# Tag Archives: algebraic geometry

## Commutative Algebra 62

Irreducible Subsets of Projective Space Throughout this article, k is an algebraically closed field. We wish to consider irreducible closed subsets of . For that we need the following preliminary result. Lemma 1. Let be a graded ring; a proper homogeneous … Continue reading

## 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 n-space is the set … Continue reading

## Commutative Algebra 57

Continuing from the previous article, A denotes a noetherian ring and all A-modules 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

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## Commutative Algebra 32

Torsion and Flatness Definition. Let A be a ring and M an A-module; let . If satisfies , we call it an –torsion element. If is an -torsion for some non-zero-divisor we call it a torsion element. M is said … Continue reading

## Commutative Algebra 30

Tensor Product of A-Algebras Proposition 1. Let B, C be A-algebras. Their tensor product has a natural structure of an A-algebra which satisfies . Proof Fix . The map is A-bilinear so it induces an A-linear map Now varying (b, c) gives … Continue reading

## 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

## Commutative Algebra 11

Coordinate Rings as k-algebras 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 k-algebra since Definition. An affine k-variety is a finitely generated k-algebra A which is … Continue reading

## Commutative Algebra 6

Injective and Surjective Maps Proposition 1. Let be a morphism of closed sets, with corresponding . is injective if and only if is dense. is surjective if and only if is an embedding of V as a closed subspace of … Continue reading