On the minimal prime ideals of a tensor product of two fields

Vámos, Peter

doi:10.1017/s0305004100054840

Cited by 33 publications

(34 citation statements)

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“…Let Z be the center of Q; then Z is extended noetherian and Q is a finitely generated module over Z [Row,6.1.25]. By a result of Vámos [Va,11], Z ⊗ k Z is noetherian if and only if Z is finitely generated as a field extension. Hence GKdim Z < ∞ [Row,6.3.41].…”

Section: Question 52 Is Every Noetherian Pi Hopf Algebra Gorenstein?mentioning

confidence: 99%

Noetherian PI Hopf algebras are Gorenstein

Zhang

2002

Trans. Amer. Math. Soc.

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Section: Question 52 Is Every Noetherian Pi Hopf Algebra Gorenstein?mentioning

confidence: 99%

Noetherian PI Hopf algebras are Gorenstein

Zhang

2002

Trans. Amer. Math. Soc.

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“…Since K is a subfield of the finitely generated field Frac(A 0 ), we have by [45,Theorem 11] that K is finitely generated. We can now replace k by K, and so…”

Section: The Case Of Principal Idealsmentioning

confidence: 99%

Poisson algebras via model theory and differential-algebraic geometry

Bell¹,

Launois²,

Sánchez³

et al. 2017

J. Eur. Math. Soc.

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Abstract. Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier-Moeglin equivalence in finite GK dimension. A weaker version of the Poisson Dixmier-Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.

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“…F is algebraically closed in G). There are several articles that study various aspects of tensor products of fields; see, for example, Sharp [29] and Vámos [35].…”

Section: ∆-Isomorphismsmentioning

confidence: 99%

“…A priori a given prime ∆-ideal may contain several minimal prime ∆-ideals, but not so in P. First we need a lemma, which is Vámos [35,Theorem 3,p. 28], but, for the sake of completeness, we prove it here.…”

Section: Corollary 93 Every Prime Ideal Ofmentioning

confidence: 99%

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The differential Galois theory of strongly normal extensions

Kovacic

2003

Trans. Amer. Math. Soc.

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Abstract. Differential Galois theory, the theory of strongly normal extensions, has unfortunately languished. This may be due to its reliance on Kolchin's elegant, but not widely adopted, axiomatization of the theory of algebraic groups. This paper attempts to revive the theory using a differential scheme in place of those axioms. We also avoid using a universal differential field, instead relying on a certain tensor product.We identify automorphisms of a strongly normal extension with maximal differential ideals of this tensor product, thus identifying the Galois group with the closed points of an affine differential scheme. Moreover, the tensor product has a natural coring structure which translates into the Galois group operation: composition of automorphisms.This affine differential scheme splits, i.e. is obtained by base extension from a (not differential, not necessarily affine) group scheme. As a consequence, the Galois group is canonically isomorphic to the closed, or rational, points of a group scheme defined over constants. We obtain the fundamental theorem of differential Galois theory, giving a bijective correspondence between subgroup schemes and intermediate differential fields.On the way to this result we study certain aspects of differential algebraic geometry, e.g. closed immersions, products, local ringed space of constants, and split differential schemes.

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On the minimal prime ideals of a tensor product of two fields

Cited by 33 publications

References 1 publication

Noetherian PI Hopf algebras are Gorenstein

Noetherian PI Hopf algebras are Gorenstein

Poisson algebras via model theory and differential-algebraic geometry

The differential Galois theory of strongly normal extensions

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