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A categorical quotient in the category of dense constructible subsets

Devrim Celik (2009)

Colloquium Mathematicae

A. A'Campo-Neuen and J. Hausen gave an example of an algebraic torus action on an open subset of the affine four space that admits no quotient in the category of algebraic varieties. We show that this example admits a quotient in the category of dense constructible subsets and thereby answer a question of A. Białynicki-Birula.

A characterization of p-bases of rings of constants

Piotr Jędrzejewicz (2013)

Open Mathematics

We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.

A characterization of proper regular mappings

T. Krasiński, S. Spodzieja (2001)

Annales Polonici Mathematici

Let X, Y be complex affine varieties and f:X → Y a regular mapping. We prove that if dim X ≥ 2 and f is closed in the Zariski topology then f is proper in the classical topology.

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