Local Algebra in Algebraic Geometry
نویسنده
چکیده
Introduction 1 0.1. Associated primes 2 0.2. Depth 2 1. Regular local rings and their cohomological properties 4 1.1. Notions of dimension 4 1.2. Dualizing functor 5 1.3. Local rings 6 1.4. Serre’s theorem on regular local rings 6 1.5. Cohomology of D when global dimension is finite 8 2. Depth 8 2.1. Property (Sk) 8 2.2. Serre’s criterion of normality 10 2.3. Cohen-Macaulay modules 11 3. Divisors and line bundles 12 3.1. Weil divisors 12 3.2. Cartier divisors 13 3.3. Relating Weil and Cartier divisors 15 4. Conclusion 17 4.1. Further reading 17 References 17
منابع مشابه
Math 603: Introduction to Commutative Algebra
1.1. What is this course about? The foundations of differential geometry (= study of manifolds) rely on analysis in several variables as “local machinery”: many global theorems about manifolds are reduced down to statements about what happens in a local neighborhood, and then anaylsis is brought in to solve the local problem. Analogously, algebraic geometry uses commutative algebraic as its “lo...
متن کاملStatement of Purpose
My research interest lies in the area of arithmetic geometry related to Langlands correspondence, where the theory of automorphic forms and algebraic geometry intersect. It is one of the most interesting and actively studied subject of algebraic number theory, whose origin goes back to the theory of complex multiplication. My research is currently focused on the local aspect of this theory, nam...
متن کاملClassification of Lie Subalgebras up to an Inner Automorphism
In this paper, a useful classification of all Lie subalgebras of a given Lie algebraup to an inner automorphism is presented. This method can be regarded as animportant connection between differential geometry and algebra and has many applications in different fields of mathematics. After main results, we have applied this procedure for classifying the Lie subalgebras of some examples of Lie al...
متن کاملTight closure’s failure to localize—a self-contained exposition
We give a treatment of the Brenner-Monsky example based on polynomial algebra and linear algebra. No prior knowledge of tight closure theory, Hilbert-Kunz theory, algebraic geometry or local cohomology is assumed.
متن کاملAlgebraic Geometry over Lie Algebras
What is algebraic geometry over algebraic systems? Many important relations between elements of a given algebraic system A can be expressed by systems of equations over A. The solution sets of such systems are called algebraic sets over A. Algebraic sets over A form a category, if we take for morphisms polynomial functions in the sense of Definition 6.1 below. As a discipline, algebraic geometr...
متن کاملun 2 00 4 Problems in algebra inspired by universal algebraic geometry
Let Θ be a variety of algebras. In every Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We consider also a special categorical invariant K Θ (H) of this geometry. The classical algebraic geometry deals with the variety Θ = Com − P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the g...
متن کامل