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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, howev ... [...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] |
Length Of A Module
In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 In particular, as in the case of vector spaces, the only modules of finite length are finitely generated modules. It is defined to be the length of the longest chain of submodules. Modules with ''finite'' length share many important properties with finite-dimensional vector spaces. Other concepts used to 'count' in ring and module theory are depth and height; these are both somewhat more subtle to define. Moreover, their use is more aligned with dimension theory whereas length is used to analyze finite modules. There are also various ideas of ''dimension'' that are useful. Finite length commutative rings play an essential role in functorial treatments of formal algebraic geometry and deformation theory where Artin rings are used extensively. Definition Length of a module Let M be a (left or right) module over some ring R. Given a chain of submo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection Theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form. There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov-Witten theory and the extension of intersection theory from schemes to stacks. Topological intersection form For a connected oriented manifold of dimension the intersection form is defined on the -th cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class in . Stated precisely, there is a bilinear form :\lambda_M \colon H^n(M,\partial M) \times H^n(M,\partial M)\to \mathbf given by :\lambda ... [...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 ideals, and 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 integers. Commutative algebra is the main technical tool in the local study of schemes. 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. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have led to the no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Conjectures In Commutative Algebra
In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectures relating various homological properties of a commutative ring to its internal ring structure, particularly its Krull dimension and depth. The following list given by Melvin Hochster is considered definitive for this area. In the sequel, A, R, and S refer to Noetherian commutative rings; R will be a local ring with maximal ideal m_R, and M and N are finitely generated R-modules. # The Zero Divisor Theorem. If M \ne 0 has finite projective dimension and r \in R is not a zero divisor on M, then r is not a zero divisor on R. # Bass's Question. If M \ne 0 has a finite injective resolution then R is a Cohen–Macaulay ring. # The Intersection Theorem. If M \otimes_R N \ne 0 has finite length, then the Krull dimension of ''N'' (i.e., the dimension of ''R'' modulo the annihilator of '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christophe Soulé
Christophe Soulé (born 1951) is a French mathematician working in arithmetic geometry. Education Soulé started his studies in 1970 at École Normale Supérieure in Paris. He completed his Ph.D. at the University of Paris in 1979 under the supervision of Max Karoubi and Roger Godement, with a dissertation titled ''K-Théorie des anneaux d'entiers de corps de nombres et cohomologie étale''. Awards and recognition In 1979, he was awarded a CNRS Bronze Medal. He received the Prix J. Ponti in 1985 and the Prize Ampère in 1993. Since 2001, he is member of the French Academy of Sciences. In 1983, he was invited speaker at the International Congress of Mathematicians (ICM) in Warsaw. Publications * Christophe Soulé, with the collaboration of Dan Abramovich, Jean-François Burnol, and Jürg Kramer: ''Lectures on Arakelov Geometry.'' Cambridge Studies in Advanced Mathematics 33. Cambridge University Press, 1992. , * Henri Gillet, Christophe Soulé: ''An arithmetic Riemann–Roch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Gillet
Henri Antoine Gillet (born 8 July 1953, Tangier) is a European-American mathematician, specializing in arithmetic geometry and algebraic geometry. Education and career Gillet received in 1974 his bachelor's degree from King’s College London and in 1978 his Ph.D. from Harvard University under David Mumford with thesis ''Applications of Algebraic K-Theory to Intersection Theory''. As a postdoc he was an instructor and from 1981 an assistant professor at Princeton University. He became in 1984 an assistant professor, in 1986 an associate professor, and in 1988 a full professor at the University of Illinois at Chicago, where he was from 1996 to 2001 the head of the department of mathematics, statistics, and computer science. He was a visiting scholar at the Tata Institute of Fundamental Research (2006), the Institute for Advanced Study (1987), the IHES (1985, 1986, 1988), in Barcelona, at the Fields Institute in Toronto and at the Isaac Newton Institute (1998). Gillet's research d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul C
Paul may refer to: *Paul (given name), a given name (includes a list of people with that name) *Paul (surname), a list of people People Christianity *Paul the Apostle (AD c.5–c.64/65), also known as Saul of Tarsus or Saint Paul, early Christian missionary and writer *Pope Paul (other), multiple Popes of the Roman Catholic Church *Saint Paul (other), multiple other people and locations named "Saint Paul" Roman and Byzantine empire *Lucius Aemilius Paullus Macedonicus (c. 229 BC – 160 BC), Roman general *Julius Paulus Prudentissimus (), Roman jurist *Paulus Catena (died 362), Roman notary *Paulus Alexandrinus (4th century), Hellenistic astrologer *Paul of Aegina or Paulus Aegineta (625–690), Greek surgeon Royals *Paul I of Russia (1754–1801), Tsar of Russia *Paul of Greece (1901–1964), King of Greece Other people *Paul the Deacon or Paulus Diaconus (c. 720 – c. 799), Italian Benedictine monk *Paul (father of Maurice), the father of Maurice, Byzan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ofer Gabber
Ofer Gabber (עופר גאבר; born May 16, 1958) is a mathematician working in algebraic geometry. Life In 1978 Gabber received a Ph.D. from Harvard University for the thesis ''Some theorems on Azumaya algebras,'' written under the supervision of Barry Mazur. Gabber has been at the Institut des Hautes Études Scientifiques in Bures-sur-Yvette in Paris since 1984 as a CNRS senior researcher. He won the Erdős Prize in 1981 and the Prix Thérèse Gautier from the French Academy of Sciences in 2011. In 1981 Gabber with Victor Kac published a proof of a conjecture stated by Kac in 1968. Books * With Lorenzo Ramero: ''Almost Ring Theory'', Springer, Lecture Notes in Computer Science, vol 1800, 2003. * With Brian Conrad, Gopal Prasad: ''Pseudo-reductive Groups'', Cambridge University Press, 20102015, 2nd editionref> See also *almost ring theory *Theorem of absolute purity In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue Field
In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a local ring and ''m'' is then its unique maximal ideal. This construction is applied in algebraic geometry, where to every point ''x'' of a scheme ''X'' one associates its residue field ''k''(''x''). One can say a little loosely that the residue field of a point of an abstract algebraic variety is the 'natural domain' for the coordinates of the point. Definition Suppose that ''R'' is a commutative local ring, with maximal ideal ''m''. Then the residue field is the quotient ring ''R''/''m''. Now suppose that ''X'' is a scheme and ''x'' is a point of ''X''. By the definition of scheme, we may find an affine neighbourhood ''U'' = Spec(''A''), with ''A'' some commutative ring. Considered in the neighbourhood ''U'', the point ''x'' correspond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tor Functor
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor1 and the torsion subgroup of an abelian group. In the special case of abelian groups, Tor was introduced by Eduard Čech (1935) and named by Samuel Eilenberg around 1950. It was first applied to the Künneth theorem and universal coefficient theorem in topology. For modules over any ring, Tor was defined by Henri Cartan and Eilenberg in their 1956 book ''Homological Algebra''. Definition Let ''R'' be a ring. Write ''R''-Mod for the category of left ''R''-modules and Mod-''R'' for the category of right ''R''-modules. (If ''R'' is commutat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003. Biography Personal life Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil. The French mathematician Denis S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
