Algebraic Geometry and Commutative Algebra

Ebook "Homological dimensions of modules" were prepared for a series of ten lectures given at Regional Conference of the Conference Board of the Mathematical Sciences in June 1971. The bulk of the lectures were on projective dimensions of "very large" modules. Although the material in these notes is not new, there are several places where existing work has been simplified. For example, a commutative local nondomain of global dimension 3 is described without reference to analysis, and the dimension of a quotient field of a polynomial ring rather than a regular local ring is calculated.

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The paper "The multiplicity and the Cohen–Macaulayness of extended Rees algebras of equimultiple ideals" gives some multiplicity formulas of the extended Rees algebras \({\rm{T = A[It}}{\rm{,}}{{\rm{t}}^{ - 1}}{\rm{]}}\). In the case A generalized Cohen–Macaulay, we determine when T is Cohen–Macaulay and as an immediate consequence we obtain e.g., some criteria for the Cohen–Macaulayness of Rees algebra R(I) over a Cohen–Macaulay ring in terms of reduction numbers and ideals.

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The local cohomology theory plays an important role in commutative algebra and algebraic geometry. The I-cofiniteness of local cohomology modules is one of interesting properties which has been studied by many mathematicians. The I-cominimax modules is an extension of I-cofinite modules which was introduced by Hartshorne.

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Tài liệu Algebraic geometry and arithmetic curves as the main contents of the document: Some topics in commutative algebra, General properties of schemes, Morphisms and base change, Some local properties, Coherent sheaves and Cech cohomology, Sheaves of differentials,...

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Commutative algebra is the theory of commutative rings and their modules. Although it is an interesting theory in itself, it is generally seen as a tool for geometry and number theory. This is my point of view. In this book I try to organize and present a cohesive set of methods in commutative algebra, for use in geometry. As indicated in the title, I maintain throughout the text a view towards complex projective geometry. In many recent algebraic geometry books, commutative algebra is often treated as a poor relation. One occasionally refers to it, but only reluctantly.

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One of the major advances of science in the 20th century was the discovery of a mathematical formulation of quantum mechanics by Heisenberg in 1925 [94].1 From a mathematical point of view, this transition from classical mechanics to quantum mechanics amounts to, among other things, passing from the commutative algebra of classical observables to the noncommutative algebra of quantum mechanical observables. To understand this better we recall that in classical mechanics an observable of a system (e.g. energy, position, momentum, etc.

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