Tag Archives: algebraic geometry

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

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Commutative Algebra 6

Injective and Surjective Maps Proposition. 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 W. … Continue reading

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Commutative Algebra 5

Morphisms in Algebraic Geometry Next we study the “nice” functions between closed subspaces of . Definition. Suppose and are closed subsets. A morphism is a function which can be expressed as: for some polynomials . We also say f is a regular … Continue reading

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Commutative Algebra 4

More Concepts in Algebraic Geometry Recall that we have a bijection between radical ideals of and closed subsets of . The bijection reverses the inclusion so if and only if . Not too surprisingly, operations on ideals translate to operations … Continue reading

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Commutative Algebra 3

Algebraic Geometry Concepts We have decided to introduce, at this early point, some basics of algebraic geometry in order to motivate the later concepts. In summary, algebraic geometry studies solutions to polynomial equations over a field. First we consider a … Continue reading

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