Rahim Moosa
University of Waterloo
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Featured researches published by Rahim Moosa.
Journal of The Institute of Mathematics of Jussieu | 2010
Rahim Moosa; Thomas Scanlon
The notion of prolongation of an algebraic variety is developed in an abstract setting that generalises the difference and (Hasse) differential contexts. An interpolating map that compares the prolongation spaces with algebraic jet spaces is introduced and studied.
arXiv: Algebraic Geometry | 2011
Rahim Moosa; Thomas Scanlon
Building on the abstract notion of prolongation developed in [10], the theory of iterative Hasse-Schmidt rings and schemes is introduced, simultaneously generalising difference and (Hasse-Schmidt) differential rings and schemes. This work provides a unified formalism for studying difference and differential algebraic geometry, as well as other related geometries. As an application, Hasse-Schmidt jet spaces are constructed generally, allowing the development of the theory for arbitrary systems of algebraic partial difference/differential equations, where constructions by earlier authors applied only to the finite-dimensional case. In particular, it is shown that under appropriate separability assumptions a Hasse-Schmidt variety is determined by its jet spaces at a point.
Crelle's Journal | 2005
Rahim Moosa
Abstract A hypothesis is introduced under which a compact complex analytic space, X, viewed as a structure in the language of analytic sets, is essentially saturated. It is shown that this condition is met exactly when the irreducible components of the restricted Douady spaces of all the cartesian powers of X are compact. Some implications of saturation on Kähler-type spaces, which by a theorem of Fujiki meet the above condition, are discussed. In particular, one obatins a model-theoretic proof of the fact that relative algebraic reductions exist in the class of Kähler-type spaces.
Transactions of the American Mathematical Society | 2004
Rahim Moosa
We study elementary extensions of compact complex spaces and deduce that every complete type of dimension 1 is internal to projective space. This amounts to a nonstandard version of the Riemann Existence Theorem, and answers a question posed by Anand Pillay.
American Journal of Mathematics | 2004
Rahim Moosa; Thomas Scanlon
Motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields, we prove a quantifier elimination and stability result for finitely generated modules over certain finite simple extensions of the integers given together with predicates for cycles of the distinguished generator of the ring.
Crelle's Journal | 2008
Rahim Moosa; Anand Pillay; Thomas Scanlon
Abstract We introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety in characteristic zero is determined by its arcs at a point. Using differential arcs, we show that if (K, +, ×, δ1, . . ., δ n ) is a differentially closed field of characteristic zero with n commuting derivations and p ∈ S(K) is a regular type over K, then either p is locally modular or there is a definable subgroup G ≦ (K, +) of the additive group having a regular generic type that is nonorthogonal to p.
Journal of the European Mathematical Society | 2017
Jason P. Bell; Stephane Launois; Omar León Sánchez; Rahim Moosa
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
Journal of Mathematical Logic | 2014
Rahim Moosa; Thomas Scanlon
\operatorname{GK}
International Mathematics Research Notices | 2006
Dragos Ghioca; Rahim Moosa
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.
Bulletin of The London Mathematical Society | 2008
Rahim Moosa; Ruxandra Moraru; Matei Toma
Generalising and unifying the known theorems for difference and differential fields, it is shown that for every finite free