1,198 research outputs found
Poisson algebras via model theory and differential-algebraic geometry
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
On a theory of the -function in positive characteristic
We present a theory of the -function (or Bernstein-Sato polynomial) in
positive characteristic. Let be a non-constant polynomial with coefficients
in a perfect field of characteristic Its -function is
defined to be an ideal of the algebra of continuous -valued functions on
The zero-locus of the -function is thus naturally
interpreted as a subset of which we call the set of roots of
We prove that has finitely many roots and that they are negative
rational numbers. Our construction builds on an earlier work of Musta\c{t}\u{a}
and is in terms of -modules, where is the ring of Grothendieck
differential operators. We use the Frobenius to obtain finiteness properties of
and relate it to the test ideals of Comment: Final versio
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