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Matlis Duality
In algebra, Matlis duality is a duality between Artinian and Noetherian modules over a complete Noetherian local ring. In the special case when the local ring has a field mapping to the residue field it is closely related to earlier work by Francis Sowerby Macaulay on polynomial rings and is sometimes called Macaulay duality, and the general case was introduced by . Statement Suppose that ''R'' is a Noetherian complete local ring with residue field ''k'', and choose ''E'' to be an injective hull of ''k'' (sometimes called a Matlis module). The dual ''D''''R''(''M'') of a module ''M'' is defined to be Hom''R''(''M'',''E''). Then Matlis duality states that the duality functor ''D''''R'' gives an anti-equivalence between the categories of Artinian and Noetherian ''R''-modules. In particular the duality functor gives an anti-equivalence from the category of finite-length modules to itself. Examples Suppose that the Noetherian complete local ring ''R'' has a subfield ''k'' that map ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variable (mathematics), variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in mathematical education, education, to the study of algebraic structures such as group (mathematics), groups, ring (mathematics), rings, and field (mathematics), fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pontryagin Dual
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive group of the integers (also with the discrete topology), the real numbers, and every finite dimensional vector space over the reals or a -adic field. The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological group formed by the continuous group homomorphisms from the group to the circle group with the operation of pointwise multiplication and the topology of uniform convergence on compact sets. The Pontryagin duality theorem establishes Pontryagin duality by stating that any locally compact abelian group is naturally isomorphic with its bidual (the dual of its dual). The Fourier inversion theorem is a special case of this the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pacific Journal Of Mathematics
The Pacific Journal of Mathematics is a mathematics research journal supported by several universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisation, and the University of California, Berkeley. It was founded in 1951 by František Wolf and Edwin F. Beckenbach and has been published continuously since, with five two-issue volumes per year and 12 issues per year. Full-text PDF versions of all journal articles are available on-line via the journal's website with a subscription. The journal is incorporated as a 501(c)(3) organization A 501(c)(3) organization is a United States corporation, Trust (business), trust, unincorporated association or other type of organization exempt from federal income tax under section 501(c)(3) of Title 26 of the United States Code. It is one of t .... References Mathematics journals Publications established in 1951 Mathematical Sciences Publish ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Spo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Local Duality
In commutative algebra, Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves. Statement Suppose that ''R'' is a Cohen–Macaulay local ring of dimension ''d'' with maximal ideal ''m'' and residue field ''k'' = ''R''/''m''. Let ''E''(''k'') be a Matlis module, an injective hull of ''k'', and let be the completion of its dualizing module. Then for any ''R''-module ''M'' there is an isomorphism of modules over the completion of ''R'': : \operatorname_R^i(M,\overline\Omega) \cong \operatorname_R(H_m^(M),E(k)) where ''H''''m'' is a local cohomology group. There is a generalization to Noetherian local rings that are not Cohen–Macaulay, that replaces the dualizing module with a dualizing complex In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dualizing Object
In mathematics, a *-autonomous (read "star-autonomous") category C is a symmetric monoidal closed category equipped with a dualizing object \bot. The concept is also referred to as Grothendieck—Verdier category in view of its relation to the notion of Verdier duality. Definition Let C be a symmetric monoidal closed category. For any object ''A'' and \bot, there exists a morphism :\partial_:A\to(A\Rightarrow\bot)\Rightarrow\bot defined as the image by the bijection defining the monoidal closure :\mathrm((A\Rightarrow\bot)\otimes A,\bot)\cong\mathrm(A,(A\Rightarrow\bot)\Rightarrow\bot) of the morphism :\mathrm_\circ\gamma_ : (A\Rightarrow\bot)\otimes A\to\bot where \gamma is the ''symmetry'' of the tensor product. An object \bot of the category C is called dualizing when the associated morphism \partial_ is an isomorphism for every object ''A'' of the category C. Equivalently, a *-autonomous category is a symmetric monoidal category ''C'' together with a functor (-)^*:C^\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internal Hom
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics. Formal definition Let ''C'' be a locally small category (i.e. a category for which hom-classes are actually sets and not proper classes). For all objects ''A'' and ''B'' in ''C'' we define two functors to the category of sets as follows: : The functor Hom(–, ''B'') is also called the ''functor of points'' of the object ''B''. Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms. The pair of functors Hom(''A'', –) and Hom(–, ''B'') are related in a natural manner. For any pair of morphisms ''f'' : ''B'' → '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Balmer
Paul Balmer (born 1970) is a Swiss mathematician, working in algebra. He is a professor of mathematics at the University of California, Los Angeles. Balmer received his Ph.D. from the University of Lausanne in 1998, under the supervision of Manuel Ojanguren, with a thesis entitled ''Groupes de Witt dérivés des Schémas'' (in French). His research centers around triangulated categories. More specifically, he is a proponent of tensor-triangular geometry, an umbrella topic which covers geometric aspects of algebraic geometry, modular representation theory, stable homotopy theory, and other areas, by means of relevant tensor-triangulated categories. Balmer was an Invited Speaker at the International Congress of Mathematicians in Hyderabad in 2010, with a talk on ''Tensor Triangular Geometry''. In 2012, he became a fellow of the American Mathematical Society. He was awarded the Humboldt Prize The Humboldt Prize, the Humboldt-Forschungspreis in German, also known as the Humbol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proceeds on the basis that the objects of ''D''(''A'') should be chain complexes in ''A'', with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences. The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Functor
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories \mathcal and \mathcal is a pair of functors (assumed to be covariant) :F: \mathcal \rightarrow \mathcal and G: \mathcal \rightarrow \mathcal and, for all objects X in \mathcal and Y in \mathcal a bijection between the respective morphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dualizing Module
In abstract algebra, a dualizing module, also called a canonical module, is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety. It is used in Grothendieck local duality. Definition A dualizing module for a Noetherian ring ''R'' is a finitely generated module ''M'' such that for any maximal ideal ''m'', the ''R''/''m'' vector space vanishes if ''n'' ≠ height(''m'') and is 1-dimensional if ''n'' = height(''m''). A dualizing module need not be unique because the tensor product of any dualizing module with a rank 1 projective module is also a dualizing module. However this is the only way in which the dualizing module fails to be unique: given any two dualizing modules, one is isomorphic to the tensor product of the other with a rank 1 projective module. In particular if the ring is local the dualizing module is unique up to isomorphism. A Noetherian ring does not necessarily have a dualizing module. Any ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
