In commutative algebra, given a homomorphismA → B of commutative rings, B is called an A-algebra of finite type if B is a finitely generated as an A-algebra. It is much stronger for B to be a finiteA-algebra, which means that B is finitely generated as an A-module. For example, for any commutative ring A and natural number n, the polynomial ringA[x1, ..., xn] is an A-algebra of finite type, but it is not a finite A-module unless A = 0 or n = 0. Another example of a finite-type homomorphism that is not finite is .
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The analogous notion in terms of schemes is: a morphismf: X → Y of schemes is of finite type if Y has a covering by affineopen subschemes Vi = Spec Ai such that f−1(Vi) has a finite covering by affine open subschemes Uij = Spec Bij with Bij an Ai-algebra of finite type. One also says that X is of finite type over Y.
For example, for any natural number n and fieldk, affine n-space and projective n-space over k are of finite type over k (that is, over Spec k), while they are not finite over k unless n = 0. More generally, any quasi-projective scheme over k is of finite type over k.
The Noether normalization lemma says, in geometric terms, that every affine scheme X of finite type over a field k has a finite surjective morphism to affine space An over k, where n is the dimension of X. Likewise, every projective schemeX over a field has a finite surjective morphism to projective spacePn, where n is the dimension of X.
morphism, finite, type, commutative, algebra, given, homomorphism, commutative, rings, called, algebra, finite, type, finitely, generated, algebra, much, stronger, finite, algebra, which, means, that, finitely, generated, module, example, commutative, ring, na. In commutative algebra given a homomorphism A B of commutative rings B is called an A algebra of finite type if B is a finitely generated as an A algebra It is much stronger for B to be a finite A algebra which means that B is finitely generated as an A module For example for any commutative ring A and natural number n the polynomial ring A x1 xn is an A algebra of finite type but it is not a finite A module unless A 0 or n 0 Another example of a finite type homomorphism that is not finite is C t C t x y y2 x3 t displaystyle mathbb C t to mathbb C t x y y 2 x 3 t This article needs attention from an expert in Mathematics See the talk page for details WikiProject Mathematics may be able to help recruit an expert August 2023 The analogous notion in terms of schemes is a morphism f X Y of schemes is of finite type if Y has a covering by affine open subschemes Vi Spec Ai such that f 1 Vi has a finite covering by affine open subschemes Uij Spec Bij with Bij an Ai algebra of finite type One also says that X is of finite type over Y For example for any natural number n and field k affine n space and projective n space over k are of finite type over k that is over Spec k while they are not finite over k unless n 0 More generally any quasi projective scheme over k is of finite type over k The Noether normalization lemma says in geometric terms that every affine scheme X of finite type over a field k has a finite surjective morphism to affine space An over k where n is the dimension of X Likewise every projective scheme X over a field has a finite surjective morphism to projective space Pn where n is the dimension of X See also editFinitely generated algebraReferences editBosch Siegfried 2013 Algebraic Geometry and Commutative Algebra London Springer pp 360 365 ISBN 9781447148289 nbsp This algebraic geometry related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Morphism of finite type amp oldid 1203292139, wikipedia, wiki, book, books, library,