Tag Archives: algebras

Commutative Algebra 43

Posted on May 1, 2020 by limsup

Catenary Rings Let us look at prime chains in greater detail. Definition. Let be a chain of prime ideals of a ring A. We say the chain is saturated if for any prime ideal of A, ; maximal if it … Continue reading →

Posted in Advanced Algebra | Tagged , , , , | 2 Comments

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 →

Posted in Advanced Algebra | Tagged , , , , , | 2 Comments

Commutative Algebra 24

Posted on April 11, 2020 by limsup

Quotient vs Localization Taking the quotient and localization are two sides of the same coin when we look at . Quotient removes the “small” prime ideals in – it only keeps the prime ideals containing . Localization removes the “large” … Continue reading →

Posted in Advanced Algebra | Tagged , , , , | 2 Comments

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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Commutative Algebra 10

Posted on March 28, 2020 by limsup

Algebras Over a Ring Let A be any ring; we would like to look at A-modules with a compatible ring structure. Definition. An –algebra is an -module , together with a multiplication operator such that becomes a commutative ring (with 1); multiplication … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , , | 5 Comments