is due to Zariski. Definition and first consequences A ring ''R'' is a local ring if it has any one of the following equivalent properties: * ''R'' has a unique maximal left ideal. * ''R'' has a unique maximal right ideal. * 1 ≠ 0 and the sum of any two non- units in ''R'' is a non-unit. * 1 ≠ 0 and if ''x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Algebra
In algebra, a commutative ''k''-algebra ''A'' is said to be 0-smooth if it satisfies the following lifting property: given a ''k''-algebra ''C'', an ideal ''N'' of ''C'' whose square is zero and a ''k''-algebra map u: A \to C/N, there exists a ''k''-algebra map v: A \to C such that ''u'' is ''v'' followed by the canonical map. If there exists at most one such lifting ''v'', then ''A'' is said to be 0-unramified (or 0-neat). ''A'' is said to be 0-étale if it is 0-smooth and 0-unramified. The notion of 0-smoothness is also called formal smoothness. A finitely generated ''k''-algebra ''A'' is 0-smooth over ''k'' if and only if Spec ''A'' is a smooth scheme over ''k''. A separable algebraic field extension ''L'' of ''k'' is 0-étale over ''k''. The formal power series ring k _1,_\ldots,_t_n.html" ;"title="![t_1, \ldots, t_n">![t_1, \ldots, t_n!/math> is 0-smooth only when \operatornamek = p > 0 and [k: k^p] and I = (t_1, \ldots, t_n). Then ''B'' is ''I''-smooth over ''A''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
