Tag Archives: varieties

Commutative Algebra 61

Posted on May 22, 2020 by limsup

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 →

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Commutative Algebra 32

Posted on April 20, 2020 by limsup

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 →

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Commutative Algebra 30

Posted on April 17, 2020 by limsup

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 →

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Commutative Algebra 11

Posted on March 29, 2020 by limsup

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 →

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