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Tag Archives: algebraic geometry
Commutative Algebra 61
In this article, we will consider algebraic geometry in the projective space. Throughout this article, k denotes an algebraically closed field. Projective Space Definition. Let . On the set , we consider the equivalence relation: The projective nspace is the set … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, closed subsets, cones, graded rings, nullstellensatz, projective varieties, varieties
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Commutative Algebra 57
Continuing from the previous article, A denotes a noetherian ring and all Amodules are finitely generated. As before all completions are taken to be stable for a fixed ideal . Noetherianness We wish to prove that the adic completion of … Continue reading
Posted in Advanced Algebra
Tagged aadic filtrations, algebraic geometry, analysis, completion, filtrations, hensels lemma, local rings, padic
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Commutative Algebra 32
Torsion and Flatness Definition. Let A be a ring and M an Amodule; let . If satisfies , we call it an –torsion element. If is an torsion for some nonzerodivisor we call it a torsion element. M is said … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, flat modules, torsion, varieties, zerodivisors
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Commutative Algebra 30
Tensor Product of AAlgebras Proposition 1. Let B, C be Aalgebras. Their tensor product has a natural structure of an Aalgebra which satisfies . Proof Fix . The map is Abilinear so it induces an Alinear map Now varying (b, c) gives … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, algebras, coproducts, fibres, tensor product, varieties
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Commutative Algebra 23
Localization and Spectrum Recall that the ideals of correspond to a subset of the ideals of A. If we restrict ourselves to prime ideals, we get the following nice bijection. Theorem 1. The above gives a bijection between Useful trick If … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, local rings, localization, prime ideals, rational functions, spectrum, zariski topology
12 Comments
Commutative Algebra 11
Coordinate Rings as kalgebras Let k be an algebraically closed field. Recall that a closed subset is identified by its coordinate ring k[V], which is a finitely generated kalgebra since Definition. An affine kvariety is a finitely generated kalgebra A which is … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, algebras, cotangent spaces, maximal ideals, tangent spaces, varieties
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